Universität Paderborn

Fakultät für Elektrotechnik, Informatik und Mathematik

Dissertation

Realisability and Localisation

Realisierbarkeit und Lokalisierung

Birgit Huber

REALISABILITY AND LOCALISATION

BIRGIT HUBER

Abstract. Let A be a differential graded algebra with cohomology ring H^{∗}A. A graded module over H^{∗}A is called realisable if it is (up to direct summands) of the form H^{∗}M for some differential graded A-module M. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the A_{∞}-algebra structure of H^{∗}A.

In this thesis we mainly consider differential graded algebras A with graded-com-mutative cohomology ring. We show that a finitely presented graded H^{∗}A-module X is realisable if and only if its 𝔭-localisation X_{𝔭} is realisable for all graded prime ideals 𝔭 of H^{∗}A.

In order to obtain such a local-global principle also for the global obstruction, we define the localisation of a differential graded algebra A at a graded prime 𝔭 of H^{∗}A, denoted by A_{𝔭}, and show the existence of a morphism of differential graded algebras inducing the canonical map H^{∗}A → (H^{∗}A)_{𝔭} in cohomology. The latter result actually holds in a much more general setting: we prove that every smashing localisation on the derived category of a differential graded algebra is induced by a morphism of differential graded algebras.

Finally we discuss the relation between realisability of modules over the group cohomology ring and the Tate cohomology ring.

Zusammenfassung. Sei A eine differenziell graduierte Algebra mit Kohomologiering H^{∗}A. Ein graduierter H^{∗}A-Modul heißt realisierbar, falls man ihn (bis auf direkte Summanden) mit einem H^{∗}A-Modul von der Form H^{∗}M identifizieren kann, wobei M ein differenziell graduierter A-Modul ist. Benson, Krause und Schwede haben ein lokales und ein globales Hindernis für Realisierbarkeit angegeben. Das globale Hindernis ist durch eine Hochschild Klasse gegeben, welche durch die sekundäre Mulitplikation der A_{∞}-Algebra-Struktur von H^{∗}A bestimmt ist.

In dieser Doktorarbeit betrachten wir hauptsächlich differenziell graduierte Algebren A mit graduiert-kommutativen Kohomologieringen. Wir zeigen, dass ein endlich präsentierter graduierter H^{∗}A-Modul X genau dann realisierbar ist, wenn dessen 𝔭-Lokalisierung X_{𝔭} für alle graduierten Primideale 𝔭 von H^{∗}A realisierbar ist.

Um ein solches Lokal-Global Prinzip auch für globale Realisierbarkeit zu formulieren, definieren wir die Lokalisierung einer differenziell graduierten Algebra A an einem Primideal 𝔭 von H^{∗}A und bezeichnen sie mit A_{𝔭}. Wir zeigen die Existenz eines Morphismus von differenziell graduierten Algebren, der in der Kohomologie die kanonische Abbildung H^{∗}A → (H^{∗}A)_{𝔭} induziert. Letzteres Resultat beweisen wir in einem wesentlich allgemeineren Kontext: Wir zeigen, dass jede mit direkten Summen kommutierende Lokalisierung der derivierten Kategorie einer differenziell graduierten Algebra von einem Morphismus von differenziell graduierten Algebren induziert ist.

Abschließend diskutieren wir den Zusammenhang von Realisierbarkeit von Moduln über dem Gruppen-Kohomologiering und dem Tate-Kohomologiering.

Contents

Danksagung 4 1. Introduction 6 Organisation 10 Notations and conventions 11 2. Triangulated categories 12 3. Graded rings and modules 15 3.1. Graded-commutative rings 17 4. Group and Tate cohomology rings 19 4.1. Group cohomology rings 20 4.2. Tate cohomology rings 21 5. The Hochschild Cohomology of a graded ring 22 5.1. Functoriality 23 5.2. The graded Bar resolution 24 5.3. Ring structure 24 5.4. The cup product pairing 26 6. Differential graded algebras and their derived categories 27 6.1. Differential graded algebras and modules 27 6.2. K(A) as stable category of a Frobenius category 29 6.3. Homotopically projective and homotopically injective dg modules 29 6.4. Derived functors 31 6.5. Cofibrant differential graded algebras 32 7. A_{∞}-algebras 33 8. Localisation in triangulated categories 37 8.1. Categories of fractions 37 8.2. Localisation functors 38 8.3. Quotient categories 38 8.4. Localisation sequences 39 8.5. Recollements 40 8.6. Cohomological localisation 41 9. Realising smashing localisations by morphisms of dg algebras 44 9.1. Construction of a dg algebra morphism 45 9.2. The 𝔭-localisation of a dg algebra 51 10. Realisability 54 10.1. A local obstruction for realisability 54 10.2. Realisability and dg algebras 56 10.3. A global obstruction for realisability 57 11. Realisability and 𝔭-localisation 57 11.1. A motivation for 𝔭-localisation 58 11.2. Realisability is a local property 59 12. Localising the global obstruction 62 12.1. A map of Hochschild cohomology rings 63 12.2. Local-global principle for the global obstruction 65 13. Comparing realisability over group and Tate cohomology 68 13.1. Local realisability 68 13.2. Examples for the global obstruction 69

REALISABILITY AND LOCALISATION 3

13.3. Lifting H^{∗}(G, k) → Ĥ^{∗}(G, k) to a morphism of dg algebras 74 13.4. Relating the global obstructions of H^{∗}(G, k) and Ĥ^{∗}(G, k) 76 References 78 Danksagung. An erster Stelle möchte ich mich ganz besonders herzlich bei meinem Dissertationsbetreuer Henning Krause für die Vergabe dieses Themas, die engagierte Betreuung dieser Doktorarbeit und viele wichtige Anregungen für meine Arbeit bedanken. Besonders danken möchte ich ihm auch für die Unterstützung außerhalb der Mathematik, insbesondere für seinen großen Einsatz, die Finanzierung meiner Arbeit stets zu sichern. Erwähnen möchte ich auch meine großartigen Dienstreisen zu Konferenzen oder Forschungsaufenthalten, die dank seines Einsatzes vom mathematischen Institut oder der Forschungskommision der Universität Paderborn finanziert wurden.

Während eines dreimonatigen Forschungsaufenthalt an der NTNU Trondheim in Norwegen wurde meine Arbeit von Øyvind Solberg betreut, und auch ihm möchte ich für sein großes Engagement und seine Gastfreundschaft sehr herzlich danken.

Besonders dankbar bin ich auch Dave Benson für eine Einladung zu einem einwöchigen Forschungsaufenthalt nach Aberdeen. Ich möchte ihm sehr herzlich für die vielen und langen Diskussionen danken, in denen ich sehr viel gelernt habe und durch die ich die Kapitel 4 and 13 über Gruppenkohomologie wesentlich verbessern konnte.

Mein sehr herzlicher Dank gilt auch Bernhard Keller, der sich bei jeder Konferenz, auf der wir uns trafen, Zeit genommen hat, meine Fragen mit großer Geduld und Sorgfalt zu beantworten. Dies hat entscheidend zur Entstehung des neunten Kapitels beigetragen.

Auch bei Ragnar-Olaf Buchweitz möchte ich mich herzlich für Diskussionen während seines Gastaufenthalts in Paderborn bedanken. Hierbei habe ich viel über Kommutative Algebra gelernt und Anregungen erhalten, die mich bei dem Beweis von Ergebnissen in Abschnitt 12.2 einen großen Schritt weiter gebracht haben.

Für anregende mathematische Diskussionen möchte ich mich auch herzlich bei Helmut Lenzing und bei Dirk Kussin bedanken.

Herzlich danken möchte ich Wolfgang Zimmermann für die Einführung in das Gebiet der Algebra und Darstellungstheorie, die ich von ihm an der Universität München erhalten habe.

Ich möchte allen Mitgliedern und ehemaligen Mitgliedern der Paderborner Darstellungstheorie Arbeitsgruppe und allen anderen wissenschaftlichen und nicht-wissenschaft-lichen Mitarbeitern für die angenehme und freundliche Atmosphäre am Mathematischen Institut der Universität Paderborn danken. Besonders danke ich in diesem Zusammenhang auch Kristian Brüning und Karsten Schmidt für die nette Atmosphäre in unserem Büro, viele mathematische Diskussionen und die ”L^{A}TEX-Unterstützung“.

Herzlich danken möchte ich auch Jesús Soto und Stefan Wolf für das Korrekturlesen meines Manuskripts.

Ich danke der Konrad-Adenauer-Stiftung für ein Graduiertenstipendium über 2,5 Jahre und besonders für die Finanzierung des Flugs zu einer Konferenz nach Mexiko.

Während meines dreimonatigem Aufenthalts in Trondheim wurde ich durch ein Marie-Curie-Stipendium finanziert, und ich möchte dafür dem Liegrits-Netzwerk und Idun Reiten sehr herzlich danken.

Besonders danken möchte ich auch der Universität Paderborn für ein sechsmonatiges Abschlussstipendium, welches mir ermöglicht hat, meine Doktorarbeit ohne finanzielle Sorgen in Ruhe abzuschließen.

Sehr herzlich danke ich Claus Ringel für die Einladung zu einem zweimonatigen Gastaufenthalt am Sonderforschungsbereich ”Spektrale Strukturen und Topologische Methoden in der Mathematik“ (SFB 701) an der Universität Bielefeld.

Last but not least möchte ich mich bei meinen Eltern und bei Martín Jiménez für die großartige und liebevolle Unterstützung außerhalb der Mathematik bedanken.

Ich danke den Gutachtern

Prof. Dave Benson University of Aberdeen

Prof. Dr. Henning Krause Universität Paderborn

Prof. Dr. Alexander Zimmermann Université de Picardie, Amiens

1. Introduction

In this thesis we connect two algebraic concepts which seem unrelated at first sight: realisability and localisation. Using some advanced methods from Homological Algebra, we establish a local-global principle for realisability. Before we discuss our main results in detail, let us first explain the concepts we deal with.

The starting point is a cohomology theory which assigns to a mathematical object X its cohomology group H^{∗}X. Such cohomology theories arise for example in Algebraic Topology, Algebraic Geometry, or in Representation Theory. Usually there exists some commutative cohomology ring E such that H^{∗}X is naturally an E-module. Then an E-module is realisable if it is up to isomorphism of the form H^{∗}X for some object X.

The assignment described above can be expressed in the language of categories and functors: If H^{∗}: C → D is a functor between categories C and D, then realisability is concerned with deciding whether an object D ∈ D is isomorphic to an object in the image of the functor H^{∗}.

In some specific representation theoretic context, Benson, Krause, and Schwede [5] established a criterion for realisability. They investigated for a finite group G and a field k the Tate cohomology functor

Ĥ^{∗}(G, −): \underline{Mod} kG → Mod_{gr} Ĥ^{∗}(G, k)

from the stable module category \underline{Mod} kG into the category of ℤ-graded modules over the Tate cohomology ring Ĥ^{∗}(G, k). In this setting, realisability deals with deciding whether a graded Ĥ^{∗}(G, k)-module X can be written as Ĥ^{∗}(G, M) for some module M over the group algebra kG.

The stable module category \underline{Mod} kG has some additional structure: it is a triangulated category. The functor Ĥ^{∗}(G, −) commutes both with arbitrary direct sums and with arbitrary products.

More generally, Benson, Krause and Schwede [5] consider a compactly generated triangulated category T admitting arbitrary direct sums, and a cohomological functor

H^{∗}: T → Mod_{gr} E

into the category of ℤ-graded modules over a cohomology ring E which preserves arbitrary direct sums and products.

For an arbitrary ℤ-graded E-module X, they have given a local obstruction

κ(X) ∈ Ext_{E}^{3,−1}(X, X)

which is trivial if and only if X is a direct summand of H^{∗}M for some object M ∈ T .

Moreover, Benson, Krause and Schwede [5] show that if there exists an infinite sequence of obstructions

κ_{n}(X) ∈ Ext_{E}^{n,2−n}(X, X), n ≥ 3,

where the class κ_{n}(X) is defined provided that the previous one κ_{n−1}(X) vanishes, then it even holds X ≅ H^{∗}M. In this sequence of obstructions all but the first one depend on choices. Only κ_{3}(X) is uniquely determined and actually, it equals the local obstruction κ(X). It is remarkable that, despite of the necessity of an infinite sequence of obstructions to decide if X ≅ H^{∗}M, the first obstruction already tells whether X is a direct summand of H^{∗}M.

Since we mainly deal with the latter question, we call a ℤ-graded E-module X realisable if it is a direct summand of H^{∗}M for some M ∈ T . If X ≅ H^{∗}M, then we refer to a strictly realisable module.

The triangulated categories for which realisability is particularly interesting arise as derived categories of differential graded algebras, or shortly, dg algebras. Such algebras are complexes with an additional multiplicative structure. They have their origin in Algebraic Topology [14] and encode topological invariants. Derived categories of dg algebras were first studied systematically by Bernhard Keller [27].

If A is a dg algebra, then realisability is concerned with deciding whether a graded module over the cohomology ring H^{∗}A is (up to direct summands) isomorphic to a cohomology module H^{∗}M, where M is a dg A-module. The functor in question is

H^{∗}: D(A) → Mod_{gr} H^{∗}A,

where D(A) denotes the derived category of the dg algebra A. Benson, Krause and Schwede [5] show that this setting even admits a global obstruction for realisability. For this purpose, they use a result of Kadeishvili [26] saying that H^{∗}A admits an A_{∞}-algebra structure. The global obstruction arises as the Hochschild class μ_{A} ∈ HH^{3,−1}(H^{∗}A) determined by the secondary multiplication m_{3}^{H^{∗}A} of H^{∗}A: if the canonical class μ_{A} is trivial, then all ℤ-graded H^{∗}A-modules are realisable [5].

The other main concept we consider is localisation. This is an algebraic concept which has its origin in Geometry. In Commutative Algebra, the localisation of a commutative ring R by a multiplicatively closed subset S of R is a uniquely determined ring of fractions S^{−1}R with the property that each s ∈ S is made invertible in S^{−1}R. Similarly, one defines a module S^{−1}M which is a module over S^{−1}R. One considers in particular multiplicatively closed subsets S ⊆ R which are the complement of a prime ideal 𝔭 of R. The ring of fractions is then denoted by R_{𝔭} and the module S^{−1}M by M_{𝔭}. The ring of fractions R_{𝔭} is a local ring, and many results on commutative rings or modules over commutative rings can be proven more easily under the assumption that the ring in question is local. The local-global principle says that an assertion holds if and only it holds in localisation at every prime ideal. It is a classical principle of Commutative Algebra.

Many rings which arise in Representation Theory of Groups or Algebraic Topology are not commutative, but still, their elements commute up to a sign which depends on the degree of the elements. Therefore these rings are called graded-commutative. Examples are the group and Tate cohomology ring of a group or the singular cohomology ring H^{∗}X of a topological space X. These examples arise as cohomology of a dg algebra. Also many other dg algebras do not have a commutative, but still a graded-commutative cohomology ring.

Although these rings are not strictly commutative, still many results from Commutative Algebra can also be proven in this more general setting. In particular, localisation at prime ideals can be done similarly as in classical Commutative Algebra. This is folklore knowledge for some experts, but there seems to be no published account. For this reason, we provide some material on rings of fractions in the graded-commutative setting in Section 3.1.

We will also consider localisation of triangulated categories. In the 1960s, Gabriel and Zisman [20] introduced the Calculus of Fractions for arbitrary categories, generalising the classical localisation of modules. This was used by Verdier in his thèse [54] to study localisation of triangulated categories and in particular, to construct the Verdier quotient.

In this thesis we show that there is a strong relation between realisability and localisation. We prove relations in several settings.

In Chapter 11 we consider differential graded algebras A having a graded-commutative cohomology ring H^{∗}A. We show that if a graded H^{∗}A-module X is realisable, then so is X_{𝔭} for every graded prime ideal of H^{∗}A. Our main result of Chapter 11 is

Theorem 1.1 (Local-global principle). Let A be a dg algebra over a commutative ring such that H^{∗}A is graded-commutative and coherent. The following conditions are equivalent for a finitely presented, graded H^{∗}A-module X:

(1) X is realisable.

(2) X_{𝔭} is realisable for all graded prime ideals 𝔭 of H^{∗}A.

(3) X_{𝔪} is realisable for all graded maximal ideals 𝔪 of H^{∗}A.

We will prove such a local-global principle also for global realisability in Chapter 12. Before we give a precise formulation of this result, let us first explain the contents of Chapter 9.

The results stated in Chapter 9 are joint work with K. Brüning [8] and also deal with a realisability problem, however in a different setting. We consider smashing localisation functors on derived categories of dg algebras, that is, localisation functors of triangulated categories which commute with arbitrary direct sums. We show that every smashing localisation on the derived category of a dg algebra can be realised by a morphism of dg algebras. More precisely, we will prove

Theorem 1.2 (joint work with K. Brüning). Let A be a dg algebra over a commutative ring and L: D(A) → D(A) a smashing localisation. Then there exists a dg algebra A_{L} with the property that D(A_{L}) ≃ D(A)/ Ker L, and the map

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}, f ↦ L(f),

is induced by a zigzag of dg algebra maps

A ⟵^{∼} A′ ⟶^{φ} A_{L}.

That is, there exists a dg algebra A′ quasi-isomorphic to A and a morphism of dg algebras φ: A′ → A_{L} such that in cohomology, we have the commutative diagram

\xymatrix{H^{∗}A′ \ar[d]_{≅} \ar[dr]^{H^{∗}φ} \\ D(A)(A, A)^{∗} \ar[r]^{L} & D(A)(LA, LA)^{∗}.}

Moreover, if A is a cofibrant dg algebra, then there exists a morphism A → A_{L} which induces the algebra map D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗} in cohomology.

We focus in particular on the following special case: Let A be a dg algebra with gradedcommutative cohomology ring, 𝔭 a graded prime ideal of H^{∗}A and L_{𝔭}: D(A) → D(A) the smashing localisation with the L_{𝔭}-acyclic objects being those X ∈ D(A) such that (H^{∗}X)_{𝔭} = 0. Then we denote

A_{𝔭} = A_{L_{𝔭}}

and call the dg algebra A_{𝔭} localisation of A at a prime 𝔭 in cohomology. The cohomology of A_{𝔭} satisfies H^{∗}(A_{𝔭}) ≅ (H^{∗}A)_{𝔭} as graded rings, and with this identification, the canonical map H^{∗}A → (H^{∗}A)_{𝔭} is the cohomology of a zigzag of dg algebras

A ⟵^{∼} A′ ⟶^{φ} A_{𝔭}.

In Chapter 12 we consider dg algebras A with graded-commutative cohomology ring. Our main result in this chapter is a local-global principle for global realisability. For this purpose, we state a global obstruction for the 𝔭-local modules, i.e. those graded H^{∗}Amodules with the property that X ≅ X_{𝔭}. Here we use the dg algebra A_{𝔭} constructed in Chapter 9: this obstruction arises from the A_{∞}-structure of (H^{∗}A)_{𝔭} ≅ H^{∗}(A_{𝔭}). Actually, we show that the canonical Hochschild class μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}(A_{𝔭})), which is a global obstruction for the graded H^{∗}(A_{𝔭})-modules due to Benson, Krause and Schwede [5], is also a global obstruction for the 𝔭-local H^{∗}A-modules.

In order to relate the global obstruction μ_{A} ∈ HH^{3,−1}(H^{∗}A) for the H^{∗}A-modules and the global obstruction μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}A_{𝔭}) for the 𝔭-local H^{∗}A-modules, we construct a map of Hochschild cohomology rings

Γ: HH^{∗,∗}(H^{∗}A) ⟶ HH^{∗,∗}(H^{∗}A_{𝔭})

which has the property Γ(μ_{A}) = μ_{A_{𝔭}}. This is the key to prove that also the global obstruction behaves well under 𝔭-localisation:

Theorem 1.3 (Local-global principle). Let A be a differential graded algebra over a field k such that H^{∗}A is graded-commutative. Assume that the algebra H^{∗}A^{op} ⊗_{k} (H^{∗}A) is Noetherian. Then the following conditions are equivalent:

(1) μ_{A} ∈ HH^{3,−1}(H^{∗}A) is trivial.

(2) μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}A_{𝔭}) is trivial for all graded prime ideals 𝔭 of H^{∗}A.

(3) μ_{A_{𝔪}} ∈ HH^{3,−1}(H^{∗}A_{𝔪}) is trivial for all graded maximal ideals 𝔪 of H^{∗}A.

In the last chapter of this thesis, we focus on realisability in the context of group representation theory. We study the relation between realisability over the group cohomology ring H^{∗}(G, k) and the Tate cohomology ring Ĥ^{∗}(G, k), where k is a field and G a finite group. Note that H^{∗}(G, k) can be viewed as a subalgebra of Ĥ^{∗}(G, k).

For the group cohomology ring, the appropriate realisability setting is given by the functor

\xymatrix{K(Inj kG) \ar[r]^{K(Inj kG)(ik,−)^{∗}} & Mod_{gr} H^{∗}(G, k),}

where K(Inj kG) is the homotopy category of injective kG-modules.

The group cohomology ring H^{∗}(G, k) has better properties than the Tate cohomology ring Ĥ^{∗}(G, k) which, for instance, is not Noetherian in general. However, when it comes to the source categories of realisability, the stable module category \underline{Mod} kG is more “handsome” than the homotopy category K(Inj kG). This is the reason why we are interested in studying the relation of realisability over group and Tate cohomology.

The triangulated categories K(Inj kG) and \underline{Mod} kG are related by a smashing localisation

\xymatrix{K(Inj kG) \ar[r]_{Q} & \underline{Mod} kG \ar[l]_{R}}

and we are now concerned with finding a relation between realisability and this localisation of triangulated categories.

We study realisability of fixed modules as well as global realisability. Note for the latter that both H^{∗}(G, k) and Ĥ^{∗}(G, k) are the cohomology of a dg algebra and thus, they admit an A_{∞}-structure yielding global obstructions denoted by μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) and μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)).

The canonical class μ̂_{G} has been computed for some groups G by Benson, Krause and Schwede [5], and by Langer [37]. We consider the same groups and compute the global obstructions for the group cohomology rings. In many cases, the Hochschild classes μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) and μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)) turn out to behave surprisingly similar. As a first explanation, we show that the algebra morphism H^{∗}(G, k) → Ĥ^{∗}(G, k) is the cohomology of a zigzag of dg algebra morphisms. Then we are ready to prove the main result of this chapter, which is, in some parts, also an application of our results on Hochschild cohomology from Chapter 12:

Theorem 1.4. Let G be finite group, k a field of characteristic p > 0 and assume that p divides the order of G. If the Hochschild class μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)) is trivial, then so is the Hochschild class μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)). If the p-rank of the group G equals one, then μ̂_{G} is trivial if and only if μ_{G} is trivial.

In general, the last statement is not true for groups with p-rank at least two, as we show by giving a counter-example.

Organisation. Our main results, as stated above, can be found in the Chapters 9, 11, 12 and 13. At the end of each of these chapters, we point out related open questions.

In the second chapter we recall facts about triangulated categories and introduce briefly those triangulated categories that we deal with in this thesis.

A short review on graded rings and modules can be found in the third chapter. We focus on localisation of graded-commutative rings in Section 3.1.

In Chapter 4 we introduce group and Tate cohomology rings and state their basic properties.

Hochschild cohomology of graded rings is discussed in Chapter 5. In particular, we study the multiplicative structure of the Hochschild cohomology ring HH^{∗,∗}(Λ) of a graded algebra Λ. We show that for elements ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ), we have the commutativity relation ζ · η = (−1)^{mn}(−1)^{ij}η · ζ, where the multiplication is given by the Yoneda or the cup product. Hochschild cohomology rings of non-graded rings are well-known to be graded-commutative, but we do not know of a published source of this more general result. We will apply it to prove some of our results in Chapter 12.

In Chapter 6 we introduce differential graded algebras and discuss properties of their derived categories.

A short introduction to A_{∞}-algebas is given in Chapter 7. In particular, for a dg algebra A, we present Kadeishvili’s construction [26] of the secondary multiplication m_{3}^{H^{∗}A}: H^{∗}A^{⊗3} → H^{∗}A of the A_{∞}-algebra H^{∗}A.

Chapter 8 is about localisation in triangulated categories and contains important requisites for our main results. After a short discussion of the Calculus of Fractions for arbitrary categories due to Gabriel and Zisman [20] we focus on triangulated categories. In particular, we introduce the Verdier quotient [54] and consider localisation sequences of triangulated categories. In Section 8.6 we state a theorem of Krause [36] on cohomological localisation and prove some results together with K. Brüning [8] which apply in particular to cohomological 𝔭-localisation.

In Chapter 10 we give a short review of the results of Benson, Krause and Schwede [5] and introduce their local and global obstruction for realisability, as discussed above. We focus especially on realisability in the setting of dg algebras.

Notations and conventions. Unless otherwise stated, modules are always considered to be right modules. In particular, we denote by Mod R the category of right R-modules and by mod R the category of finitely generated right R-modules over a ring R. If R is self-injective, then \underline{Mod} R resp. \underline{mod} R denote the stable module categories of Mod R resp. mod R.

When we talk about graded rings and modules, we always mean ℤ-graded rings and modules. If R is a graded ring and it is clear from the context that we mean graded Rmodules, then we sometimes speak of R-modules. We denote by Mod_{gr} R the category of graded right R-modules, where the morphisms are the homogeneous graded R-linear maps of degree zero. By Hom_{R}^{i}(M, N) we denote the homogeneous graded R-linear maps M → N rising the degree by i ∈ ℤ, and we write Hom_{R}(M, N) for Hom_{R}^{0}(M, N). Moreover, we set Hom_{R}^{∗}(M, N) = ∐_{n∈ℤ} Hom_{R}^{n}(M, N).

Cohomology of graded modules over a graded ring R is bigraded; the first index gives the cohomological degree and the second, internal degree arises from the grading of R. For example, for i ≥ 0 and j ∈ ℤ we have

Ext_{R}^{i,j}(M, N) = Ext_{R}^{i}(M, N[j]),

where [j] denotes the j-fold shift on Mod_{gr} R.

In particular, Hochschild cohomology of a graded algebra Λ, denoted by HH^{∗,∗}(Λ), is bigraded. Note that only the first grading is changed by the differential.

For the homotopy category of complexes in an additive category A we write K(A), and the derived category of an abelian category A is denoted by D(A). For the derived category D(Mod R) of a ring R we write shortly D(R).

All dg algebras considered in this thesis are supposed to have a differential of degree +1. So the homology of these dg algebras is, in fact, cohomology, and throughout this thesis, we speak of cohomology. If A is a dg algebra, then we denote its cohomology ring by H^{∗}A, and the homotopy resp. derived category of A will be denoted by K(A) resp. D(A).

The symbol ∼ indicates a quasi-isomorphism and the symbol ≅ is used for isomorphisms of objects in categories. Equivalences of categories are indicated by ≃.

The set of morphisms X → Y in a category C is denoted by C(X, Y) or Hom_{C}(X, Y). If T is a triangulated category, then we denote its suspension functor by Σ or [1]. For i ∈ ℤ, we write T (X, Y)^{i} for T (X, Σ^{i}Y) and we denote

T (X, Y)^{∗} = ∐_{i∈ℤ}T (X, Y)^{i}.

The composition of maps f: A → B and g: B → C is denoted by g∘f or gf; similarly for functors.

2. Triangulated categories

Triangulated categories were introduced independently by Verdier in his thèse [54], and in Algebraic Topology by Puppe [44].

The purpose of this chapter is to state results on triangulated categories which we will use later on. In particular, we recall examples of triangulated categories and fix notation. For the definition of a triangulated category we refer to the book of Neeman [43] or Krause’s notes [34].

Let T be a triangulated category. We denote the suspension functor by [1]: T → T or Σ: T → T . A non-empty full subcategory S is a triangulated subcategory if

(i) S is closed under shifts, i.e. X ∈ S if and only if X[1] ∈ S.

(ii) S is closed under triangles, i.e. if in the exact triangle X → Y → Z → X[1] two

objects from {X, Y, Z} belong to S, then also the third. A triangulated subcategory S is called thick if it is closed under direct factors, that is, a decomposition X = X′ ⨿ X′′ for X ∈ S implies X′ ∈ S.

A triangulated subcategory S admitting arbitrary direct sums is called localising. If S is localising, then it is already a thick subcategory ([43, Rem. 3.2.7]).

Let N be a class of objects in T . The triangulated subcategory generated by N is the smallest full triangulated subcategory which contains N. We refer to [34, Ch. 2.8] for an explicit construction.

If T admits arbitrary direct sums, then the triangulated subcategory generated by N is the smallest full triangulated subcategory which contains N and is closed under taking arbitrary direct sums. We denote this category Loc(N) since it is the smallest localising subcategory that contains N.

An object X ∈ T is called compact if the covariant Hom functor

T (X, −): T → Ab

into the category Ab of abelian groups commutes with arbitrary direct sums. We provide a useful criterion to prove that a category is generated by compact objects.

Lemma 2.1. [50, Lemma 2.2.1] Let T be a triangulated category with arbitrary direct sums and M a set of compact objects. The following conditions are equivalent:

(1) T is generated by M, i.e. T = Loc(M).

(2) An object X ∈ T is trivial if and only if there are no graded maps from M to X, i.e. T (M, X[n]) = 0 for all M ∈ M and n ∈ ℤ.

An exact functor T → S between triangulated categories is a functor preserving exact triangles and shifts. More precisely, it is a pair (F, η), consisting of a functor F: T → S and a natural isomorphism η: F ∘ [1]_{T} ⟶^{≅} [1]_{S} ∘ F such that for every exact triangle

X ⟶^{f} Y ⟶^{g} Z ⟶^{h} X[1], the triangle

F X ⟶^{F f} F Y ⟶^{F g} F Z ⟶^{η_{X}∘F h} (F X)[1]

is exact in S. The following proposition is useful to check whether an exact functor is an equivalence. It is a version of ‘Beilinson’s Lemma’ [1].

Proposition 2.2. [48, Prop. 3.10] Let T , S be triangulated categories admitting arbitrary direct sums and F: T → S an exact functor preserving arbitrary direct sums. Suppose that T has a compact generator C such that (1) F C is a compact generator of S, and

(2) the map

F: T (C, C[n]) → S(F C, F C[n])

is bijective for all n ∈ ℤ. Then F is an equivalence of triangulated categories.

Let T be a triangulated and A an abelian category. A functor F: T → A is called cohomological if it sends each exact triangle in T to an exact sequence in A. In particular, if X → Y → Z → X[1] is an exact triangle in T , then F gives rise to an infinite exact sequence

⋯ → F (Y [−1]) → F (Z[−1]) → F X → F Y → F Z → F (X[1]) → F (Y [1]) → ⋯

For a proof of the following well-known lemma, we refer to [34, Ch. 2.3].

Lemma 2.3. Let T be triangulated and X in T . The representable functors

T (X, −): T → Ab and T (−, X): T^{op} → Ab

are cohomological.

The following result is due to Neeman. It is a consequence of the Brown Representability Theorem (see for example [34, Ch. 4.5]).

Proposition 2.4. [35, Prop. 3.3] Let F: S → T be a an exact functor between triangulated categories, and suppose that S is compactly generated.

(1) There is a right adjoint T → S if and only if F preserves arbitrary direct sums.

(2) There is a left adjoint T → S if and only if F preserves arbitrary products.

Example 2.5. We introduce briefly the triangulated categories which are considered in this thesis, in particular to fix notation. For detailed definitions, we refer to [34].

(1) Let A be an additive category and denote by C(A) the category of complexes in A. The null-homotopic maps form an ideal in C(A) and the homotopy category K(A) is the quotient of C(A) with respect to this ideal. Denote by Σ: K(A) → K(A) the equivalence which takes a complex X to its shifted complex ΣX, defined by

(ΣX)^{n} = X^{n+1} and d_{ΣX}^{n} = −d_{X}^{n+1}.

Given a map α: X → Y of complexes, the mapping cone Cone(α) is the complex defined

in degree n by X^{n+1} ⨿ Y^{n}, and endowed with the differential d_{Cone(α)}^{n} = [ \begin{matrix}−d_{X}^{n+1} & 0 \\ α^{n+1} & d_{Y}^{n} \end{matrix} ] . It fits into a mapping cone sequence

X ⟶^{α} Y ⟶^{β} Cone(α) ⟶^{γ} ΣX,

given in degree n by

X^{n} ⟶^{α^{n}} Y^{n} \underrightarrow{[ \begin{matrix}0 \\ id \end{matrix} ]} X^{n+1} ⨿ Y^{n} ⟶^{[ − id 0 ]} X^{n+1}.

K(A) is triangulated, with the exact triangles being those isomorphic to a mapping cone sequence as defined above.

(2) The derived category D(A) of an abelian category A is obtained from K(A) by formally inverting all quasi-isomorphisms. D(A) is a triangulated category, where the triangulated structure is induced by the one of K(A). More precisely, D(A) carries a unique triangulated structure such that the canonical functor K(A) → D(A) is exact.

(3) In Chapter 6 we introduce differential graded algebras. If A is a differential graded algebra, then the homotopy category K(A) and the derived category D(A) are triangulated categories. This generalises the homotopy resp. derived category of a nongraded algebra, viewed as differential graded algebra concentrated in degree zero.

(4) Let A be an exact category in the sense of Quillen [45]. Thus A is an additive category with a distinguished class of sequences

0 → X ⟶^{α} Y ⟶^{β} Z → 0

which are called exact. The exact sequences satisfy a number of axioms. In particular, the maps α and β in each exact sequence as above form a kernel-cokernel pair. That is, α is a kernel of β and β is a cokernel of α. A map in A arising as the kernel in some exact sequence is called admissible mono, and a map arising as a cokernel is called admissible epi. A full subcategory B of A is extension-closed if every exact sequence in A belongs to B, provided that its end terms belong to B.

Let A be an exact category. An object P ∈ A is called projective if the induced map Hom_{A}(P, Y) → Hom_{A}(P, Z) is surjective for every admissible epi Y → Z. Dually, an object I is injective if the induced map Hom_{A}(Y, I) → Hom_{A}(X, I) is surjective for every admissible mono X → Y. The category A has enough projectives if every object Z admits an admissible epi Y → Z with Y projective, and it has enough injectives if every object X admits an admissible mono X → Y with Y injective. Finally, A is called a Frobenius category if A has enough projectives and enough injectives and if both coincide.

The stable category of a Frobenius category A is denoted by S(A) and defined to be the quotient of A with respect to the ideal I of morphisms factoring through an injective object. Thus

Hom_{S(A)}(X, Y) = Hom_{A}(X, Y)/I(X, Y)

for all X, Y in A.

We choose for each X ∈ A an exact sequence

0 → X → I(X) → ΣX → 0

such that I(X) is injective. The morphism X → I(X) is called injective hull. One easily checks that the assignment X ↦ ΣX defines an equivalence on S(A). Every exact sequence 0 → X → Y → Z → 0 fits into a commutative diagram

[figure: 0; X; α; Y; β; Z; γ; 0; 0; X; I(X); ΣX; 0]

such that I(X) is injective. The category S(A) carries a triangulated structure, with the exact triangles being those isomorphic to a sequence of maps

X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

as in the diagram above.

(4)(a) If A is a finite dimensional self-injective algebra, then Mod A is a Frobenius category and obviously, Σ equals the first cosyzygy Ω^{−1}. We denote the stable category S(Mod A) by \underline{Mod} A and conclude that it has a triangulated structure.

(4)(b) The homotopy category K(A) of an additive category A identifies with the stable category of C(A), where the exact structure is induced by the degree-wise split short exact sequences of complexes, see [34, Ch. 7.2].

(4)(c) Similarly, if A is a differential graded algebra, then the homotopy category K(A) is the stable category of a Frobenius category. We provide more details in Section 6.2.

3. Graded rings and modules

In this section we introduce graded rings and modules, and state some properties of the category of graded modules. In particular, we fix the sign convention we will use throughout this paper. Unless otherwise stated, we mean graded right modules when we speak of graded modules.

A ℤ-graded ring is a ring R together with a decomposition of abelian groups

R = ∐_{i∈ℤ}R_{i}

such that R_{i}R_{j} ⊆ R_{i+j}.

A ℤ-graded module over a ℤ-graded ring R is an R-module M together with a decomposition of abelian groups

M = ∐_{i∈ℤ}M_{i}

satisfying M_{i}R_{j} ⊆ M_{i+j}.

A ℤ-graded algebra over some commutative ring k is a graded ring Λ which also has a graded k-module structure that makes Λ into a k-algebra. Note that the operation of k on Λ has degree zero.

Throughout this thesis, we will talk about graded rings, graded algebras and graded modules and always refer to ℤ-graded rings, algebras and modules.

Let M be a graded module over a graded ring R. The elements m ∈ M_{i} are called homogeneous elements of degree i, and we denote the degree of m by |m|. For n ∈ ℤ, the n-fold shifted graded module M[n] is given by M[n]^{i} = M^{n+i}. We use the notation Σ^{n}m when we view m ∈ M^{n+i} as an element in M[n]^{i}.

If M, N are graded R-modules and n ∈ ℤ, then an R-linear map f: M → N is called homogeneous graded map or shortly, graded map of degree n if f(M_{j}) ⊆ N_{j+n} for all j ∈ ℤ. Note that f can also be considered as a graded map M → N[n] of degree zero. We denote by Hom_{R}^{n}(M, N) the set of all graded maps M → N of degree n, and we define

Hom_{R}^{∗}(M, N) = ∐_{n∈ℤ}Hom_{R}^{n}(M, N).

The graded R-modules form a category denoted by Mod_{gr} R. The morphisms are the graded maps of degree zero and we denote

Hom_{R}(M, N) = Hom_{R}^{0}(M, N).

A graded R-module B ⊆ M is a graded submodule of M if the inclusion map is a morphism in Mod_{gr} R. In this case, the quotient M/B also carries a natural grading. If f: M → N is a morphism in Mod_{gr} R, then Ker f, Im f and Coker f are graded modules. Moreover, one can show that Mod_{gr} R is a Grothendieck category [40, Ch. 2.2].

The graded submodules of R are called graded right ideals. An arbitrary right ideal I of R is graded if and only if it is generated by homogeneous elements.

A graded R-module is graded free if it is a direct sum of shifted copies of R, or equivalently, if it has an R-basis consisting of homogeneous elements. Note that it is not enough to assume that the module is graded and free as non-graded module: Considering R = ℤ × ℤ as graded ring concentrated in degree 0, the module F = ℤ × ℤ endowed with the grading F_{0} = ℤ × 0, F_{1} = 0 × ℤ, and F_{i} = 0 otherwise, is not graded free since it cannot have an R-basis consisting of homogeneous elements.

A graded R-module is called graded projective if it is a projective object in the category Mod_{gr} R, or equivalently, if it is a direct summand of a graded free R-module. Since Mod_{gr} R is a Grothendieck category, it has enough injective objects. Those are the graded injective modules.

Every graded R-module M admits a graded free presentation F_{1} → F_{0} → M → 0, i.e. F_{0} and F_{1} are graded free modules. If both F_{0} and F_{1} can be chosen to be finitely generated, then M is called finitely presented.

A graded ring is right Noetherian if it is right Noetherian as a ring, i.e. every (not necessarily graded) ideal is finitely generated. In this case, every finitely generated graded R-module is already finitely presented.

If N is a graded R-module and i ≥ 0, j ∈ ℤ, then one defines

Ext_{R}^{i,j}(M, N) = Ext_{R}^{i}(M, N[j]).

An element of Ext_{R}^{i,j}(M, N) can be represented by an exact sequence of graded Rmodules

0 → N[j] → X_{i} → ⋯ → X_{1} → M → 0.

Assume now that R is a graded algebra over some commutative ring k. For graded modules M and N the tensor product M ⊗_{k} N is a graded module, where the degree i component is given by

(M ⊗_{k} N)_{i} = ∐_{p+q=i} M_{p} ⊗_{k} N_{q}.

Let f: M → M′ and g: N → N′ be graded maps. Note that due to the Koszul sign rule, in the tensor product f ⊗ g there appears a sign:

(3.1)  (f ⊗ g)(m ⊗ n) = (−1)^{|g|·|m|}f(m) ⊗ g(n),

where m ∈ M is homogeneous and n ∈ N.

One also needs to involve signs to define the opposite algebra: R^{op} is again a graded algebra, with multiplication

(3.2)  r · r′ = (−1)^{|r|·|r′|}r′r.

A graded right R-module can be viewed as graded left R^{op}-module by setting

r · m = (−1)^{|r||m|}mr.

If S is a graded k-algebra, then R ⊗_{k} S is a graded k-algebra with multiplication

(3.3)  (r ⊗ s)(r′ ⊗ s′) = (−1)^{|r′||s|}(rr′ ⊗ ss′).

A graded (R, S)-bimodule M is simultaneously a graded left R-module and a graded right S-module such that (rm)s = r(ms). The graded (R, S)-bimodules correspond to the graded right modules over R^{op} ⊗_{k} S.

If M is a graded (R, S)-bimodule, then M[t] is a graded (R, S)-bimodule by setting

(3.4)  r · (Σ^{t}m) · s = (−1)^{t|r|} Σ^{t}(rms).

Note that the shift functor M ↦ M[1] for graded right modules does not involve any extra sign: we have (Σm) · r = Σ(mr). However, the sign in (3.4) appears when translating graded right R-modules into graded left modules over R^{op}.

The graded rings we are particularly interested in arise as graded endomorphism rings of objects of triangulated categories.

Example 3.1. Let T be a triangulated category with arbitrary direct sums and suspension functor Σ. For objects M, N ∈ T we write T (N, M)^{i} = T (N, Σ^{i}M). Then

T (N, N)^{∗} = ∐_{i∈ℤ}T (N, N)^{i}

is a graded ring, called the graded endomorphism ring of N, and T (N, M)^{∗} is a graded T (N, N)^{∗}-module by composition of graded maps.

3.1. Graded-commutative rings. A graded ring R is called graded-commutative if rs = (−1)^{|r||s|}sr for all homogeneous elements r, s ∈ R. Although such a ring is not strictly commutative, many results about graded and commutative rings (which are studied for example in [10]) can still be carried over. However, “Commutative Algebra over graded-commutative rings” is rarely treated in literature. We provide definitions and results about localisation of graded-commutative rings that we will need later on.

We like to thank Dave Benson for pointing out Remark 3.3 and Lemma 3.5.

3.1.1. Prime and maximal ideals. A graded right ideal 𝔪 of a graded ring R is called graded maximal right ideal if 𝔪 ≠ R and moreover, for any graded right ideal 𝔞 such that 𝔪 ⊆ 𝔞 ⊆ R, it holds 𝔞 = 𝔪 or 𝔞 = R. If 𝔟 ≠ R is a graded right ideal of R, then there exists a maximal right ideal 𝔪 containing 𝔟. If R is graded-commutative, a graded (maximal) right ideal is a graded (maximal) ideal.

For an arbitrary ring R, a prime ideal 𝔭 ⊊ R is an ideal such that for a, b ∈ R, it holds a ∈ 𝔭 or b ∈ 𝔭 whenever aRb ⊆ 𝔭. If R is graded-commutative, then this definition simplifies as in the case of strictly commutative rings:

Definition 3.2. Let R be a graded-commutative ring. An ideal 𝔭 ⊊ R is a prime ideal if ab ∈ 𝔭 implies that a ∈ 𝔭 or b ∈ 𝔭. The set of graded prime ideals 𝔭 ⊆ R is called the graded spectrum of R and denoted by Spec_{gr}(R).

Actually, the prime spectrum of a graded-commutative ring can be identified with the prime spectrum of a graded, strictly commutative ring:

Remark 3.3. Let R be a graded-commutative ring, and denote by 𝔫 the ideal generated by the homogeneous nilpotent elements. If x is a homogeneous element of odd degree, it holds 2x^{2} = 0, and thus, 2x is nilpotent. Hence x ≡ −x mod 𝔫, and the factor ring R/𝔫 is a graded, strictly commutative ring. Since 𝔫 is contained in all graded prime ideals, it follows that

Spec_{gr}(R) = Spec_{gr}(R/𝔫).

Lemma 3.4. Let R be a graded-commutative ring and 𝔞 ⊆ R a graded ideal. (1) 𝔞 is prime if and only if R/𝔞 is a domain.

(2) 𝔞 is a graded maximal ideal if and only if every non-zero homogeneous element of R/𝔞 is invertible.

Proof. (1) is trivial. For (2) note that all non-zero homogeneous elements of a gradedcommutative ring T are invertible if and only if the only graded ideals contained in T are (0) and T. □

If R is graded commutative, then all non-zero homogeneous elements being invertible implies that R is a domain. Thus every graded maximal ideal is prime.

3.1.2. Rings and modules of fractions. Let R be a (not necessarily graded) ring and S a multiplicative subset with 1 ∈ S. We define the right ring of fractions of R with respect to S as a ring R[S^{−1}] together with a ring homomorphism ρ: R → R[S^{−1}] satisfying

(F1) ρ(s) is invertible for each s ∈ S.

(F2) Every element in R[S^{−1}] has the form ρ(r)ρ(s)^{−1} with s ∈ S.

(F3) ρ(r) = 0 if and only if there exists an element s ∈ S such that rs = 0. It is not immediately clear from these axioms that R[S^{−1}] is uniquely determined, but it is, in fact, the case. We refer to [52, Ch. II].

Let S be a multiplicatively closed subset of R. The ring R[S^{−1}] exists if and only if the following conditions, called right Ore conditions, are satisfied:

(O1) If s ∈ S and r ∈ R, then there exist s′ ∈ S and r′ ∈ R such that sr′ = rs′ (O2) If r ∈ R and s ∈ S with sr = 0, then there exists s′ ∈ S such that rs′ = 0.

If (O1) and (O2) are satisfied, then

R[S^{−1}] = R × S/ ∼,

where the equivalence relation ∼ is given by

\frac{r}{s} ∼ \frac{r′}{s′}

if and only if there exist u, v ∈ R such that ru = r′v and su = s′v.

If the analogous left Ore conditions are satisfied, then there exists the left ring of fractions [S^{−1}]R. Furthermore, if both R[S^{−1}] and [S^{−1}]R exist, then they are isomorphic [52, II, Cor. 1.3].

If we now assume that R is graded and S a multiplicative subset of homogeneous elements with 1 ∈ S, then it suffices to check the Ore conditions on homogeneous elements. Moreover, if R[S^{−1}] exists, then it is a graded ring, where

deg (\frac{r}{s}) = deg(r) − deg(s)

for any homogeneous element r ∈ R and s ∈ S. [40, Ch. 8.1]

If R is graded-commutative, then the Ore conditions are trivially satisfied and the right ring of fractions R[S^{−1}] exists. One might want to define the equivalence relation as for strictly commutative rings, but the transitivity fails for elements s ∈ S of odd degree. However, one easily checks

Lemma 3.5. Let R be a graded-commutative ring and S is a multiplicative closed subset of homogeneous elements of R. Let S_{ev} ⊆ S the subset of even-degree elements of S. Then R[S^{−1}] ≅ R[S_{ev}^{−1}] as graded rings and the equivalence relation simplifies into \frac{r}{s} ∼ \frac{r′}{s′} with r, r′ ∈ R, s, s′ ∈ S_{ev} if and only if there exists t ∈ S_{ev} such that rs′t = r′st. □

Consequently, we may define the localisation of a graded-commutative ring R with respect to a multiplicative subset S of homogeneous elements to be the ring

R[S_{ev}^{−1}]

with the equivalence relation used in the strictly commutative case. Then addition and multiplication are also defined as in that well-known case. We write S^{−1}R for R[S_{ev}^{−1}].

If M is a graded R-module, we define M_{S}, the localisation of M with respect to S as

S^{−1}M = R × S_{ev}/ ∼,

with \frac{m}{s} ∼ \frac{m′}{s′} if and only if ms′t = m′st for some t ∈ S_{ev}. Obviously, S^{−1}M is a graded S^{−1}R-module with the canonical structure and grading

deg (\frac{m}{s}) = deg(m) − deg(s),

where m ∈ M is homogeneous and s ∈ S_{ev}.

Note that S^{−1}R is flat as both left and right R-module, and that M ⊗_{R}S^{−1}R ≅ S^{−1}M as graded S^{−1}R-modules.

Let 𝔞 be a graded ideal of R and let S be the subset of homogeneous elements of R\𝔞. Then we set M_{𝔞} = S^{−1}M. Similarly as in classical Commutative Algebra, we have

Proposition 3.6 (Local-global principle). Let M be a graded R-module. The following conditions are equivalent:

(1) M = 0.

(2) M_{𝔭} = 0 for all graded prime ideals 𝔭.

(3) M_{𝔪} = 0 for all graded maximal ideals 𝔪.

Proof. We only need to show that (3) implies (1). Let x ∈ M be any element. We consider Ann(x)^{∗}, the largest graded ideal contained in Ann(x) = {r ∈ R | rx = 0}. Assuming that Ann(x)^{∗} is a proper graded ideal of R, we obtain a graded maximal ideal 𝔪 which contains Ann(x)^{∗}. Since M_{𝔪} = 0, there exists a homogeneous element s in R \ 𝔪 such that sx = 0. So s is contained in Ann(x)^{∗} and thus in 𝔪, which is a contradiction. □

4. Group and Tate cohomology rings

Homology and Cohomology of groups has been considered since the 1940s. Inspired by a work of Hopf [23] from 1941 in which he considers what today is called the second homology group H_{2}(G, ℤ) of a group G, Eilenberg and Mac Lane [18] started to study systematically homology and cohomology of groups.

The Tate cohomology ring was invented by Tate, but these results were never published by himself; the first published account is contained in the book of Cartan and Eilenberg [15].

In the first section of this chapter we study group cohomology rings, and in the second Tate cohomology rings.

We thank Dave Benson for many useful comments and pointing out references to the author.

4.1. Group cohomology rings. Let k be a commutative ring and G a finite group. The ring k becomes a kG-module by trivial action of G. This module is called the trivial module and also denoted by k.

If M is a kG-module and n ≥ 0, then the n-th cohomology of G with coefficients in M is defined to be

H^{n}(G, M) = Ext_{kG}^{n}(k, M).

The Yoneda splice multiplication yields a k-bilinear, associative map (see [13, Sect. 6])

H^{n}(G, M) × H^{l}(G, k) → H^{n+l}(G, M),

defined as follows: If ζ ∈ H^{n}(G, M) is represented by

E_{ζ}: 0 → M → X_{0} → ⋯ → X_{n} ⟶^{π} k → 0

and η ∈ H^{l}(G, k) by

E_{η}: 0 → k ⟶^{ι} Y_{0} → ⋯ → Y_{l} → k → 0,

then the Yoneda splice product of ζ and η is the class which is represented by the exact sequence obtained by splicing together E_{ζ} and E_{η}:

\xymatrix{0 \ar[r] & M \ar[r] & X_{0} \ar[r] & ⋯ \ar[r] & X_{n} \ar[dr]^{π} \ar[rr]^{ι∘π} &  & Y_{0} \ar[r] & ⋯ \ar[r] & Y_{l} \ar[r] & k \ar[r] & 0 \\  &  &  &  &  & k \ar[ur]^{ι}}

We denote by K(Inj kG) the homotopy category of Inj kG, which is the full subcategory Mod kG formed of the injective kG-modules. The category Inj kG is additive, and it is closed under arbitrary direct sums, provided that kG is noetherian.

Write iM ∈ K(Inj kG) for an injective resolution of a kG-module M. With the well-known identification

Ext_{kG}^{m}(k, X) ≅ K(Inj kG)(ik, Σ^{m}(iX))

for any X ∈ Mod kG and m ≥ 0, one can also form a k-bilinear, associative product by composition of chain maps of injective resolutions:

H^{n}(G, M) × H^{l}(G, k) → H^{n+l}(G, M), (f, g) ↦ Σ^{l}(g) ∘ f.

This product coincides with the Yoneda splice product (see [13, Sect. 6]).

With any of the two multiplications,

H^{∗}(G, k) = ∐_{n≥0} H^{n}(G, k)

is a graded ring (concentrated in non-negative degrees) and

H^{∗}(G, M) = ∐_{n≥0} H^{n}(G, M)

is a graded module over H^{∗}(G, k).

Note also that H^{∗}(G, k) is a graded-commutative ring [13, Cor. 6.9]. Due to Evens and Venkov, it is Noetherian whenever k is.

Theorem 4.1 (Evens, Venkov). If k is Noetherian, then H^{∗}(G, k) is a finitely generated k-algebra.

4.2. Tate cohomology rings. Let k be a field of characteristic p > 0 and G be a finite group such that p divides the order of G. We denote by \underline{Mod} kG the stable module category of kG. The objects are the same as in the module category Mod kG, but the morphisms are given by

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/I(M, N),

where I(M, N) denotes the morphisms factoring through an injective object. The category \underline{Mod} kG is triangulated with shift functor Ω^{−1}, the first cosyzygy, see Example 2.5(4)(a). An elementary proof for the fact that \underline{Mod} kG is triangulated can be found in Carlson’s book [13, Thms. 5.6, 11.4].

A Tate resolution or complete resolution of a kG-module X is an exact sequence of projectives

tX: ⋯ → P_{2} → P_{1} → P_{0} ⟶^{δ} P_{−1} → P_{−2} → ⋯

with Im δ = X. It can be constructed by splicing together a projective and an injective resolution of X.

If M denotes a kG-module and n ∈ ℤ, then the n-th Tate cohomology group of G with coefficients in M is defined to be the n-th cohomology of the complex Hom_{kG}(tk, M) and denoted by

Ĥ^{n}(G, M) = Êxt_{kG}^{n}(k, M).

The Tate cohomology groups identify with morphism groups in the stable module category of kG: it holds

Ĥ^{n}(G, M) ≅ \underline{Hom}_{kG}(k, Ω^{−n}M),

see [13, Ch. 6]. Thus Ĥ^{∗}(G, k) = ∐_{n∈ℤ} Ĥ^{n}(G, k) becomes a graded ring with multiplication

Ĥ^{n}(G, k) × Ĥ^{m}(G, k) → Ĥ^{n+m}(G, k), (f, g) ↦ Ω^{−n}(f) ∘ g,

and Ĥ^{∗}(G, M) is a graded module over Ĥ^{∗}(G, k), also by composition of graded maps.

Since the non-negative part of the Tate resolution tk is a projective resolution of k, we have Ĥ^{n}(G, M) = H^{n}(G, M) for n > 0, and we obtain an exact sequence

0 → I(k, M) → H^{∗}(G, M) → Ĥ^{∗}(G, M) → H^{−}(G, M) → 0,

where H^{−}(G, M) denotes the negatively graded part of Tate cohomology. In particular, since I(k, M) = 0 whenever M has no projective direct summands, we can view the group cohomology ring as subring of the Tate cohomology ring. In fact, for the positively graded part of Ĥ^{∗}(G, k), Yoneda splice multiplication and composition of graded maps in the stable module category coincide [13, Sect. 6].

Proposition 4.2 (Tate duality). [15, XII, Cor. 6.5], [4, Sect. 2] Let D = Hom_{k}(−, k). For any kG-module M, it holds

Ĥ^{n−1}(G, DM) ≅ DĤ^{−n}(G, M).

The Tate cohomology ring is graded-commutative [15, XII, Prop. 5.2]. In general, it is not Noetherian. However, this is true in the so-called periodic case that we discuss below. We omit the proof of the following well-known characterisation.

Lemma 4.3. The following are equivalent:

(1) The trivial module k admits a periodic projective resolution.

(2) There exists n > 0 and an element x ∈ Ĥ^{n}(G, k) such that the map

Ĥ^{m}(G, k) → Ĥ^{m+n}(G, k), γ ↦ γ · x,

is an isomorphism for all m ∈ ℤ.

We call an element x ∈ Ĥ^{n}(G, k) satisfying (2) a periodicity generator, and from Lemma 4.3, we infer

Lemma 4.4. In the periodic case, the Tate cohomology ring is a localisation of the group cohomology ring:

Ĥ^{∗}(G, k) = S^{−1}H^{∗}(G, k),

where S is the multiplicative subset generated by the periodicity generator of lowest degree. In particular, Ĥ^{∗}(G, k) is Noetherian whenever it is periodic. □

Whether Tate cohomology is periodic or not depends on the p-rank of G, which is the maximal rank of an elementary abelian p-subgroup of G and denoted by r_{p}(G).

Theorem 4.5. [15, XII, Prop. 11.1] Let k a be field of characteristic p and G a group with order divisible by p. Then the trivial module k admits a periodic resolution if and only if r_{p}(G) = 1.

The p-groups of p-rank one are characterised as follows:

Theorem 4.6. [21, Chapter 5, Theorem 4.10] Let G be a p-group. Then r_{p}(G) = 1 if and only if G is cyclic or p = 2 and G is a generalised quaternion group.

The generalised quaternion group Q_{2^{n}} is given by the defining relations

Q_{2^{n}} = ⟨h, g | h^{2^{n−1}} = g^{2} = b, b^{2} = 1, g^{−1}hg = h^{−1}⟩

and has order 2^{n+1}, see [21, Ch. 2.6]. In the special case n = 2, the group is called quaternion group.

Remark 4.7. The Tate cohomology ring comes in three different types: It is periodic if and only if the p-rank of the group G equals one. Whenever the depth of the group cohomology ring H^{∗}(G, k) is at least two, then Ĥ^{∗}(G, k) is a trivial extension of H^{∗}(G, k) by the negatively graded part Ĥ^{−}(G, k) of Ĥ^{∗}(G, k) [3, Thm. 3.3]. That is in particular the case if the p-rank of the centre of a p-Sylow subgroup of G is at least two ([17], see also [3, Thm. 3.2]). But there are also cases where the Tate cohomology ring is neither periodic nor a trivial extension: The smallest example is the semidihedral group of order 16

SD_{16} = ⟨g, h | g^{8} = 1, h^{2} = 1, h^{−1}gh = g^{3}⟩.

The 2-rank of SD_{16} is two and the depth of H^{∗}(SD_{16}, k) equals one (see [3]).

5. The Hochschild Cohomology of a graded ring

Hochschild cohomology HH^{∗}(R) of an algebra R over some commutative ring k was first defined by Hochschild in 1945 using the Bar resolution [22]. Cartan and Eilenberg showed that HH^{∗}(R) is isomorphic to Ext_{R^{e}}^{∗}(R, R) whenever R is projective over k [15, Ch. IX.6].

Hochschild cohomology of a graded algebra Λ is bigraded; it is denoted by HH^{∗,∗}(Λ), where the first index is the cohomological degree which is changed by the differential and the second, internal one, arises from the grading of Λ. The difference between “usual” Hochschild cohomology and Hochschild cohomology of a graded algebras consists basically in the occurrence of additional signs. However, sometimes it can be tedious to figure them out.

In this chapter we introduce Hochschild cohomology of a graded algebra Λ with the sign conventions as in [5]. In addition, we study in Section 5.3 the ring structure of HH^{∗,∗}(Λ). Hochschild cohomology HH^{∗}(R) of a non-graded algebra R is well-known to be a graded-commutative ring (in the sense of Section 3.1) with multiplication given by cup or Yoneda product. We show that for bigraded Hochschild cohomology HH^{∗,∗}(Λ), one additionally needs to take into account the internal degree: for ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ), we prove that ζη = (−1)^{mn}(−1)^{ij}ηζ.

Throughout this chapter let Λ be a graded algebra over a field k. We write Λ^{⊗n} for the n-fold tensor product over k and denote a tuple λ_{1} ⊗⋯⊗λ_{n} ∈ Λ^{⊗n} by (λ_{1}, ⋯, λ_{n}).

For any graded (Λ, Λ)-bimodule M, the Hochschild cohomology of Λ with coefficients in M is the cohomology of the bigraded complex C^{∗,∗}(Λ, M) given by

(5.1)  C^{n,m}(Λ, M) = Hom_{k}^{m}(Λ^{⊗n}, M),

where n ≥ 0 and m ∈ ℤ. Note that the differential δ: C^{n,m}(Λ, M) → C^{n+1,m}(Λ, M) changes only the first grading. It is given by

(δφ)(λ_{1}, …, λ_{n+1}) = (−1)^{m|λ_{1}|}λ_{1}φ(λ_{2}, …, λ_{n+1}) +

∑_{i=1}^{n}(−1)^{i}φ(λ_{1}, …, λ_{i}λ_{i+1}, …, λ_{n+1}) + (−1)^{n+1}φ(λ_{1}, …, λ_{n})λ_{n+1}.

This construction obviously generalises the Hochschild complex in the non-graded case.

The Hochschild cohomology groups HH^{∗,∗}(Λ, M) are the cohomology groups of the complex C^{∗,∗}(Λ, M),

HH^{s,t}(Λ, M) = H^{s}(C^{∗,t}(Λ, M)) .

HH^{s,t}(Λ, Λ) is abbreviated by HH^{s,t}(Λ).

Using the bimodule structure of M[t] given in (3.4), there is a natural isomorphism of chain complexes

C^{∗,m}(Λ, M) ≅ C^{∗,0}(Λ, M[t]),

which induces a natural isomorphism of Hochschild cohomology groups

HH^{s,t}(Λ, M) ≅ HH^{s,0}(Λ, M[t]).

5.1. Functoriality. The complex C^{∗,∗}(Λ, M) and its cohomology groups HH^{∗,∗}(Λ, M) are covariant functors in the (Λ, Λ)-bimodule M: If f: M → N is a morphism of (Λ, Λ)- bimodules, then we have a cochain homomorphism

C^{s,t}(Λ, M) → C^{s,t}(Λ, N), φ ↦ f ∘ φ,

which induces the map

HH^{s,t}(Λ, M) → HH^{s,t}(Λ, N), [φ] ↦ [f ∘ φ].

Furthermore, the Hochschild groups are contravariant functors in the graded algebra. Let α : Γ → Λ be a map of graded k-algebras. Then a (Λ, Λ)-bimodule M carries a (Γ, Γ)-bimodule structure through the morphism α. We obtain a cochain homomorphism

C^{s,t}(Λ, M) → C^{s,t}(Γ, M), φ ↦ φ ∘ α^{⊗s},

inducing the map

HH^{s,t}(Λ, M) → HH^{s,t}(Γ, M), [φ] ↦ [φ ∘ α^{⊗s}].

In general, we cannot expect α : Γ → Λ to give rise to a map HH^{s,t}(Λ) → HH^{s,t}(Γ). However, in the case that α is a flat epimorphism of rings, we will construct an algebra homomorphism HH^{s,t}(Λ) → HH^{s,t}(Γ) induced by α in Chapter 12.1. For this construction, we need the graded Bar resolution.

5.2. The graded Bar resolution. Let Λ^{e} = Λ^{op} ⊗ Λ. With the necessary precaution on the signs (see (3.2) and (3.3)), this algebra is graded and we may identify graded (Λ, Λ)-bimodules with graded right Λ^{e}-modules.

The graded bar resolution 𝔹 = 𝔹(Λ) is a Λ^{e}-projective resolution of Λ defined as 𝔹_{n} = B_{n} = Λ^{⊗(n+2)}, with Λ^{e}-module structure given by

(λ_{0}, …, λ_{n+1})(μ, μ′) = (−1)^{|μ||λ_{0}···λ_{n+1}|} (μλ_{0}, λ_{1}, …, λ_{n}, λ_{n+1}μ′),

and with differential

d_{n}(λ_{0}, …, λ_{n+1}) = ∑_{i=0}^{n}(−1)^{i}(λ_{0}, …, λ_{i}λ_{i+1}, …, λ_{n+1}).

For the fact that 𝔹(Λ) is indeed a graded projective Λ^{e}-resolution of Λ, we refer to [5, Sect. 4] and [15, Ch. IX.6].

Lemma 5.1. The map

(5.2)  Hom_{k}^{t}(Λ^{⊗s}, M) → Hom_{Λ^{e}}^{t}(Λ^{⊗(s+2)}, M), f ↦ f̃,

where

f̃(λ_{0}, ⋯, λ_{s+1}) = (−1)^{t|λ_{0}|}λ_{0}f(λ_{1}, ⋯, λ_{s})λ_{s+1},

is an isomorphism and extends to an isomorphism of chain complexes

(5.3)  C^{∗,t}(Λ, M) → Hom_{Λ^{e}}^{t}(B_{∗}, M).

Consequently, we have

HH^{s,t}(Λ, M) = H^{s}(C^{∗,t}(Λ, M)) = H^{s}(Hom_{Λ^{e}}^{t}(𝔹, M)),

and thus, it holds

HH^{s,t}(Λ, M) = Ext_{Λ^{e}}^{s,t}(Λ, M).

Of course, the Hochschild cohomology groups can also be computed with an arbitrary graded Λ^{e}-projective resolution of Λ.

5.3. Ring structure. G. Hochschild [22] proved that Hochschild cohomology HH^{∗}(R) of an algebra R admits a ring structure, with multiplication given by the cup product. It is well-known that the cup product coincides with the Yoneda product (see [12, Prop. 1.1]) and makes HH^{∗}(R) into a graded-commutative ring (see for example [51]). In this section we study the ring structure of bigraded Hochschild cohomology HH^{∗,∗}(Λ) of a graded algebra Λ.

There is a degree zero chain map Δ: 𝔹 → 𝔹 ⊗_{Λ} 𝔹 lifting the identity map of Λ, given by

Δ(λ_{0}, ⋯, λ_{n+1}) = ∑_{i=0}^{n}(λ_{0}, ⋯ λ_{i}, 1) ⊗_{Λ} (1, λ_{i+1}, ⋯, λ_{n+1}).

For the non-graded case, this can be found for example in [51]. It carries over to our case without any additional sign adjustment.

Let ζ ∈ HH^{m,i}(Λ) be represented by a cocycle B^{m} → Λ of degree i, and represent η ∈ HH^{n,j}(Λ) by a (n, j)-cocycle B^{n} → Λ. Then the cup product of ζ and η is given by the composition of graded maps

𝔹 ⟶^{Δ} 𝔹 ⊗_{Λ} 𝔹 ⟶^{ζ⊗η} Λ ⊗_{Λ} Λ ⟶^{ν} Λ,

where ν: Λ ⊗_{Λ} Λ → Λ is the multiplication map. In order to write the cup product ζ ∪ η explicitly as a (m+n, i+j)-cocycle B^{m+n} → Λ, we need to take into account the Koszul sign rule (3.1) and obtain

(5.4) (ζ ∪ η)(λ_{0}, ⋯, λ_{m+n+1}) = (−1)^{(|λ_{0},··· ,λ_{m}|)·j}ζ(λ_{0}, ⋯ λ_{m}, 1)η(1, λ_{m+1}, ⋯ λ_{m+n+1}).

The cup product makes HH^{∗,∗}(Λ) into a bigraded ring, that is, a ℤ × ℤ-graded ring.

The Yoneda product ζ ∗ η is given through a graded lifting of η, i.e. a graded chain map 𝔹 → Σ̄^{n}𝔹 which lifts η. Here Σ̄^{n}𝔹 denotes the n-fold shift to the left of the complex 𝔹 without changing the signs in the differential. Note that the ‘internal’ degree of this chain map is the degree of η.

Now we adapt the proof for the non-graded case (see [12, Prop. 1.1]) to our setting and show that cup product and Yoneda product coincide: Proposition 5.2. Let ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ) be represented by the (m, i)- cocycle ζ: B^{m} → Λ and the (n, j)-cocycle η: B^{n} → Λ, respectively. Then it holds

ζ ∪ η = ζ ∗ η.

Proof. In [12, Prop. 1.1], it is shown that η̃: 𝔹 → Σ̄^{n}𝔹, defined by

𝔹 ⟶^{Δ} 𝔹 ⊗_{Λ} 𝔹 ⟶^{𝔹⊗η} 𝔹 ⊗_{Λ} Σ̄^{n}Λ ⟶^{ν} Σ̄^{n}𝔹,

is a lifting of η. Actually, this is a graded lifting of internal degree j. Hence we may set

ζ ∗ η = ζη̃ = ζν(𝔹 ⊗ η)Δ.

The Koszul sign rule (3.1) permits the equations

ζν = ν(ζ ⊗ Σ̄^{n}Λ) and (ζ ⊗ Σ̄^{n}Λ)(𝔹 ⊗ η) = ζ ⊗ η.

But then

ζ ∗ η = ζν(𝔹 ⊗ η)Δ
= ν(ζ ⊗ Σ̄^{n}Λ)(𝔹 ⊗ η)Δ
= ν(ζ ⊗ η)Δ
= ζ ∪ η.  □

Now we are ready to prove Theorem 5.3. HH^{∗,∗}(Λ) is a bigraded-commutative ring: For ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ), we have

ζη = (−1)^{mn}(−1)^{ij}ηζ,

where the multiplication is given by the cup or Yoneda product. ^{1} Proof. Let ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ) be represented by the (m, i)-cocycle ζ: B^{m} → Λ and the (n, j)-cocycle η: B^{n} → Λ, respectively. We carry over the proof of [51, Thm. 2.1] to our bigraded case. There it is shown that the chain map ζ′: 𝔹 → Σ̄^{n}𝔹 given by

1With another sign convention one obtains a different result: ζ ˜∪ η = (−1)^{(m+i)(n+j)}η ˜∪ ζ for ζ ∈ HH^{m,i}(Λ) and η ∈ HH^{n,j}(Λ). Here the multiplication ˜∪ arises by taking into account also the external degree of the bigraded elements in the Koszul sign rule (3.1) and thus in the cup product (5.4). However, starting with our sign convention, one can define a new multiplication by the rule η ∗ζ = (−1)^{in}ηζ. This new multiplication is then still associative, as is easily checked, and satisfies ζ ∗ η = (−1)^{(m+i)(n+j)}η ∗ ζ.

ζ′_{p}: B^{m+p} ⟶ B^{p}
(λ_{0}, ⋯, λ_{p+m+1}) ⟼ (−1)^{mp}(ζ(λ_{0}, ⋯, λ_{m}, 1), λ_{m+1}, ⋯, λ_{p+m+1})

is a lifting of ζ. Since all ζ′_{p} are homogeneous maps of degree i, we conclude that ζ′ is actually a graded lifting. Moreover, we have that

η ∗ ζ(λ_{0}, ⋯, λ_{m+n+1}) = ηζ_{n}(λ_{0}, ⋯, λ_{m+n+1})
= (−1)^{mn}η(ζ(λ_{0}, ⋯, λ_{m}, 1), λ_{m+1}, ⋯, λ_{n+m+1})
= (−1)^{mn}(−1)^{|ζ(λ_{0},··· ,λ_{m},1)|·j}ζ(λ_{0}, ⋯, λ_{m}, 1)η(1, λ_{n+1}, ⋯, λ_{n+m+1})
= (−1)^{mn}(−1)^{(i+|(λ_{0},··· ,λ_{m})|)·j}ζ(λ_{0}, ⋯, λ_{m}, 1)η(1, λ_{m+1}, ⋯, λ_{n+m+1})

On the other hand, it holds

(ζ ∪ η)(λ_{0}, ⋯, λ_{m+n+1}) = (−1)^{|(λ_{0},··· ,λ_{m})|·j}ζ(λ_{0}, ⋯ λ_{m}, 1)η(1, λ_{m+1}, ⋯ λ_{m+n+1}).

We infer

η ∗ ζ = (−1)^{mn}(−1)^{ij}ζ ∪ η,

and the claim follows since cup and Yoneda product coincide by Proposition 5.2. □ Remark 5.4. We may identify the graded ring HH^{0,∗}(Λ) with the graded centre of Λ, Z_{gr}(Λ) = {x ∈ Λ | xλ = (−1)^{|x||λ|}λx for all λ ∈ Λ}, by the evaluation map

HH^{0,∗}(Λ) → Z_{gr}(Λ), f ↦ f(1),

which is easily checked to be well-defined and bijective. 5.4. The cup product pairing. Let φ ∈ Hom_{Λ^{e}}(B_{s}, Λ) be a (s, t)-Hochschild cocycle

\xymatrix{𝔹: & ⋯ \ar[r] & B_{s} \ar[r] \ar[d]^{φ} & B_{s−1} \ar[r] & ⋯ \ar[r] & B_{0} \ar[r] & Λ \ar[r] & 0 \\  &  & Λ}

Tensoring φ over Λ with a homomorphism of graded right Λ-modules f: X → Y yields the map

f ⊗_{Λ} φ: X ⊗_{Λ} B_{s} → Y

which is homogeneous of degree t. The complex 𝔹 ⊗_{Λ} X is a projective resolution of X in Mod_{gr} Λ, and since this construction commutes with the differential, f ⊗_{Λ} φ is a (s, t)-cocycle:

\xymatrix{𝔹 ⊗_{Λ} X: & ⋯ \ar[r] & B_{s} ⊗_{Λ} X \ar[r] \ar[d]^{f⊗_{Λ}φ} & B_{s−1} ⊗_{Λ} X \ar[r] & ⋯ \ar[r] & B_{0} ⊗_{Λ} X \ar[r] & X \ar[r] & 0 \\  &  & Y}

The cohomology class f ∪ φ of f ⊗_{Λ} φ only depends on the cohomology class of the cocycle φ and hence, we obtain a well-defined map

(5.5)  ∪ : Hom_{Λ}(X, Y) ⊗_{Λ} HH^{s,t}(Λ) ⟶ Ext_{Λ}^{s,t}(X, Y),

called the cup product pairing.

6. Differential graded algebras and their derived categories

Differential graded algebras, or shortly, dg algebras were introduced by Cartan [14] in 1956. They arise as complexes with an additional multiplicative structure. For example, the endomorphism complex End(C) of a complex C carries a natural dg algebra structure. Derived categories of dg algebras or more generally, of dg categories, were first studied systematically by Bernhard Keller in ‘Deriving DG Categories’ [27].

6.1. Differential graded algebras and modules. A graded algebra over a commutative ring

A = ∐_{n∈ℤ}A^{n}

is called a differential graded algebra or dg algebra if it carries a differential d: A → A, i.e. a graded k-linear map of degree +1 with the property d^{2} = 0, which is required to satisfy the Leibniz rule

d(xy) = d(x)y + (−1)^{n}xd(y) for all x ∈ A^{n} and y ∈ A.

The cohomology of A is a graded associative algebra over k and denoted by H^{∗}A.

A dg A-module is a graded (right) A-module X endowed with a differential d: X → X satisfying the Leibniz rule

d(xy) = d(x)y + (−1)^{n}xd(y) for all x ∈ X^{n} and y ∈ A.

A morphism of dg A-modules is an A-linear map which is homogeneous of degree zero and commutes with the differential. We denote the category of dg A-modules by Mod_{dg} A.

A map f: X → Y of dg A-modules is null-homotopic if there is a graded A-linear map ρ: X → Y of degree −1 such that f = d_{Y} ^{∘} ρ + ρ^{∘} d_{X}. The null-homotopic maps form an ideal and the homotopy category K(A) is the quotient of Mod_{dg} A with respect to this ideal. The homotopy category carries a triangulated structure which is defined in the same way as for the homotopy category K(C) of an additive category C.

A map X → Y of dg A-modules is a quasi-isomorphism if it induces an isomorphism H^{n}X → H^{n}Y in each degree n ∈ ℤ. The derived category of the dg algebra A is the localisation of K(A) with respect to the class S of all quasi-isomorphisms,

D(A) = K(A)[S^{−1}].

Note that S is a multiplicative system and compatible with the triangulation. Therefore D(A) is triangulated and the localisation functor K(A) → D(A) is exact.

Remark 6.1. (1) Any graded algebra is a dg algebra A with differential zero. A nongraded algebra R can be viewed as a dg algebra A with A^{0} = R and A^{n} = 0, otherwise. In this case, Mod_{dg} A can be identified with the category of complexes of R-modules C(Mod R). Furthermore, K(A) identifies with K(Mod R), the homotopy category of C(Mod R), and D(A) with D(Mod R), the derived category of complexes of R-modules. In particular, the results we state in the following sections carry over to the derived category of a module category over a non-graded algebra.

(2) Let X, Y be complexes in some additive category C. Then the complex Hom_{C}(X, Y) is given by

∐_{n∈ℤ} ∏_{p∈ℤ}Hom_{C}(X^{p}, Y^{p+n}),

with differential

d^{n}(ϕ) = d_{Y} ^{∘} ϕ − (−1)^{n}ϕ^{∘} d_{X}

for ϕ = (ϕ^{p})_{p∈ℤ} ∈ Hom_{C}(X^{p}, Y^{p+n}). Note that

H^{n} Hom_{C}(X, Y) ≅ Hom_{K(C)}(X, Σ^{n}Y)

because Ker d^{n} identifies with Hom_{C(C)}(X, Σ^{n}Y) and Im d^{n−1} with the ideal of nullhomotopic maps X → Σ^{n}Y. The composition of graded maps yields a dg algebra structure for

End_{C}(X) = Hom_{C}(X, X),

and Hom_{C}(X, Y) is a dg module over End_{C}(X).

(3) If A is a dg algebra and X, Y dg A-modules, then the homomorphism complex Hom_{A}(X, Y) is defined in an analogous way:

Hom_{A}(X, Y) = ∐_{n∈ℤ}Hom_{A}^{n}(X, Y),

where Hom_{A}^{n}(X, Y) denotes the homogeneous graded A-linear maps X → Y rising the degree by n ∈ ℤ. The differential d^{n}: Hom_{A}^{n}(X, Y) → Hom_{A}^{n+1}(X, Y) is defined to be

d^{n}(f) = d_{Y} ∘ f − (−1)^{n}f ∘ d_{X}.

Also in this case, we have an isomorphism

H^{n} Hom_{A}(X, Y) ≅ Hom_{K(A)}(X, Σ^{n}Y).

The endomorphism ring End_{A}(X) = Hom_{A}(X, X) is a dg algebra and Hom_{A}(X, Y) a dg module over End_{A}(X) by composition of graded maps.

The following well-known lemma shows that the functor

D(A)(A, −)^{∗} = ∐_{i∈ℤ}D(A)(A, −)^{i}

is naturally isomorphic to the cohomology functor H^{∗}.

Lemma 6.2. Let A be a dg algebra and H^{∗}A its cohomology algebra. For any X ∈ D(A), the evaluation map

D(A)(A, X)^{∗} → H^{∗}X, f ↦ f(1),

is a natural isomorphism of graded H^{∗}A-modules, where D(A)(A, X)^{∗} becomes a graded H^{∗}A-module via the isomorphism D(A)(A, A)^{∗} ≅ H^{∗}A.

It follows in particular that D(A) is compactly generated by A, the free dg A-module of rank one.

J. Rickard [46] proved that the compact objects of D(Mod R), where R is a ring, are the perfect complexes. A complex of R-modules is called perfect if it is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. The full subcategory of perfect complexes of D(Mod R) is denoted by D^{per}(Mod R).

This characterisation of the compact objects was extended to the derived category D(A) of a dg algebra A by Neeman [41]. Here D^{per}(A) denotes the smallest thick subcategory of D(A) containing A.

Proposition 6.3. [41], [34, Ch. 5.5]. A dg module is compact in D(A) if and only if it is contained in D^{per}(A).

6.2. K(A) as stable category of a Frobenius category. Let A be a dg algebra over a commutative ring k. Then Mod_{dg} A is an exact category with respect to the exact sequences of dg A-modules

0 → X → Y → Z → 0

which are split considered as sequences of graded A-module maps. Furthermore, Mod_{dg} A is a Frobenius category (see Example 2.5(4) for a definition). The projective-injective objects are the dg A-modules M ⊕ M[1] with differential [ \begin{matrix}0 & id \\ 0 & 0 \end{matrix} ], where M ∈ Mod_{dg} A. Since the maps factoring through an injective object are precisely the nullhomotopic maps, the associated stable category coincides with the homotopy category K(A). We refer to [28, Sect. 8.2.3] and [27, Sect. 2.2] for more details.

Lemma 6.4. Let φ: X → Y be any morphism in K(A). Then φ can be represented by a morphism X → Ŷ in Mod_{dg} A which is a split monomorphism in the category of graded A-modules.

Proof. Since K(A) = S(Mod_{dg} A), we can choose a map of dg A-modules f: X → Y that induces φ in the stable category. Let i: X → I(X) be the injective hull and s: I(X) → X the graded A-linear map satisfying s ∘ i = id_{X}. We set Ŷ = Y ⊕ I(X). Then the map

[ \begin{matrix}f \\ i \end{matrix} ] : X → Y ⊕ I(X)

is clearly a monomorphism of dg A-modules that induces φ in the homotopy category, and the graded A-module map

[^{0} ^{s}] : Y ⊕ I(X) → X

satisfies [^{0} ^{s}] ∘ [ \begin{matrix}f \\ i \end{matrix} ] = id_{X} . □

6.3. Homotopically projective and homotopically injective dg modules. We use the terminology of Bernhard Keller, as presented in [28]. Throughout this section let A be a dg algebra over some commutative ring k. We say that a dg A-module X is homotopically projective if

K(X, Y) = 0

for all acyclic dg A-modules Y.

Dually, X is called homotopically injective if

K(Y, X) = 0

for all acyclic dg A-modules Y.

We denote by K_{p}(A) (resp. K_{i}(A)) the full subcategory of homotopically projective (resp. homotopically injective) dg A-modules of K(A).

Theorem 6.5. [28, Sect. 8.1.6] (1) For any dg A module X, there is a triangle

pX → X → aX → ΣpX

in K(A), where pX is homotopically projective and aX is acyclic. Any triangle Z → X → Y → ΣZ with homotopically projective Z and acyclic Y is isomorphic to (pX, X, aX), and there is a unique such isomorphism extending the identity of X.

(2) For any dg A module X, there is a triangle

a′X → X → iX → Σa′X

in K(A), where iX is homotopically injective and a′X is acyclic. Any triangle Z → X → Y → ΣZ with acyclic Z and homotopically injective Y is isomorphic to (a′X, X, iX), and there is a unique such isomorphism extending the identity of X.

In particular, each dg A module X is quasi-isomorphic to a homotopically projective (resp. homotopically injective) dg A-module and we call

pX → X resp. X → iX (6.1)

an homotopically projective resp. homotopically injective resolution. In particular, the assignments p and i can be shown to be functorial. Since they vanish on acyclic complexes, they extend to functors D(A) → K(A). In fact, we have

Theorem 6.6. [28, Sect. 8.2.6]

(1) The composition

K_{p}(A) ↪ K(A) ⟶^{can} D(A)

is an equivalence of triangulated categories with quasi-inverse given by

p: D(A) → K_{p}(A).

More precisely, p induces a fully faithful left adjoint to the quotient functor can: K(A) → D(A).

(2) The composition

K_{i}(A) ↪ K(A) ⟶^{can} D(A)

is an equivalence of triangulated categories with quasi-inverse given by

i: D(A) → K_{i}(A).

More precisely, i induces a fully faithful right adjoint to the quotient functor can: K(A) → D(A).

Corollary 6.7. For all dg A-modules X and Y, we have

K(A)(X, iY) ≅ D(A)(X, iY) ≅ D(A)(X, Y) ≅ D(A)(pX, Y) ≅ K(A)(pX, Y).

Remark 6.8. A dg A-module is homotopically projective if and only if it is chain homotopy equivalent to a cofibrant dg module (see [48, Rem. 3.17]). A dg A-module X is cofibrant if there exists an exhaustive increasing filtration by dg A-submodules

0 = X_{0} ⊆ X_{1} ⊆ ⋯ ⊆ X_{n} ⊆ ⋯

such that each subquotient X_{n+1}/X_{n} is a direct summand of shifted copies of A. The derived category D(A) can also be defined in the following way: The objects are the cofibrant modules, and the morphisms are chain homotopy classes of dg A-module morphisms.

6.4. Derived functors. Let A and B be two dg algebras over a commutative ring k. A dg (A, B)-bimodule is a graded (A, B)-bimodule which carries in addition a k-linear differential d of degree +1 satisfying

d(axb) = (da)xb + (−1)^{p}a(dx)b + (−1)^{p+q}ax(db)

for all a ∈ A^{p}, x ∈ X^{q}, b ∈ B.

Let M be any dg A-module. To define the tensor product M ⊗_{A} X of dg modules, we first observe that the tensor product M ⊗_{k} X is a dg B-module. As for graded rings, the degree n component is defined to be

(M ⊗_{k} X)^{n} = ∐_{p+q=n} M^{p} ⊗ X^{q}.

Additionally, we now have the differential

d(m ⊗ x) = (dm) ⊗ x + (−1)^{|m|}m ⊗ dx.

Since the k-submodule generated by all differences ma ⊗ x − m ⊗ ax is stable under both d and multiplication with elements of B, the quotient modulo this submodule is a well-defined dg B-module which we denote by M ⊗_{A} X. Moreover, this construction is functorial in M and X.

Let N be a dg B-module. Then Hom_{B}(X, N), as defined in Remark 6.1, is a right dg A-module by setting

(fa)(x) = f(ax).

Observe that − ⊗_{A} X and Hom_{B}(X, −) induce functors between K(A) and K(B) which form an adjoint pair

\xymatrix{K(A) \ar[r]_{−⊗_{A}X} & K(B) \ar[l]_{Hom_{B}(X,−)}}

We define the total left derived functor − ⊗_{A}^{L} X as the composition

D(A) ⟶^{p} K(A) ⟶^{−⊗_{A}X} K(B) ⟶^{can} D(B),

and the total right derived functor R Hom_{B}(X, −) as

D(B) ⟶^{i} K(B) ⟶^{Hom_{B}(X,−)} K(A) ⟶^{can} D(A).

Then the total derived functors also form an adjoint pair

\xymatrix{D(A) \ar[r]_{−⊗_{A}^{L}X} & D(B) \ar[l]_{R Hom_{B}(X,−)}} .

In particular, we deduce from Proposition 2.4 that − ⊗_{A}^{L} X preserves arbitrary direct sums and R Hom_{B}(X, −) preserves arbitrary direct products.

6.5. Cofibrant differential graded algebras. The category of dg algebras dga/k over a commutative ring k admits a model category structure [49]. A model category is a category with three distinguished classes of morphisms, the fibrations, cofibrations and weak equivalences. These are required to satisfy certain axioms. An object C in a model category is called cofibrant if the morphism 0 → C is a cofibration. We refer to [24, Ch. 1.1] for details.

In the category dga/k, the fibrations are the degree-wise surjective dg algebra morphisms and the weak equivalences equal the quasi-isomorphisms. A dg algebra is called cofibrant if it is a cofibrant object in the model category dga/k, that is:

Definition 6.9. A dg algebra A is cofibrant if for any morphism of dg algebras f: A → C and every surjective quasi-isomorphism of dg algebras g: B → C, there exists a lift h: A → B. That is, we have a commutative diagram

\xymatrix{ & B \ar[d] \ar@{->>}[d] \ar[d]^{g}_{∼} \\ A \ar[r]^{f} \ar@{.>}[ur]^{h} & C}

A direct consequence of the model category axioms for dga/k is

Lemma 6.10. [24, Ch. 1.1] If A is any dg algebra over a commutative ring k, then there exists a cofibrant dg algebra A^{cof} and a quasi-isomorphism

\xymatrix{A^{cof} \ar[r]^{∼} & A.}

We remark that from the model theory axioms, it follows moreover that the quasiisomorphism A^{cof} ⟶^{∼} A above is a surjective map [24, Ch. 1.1].

Examples for cofibrant dg algebras are the Sullivan algebras [19, Ch. 12] which we define in the following. From now on we assume that k is a field of characteristic zero. We recall the definition of the free graded-commutative algebra:

Let V be a graded vector space over k. The elements v ⊗ w − (−1)^{|v||w|}w ⊗ v generate an ideal I in the tensor algebra T V. The free graded-commutative algebra ΛV is quotient of the Tensor algebra T V by the ideal I,

ΛV = T V/I.

If v_{1}, ⋯, v_{n} is a k-basis of V, one also writes Λ(v_{1}, ⋯, v_{n}) for ΛV.

Definition 6.11. A Sullivan algebra is a dg algebra of the form (ΛV, d), where (1) V = ∐_{p≥1} V_{p} is a positively graded vector space

(2) V = ⋃_{l≥0} V (l), where V (0) ⊆ V (1) ⊆ ⋯ is an increasing sequence of graded subspaces such that

d = 0 on V (0) and d(V (l)) ⊆ ΛV (l − 1) for all l ≥ 1.

Lemma 6.12. [19, Lemma 12.4] Every Sullivan algebra (ΛV, d) is a cofibrant dg algebra.

For an arbitrary dg algebra A, the quasi-isomorphism A^{cof} ⟶^{∼} A which exists by Lemma 6.10 is not easy to compute. However, for a certain class of dg algebras, one can construct explicitly quasi-isomorphic Sullivan algebras:

Proposition 6.13. [19, Prop. 12.1] Assume that A is a graded-commutative dg algebra concentrated in non-negative degrees which satisfies H^{0}(A) = k. Then there exists a Sullivan algebra (ΛV, d) and a quasi-isomorphism

\xymatrix{(ΛV, d) \ar[r]^{∼} & A.}

Example 6.14. [19, Ch. 12, Exm. 4] Not every dg algebra of the form (ΛV, d) is a Sullivan algebra: Consider (Λ(v_{1}, v_{2}, v_{3}), d), where |v_{i}| = 1, and the differential is given by dv_{1} = v_{2}v_{3}, dv_{2} = v_{3}v_{1}, and dv_{3} = v_{1}v_{2}. This dg algebra is not a Sullivan algebra. However, we can state a Sullivan algebra which is quasi-isomorphic to (Λ(v_{1}, v_{2}, v_{3}), d): there is a quasi-isomorphism

\xymatrix{σ: (Λ(w), 0) \ar[r]^{∼} & (Λ(v_{1}, v_{2}, v_{3}), d),}

where w is of degree 3; the map σ is given by σ(w) = v_{1}v_{2}v_{3}.

7. A_{∞}-algebras

A_{∞}-algebras are generalisations of dg algebras. They were invented by J. Stasheff at the beginning of the 1960s as a tool in the study of ‘group-like’ topological spaces. In the 1990s, the relevance of A_{∞}-algebras in algebra became more and more apparent. We focus on a result of Kadeishvili stating that the cohomology of a dg algebra is an A_{∞}-algebra. Instead of Kadeishvili’s Russian original paper [26] we refer the reader to the articles [29] and [30] by Bernhard Keller.

Throughout this chapter let k be a field and write shortly ⊗ for ⊗_{k}.

Definition 7.1. An A_{∞}-algebra is a ℤ-graded vector space

A = ∐_{p∈ℤ}A_{p}

together with a family of homogeneous k-linear maps

m_{n}: A^{⊗n} → A, n ≥ 1,

of degree 2 − n satisfying the relations (i) m_{1}m_{1} = 0. (ii) m_{1}m_{2} = m_{2}(m_{1} ⊗ 1 + 1 ⊗ m_{1}). (iii) More generally, for all n ≥ 1,

∑(−1)^{r+st}m_{u}(id^{⊗r} ⊗m_{s} ⊗ id^{⊗t}) = 0,

where the sum runs over all decompositions n = r+s+t, and we set u = r+1+t.

Note that (A, m_{1}) is a differential complex due to (i). Condition (ii) means that m_{1} is a graded derivation with respect to the multiplication m_{2}, and equation (iii) with n = 3 shows that the multiplication m_{2} is associative only up to homotopy. The map m_{3} is called the secondary multiplication.

Remark 7.2. (1) In general, an A_{∞}-algebra is not associative. However, its cohomology H^{∗}A with respect to the differential m_{1} is an associative ℤ-graded algebra with the multiplication induced by m_{2}.

(2) If A is concentrated in degree zero, then A = A_{0} is just an associative algebra. That is because m_{n} is of degree 2 − n and consequently, all m_{n} other than m_{2} have to vanish.

(3) If m_{n} is trivial for all n ≥ 3, then A is a dg algebra. Conversely, each dg algebra carries an A_{∞}-structure with m_{1} the differential, m_{2} the multiplication, and all other m_{n} trivial.

A morphism between two A_{∞}-algebras A and B is in general not just a map A → B, but something quite more complicated:

Definition 7.3. A morphism of A_{∞}-algebras f: A → B is a family of graded maps

f_{n}: A^{⊗n} → B

of degree 1 − n such that (i) f_{1}m_{1} = m_{1}f_{1}, i.e. f_{1} : A → B is a chain map. (ii) f_{1}m_{2} = m_{2}(f_{1} ⊗ f_{1}) + m_{1}f_{2} + f_{2}(m_{1} ⊗ id + id ⊗m_{1}). (iii) More generally, for n ≥ 1, we have

∑(−1)^{r+st}f_{u}(id^{⊗r} ⊗m_{s} ⊗ id^{⊗t} = ∑(−1)^{s}m_{r}(f_{i_{1}} ⊗ f_{i_{2}} ⊗ ⋯ f_{i_{r}}),

where the first sum runs over all decompositions n = r + s + t, and we set u = r + 1 + t. The second sum runs over 1 ≤ r ≤ n and all decompositions n = i_{1} + ⋯ + i_{r}. Furthermore, the sign on the right hand side is given by

s = (r − 1)(i_{1} − 1) + (r − 2)(i_{2} − 1) + ⋯ + 2(i_{r−2} − 1) + (i_{r−1} − 1).

Note that equation (ii) means that f_{1} commutes with the multiplication m_{2} up to a homotopy given by f_{2}.

An A_{∞}-morphism f: A → B is • a quasi-isomorphism if the chain map f_{1} is a quasi-isomorphism,

• strict if f_{i} = 0 for all i ≠ 1,

• the identity morphism if f: A → A is strict with f_{1} = id_{A}. The composition of two A_{∞}-morphisms g: A → B and h: B → C is defined as

(h ∘ g)_{n} = ∑(−1)^{s}h_{r} ∘ (g_{i_{1}} ⊗ ⋯ ⊗ g_{i_{r}}),

where the sum and the sign are as in Definition 7.3 (iii).

Theorem 7.4 (Kadeishvili [26], see also [30]). Let A be an A_{∞}-algebra. Then the cohomology H^{∗}A has an A_{∞}-algebra structure such that

1) m_{1}^{H^{∗}A} = 0 and m_{2}^{H^{∗}A} is induced by m_{2}^{A}, and

2) there is a quasi-isomorphism of A_{∞}-algebras f: H^{∗}A → A lifting the identity in

cohomology, i.e. H^{∗}f_{1} = id_{H^{∗}A}. Moreover, this structure is unique up to (non unique) isomorphism of A_{∞}-algebras.

In particular, the cohomology H^{∗}A of a dg algebra A is an A_{∞}-algebra. We now show how to construct the secondary multiplication m_{3}^{H^{∗}A} of H^{∗}A and at the same time, the first three terms of the quasi-isomorphism f: H^{∗}A → A lifting the identity of H^{∗}A:

Construction 7.1. Let A be a dg algebra with differential m_{1}^{A} and multiplication m_{2}^{A}. We view H^{∗}A as a complex with zero differential. Since we are working over a field, we can choose a quasi-isomorphism f_{1} : H^{∗}A → A inducing the identity in cohomology.

This amounts to choosing a representative cocycle for each cohomology class, in a linear way. Note that f_{1} cannot be chosen to be multiplicative, but it does commute with multiplication up to coboundaries. So we can choose a k-linear map of degree −1,

f_{2} : H^{∗}A ⊗ H^{∗}A → A,

satisfying

(7.2)  m_{1}^{A}f_{2}(x, y) = f_{1}(xy) − f_{1}(x)f_{1}(y).

So it holds indeed

(7.3)  f_{1}m_{2}^{H^{∗}A} = m_{2}^{A}(f_{1} ⊗ f_{1}) + m_{1}^{A}f_{2}.

Now we look for f_{3} and m_{3} such that

f_{1} ∘ m_{3}^{H^{∗}A} + f_{2} ∘ (m_{2}^{H^{∗}A} ⊗ id − id ⊗m_{2}^{H^{∗}A})

+ f_{3} ∘ (m_{1}^{H^{∗}A} ⊗ id^{⊗2} + id ⊗m_{1}^{H^{∗}A} ⊗ id + id^{⊗2} ⊗m_{1}^{H^{∗}A})
= m_{3}^{A} ∘ (f_{1} ⊗ f_{1} ⊗ f_{1}) + m_{2}^{A} ∘ (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) + m_{1}^{A}f_{3}.

Since m_{1}^{H^{∗}A} = 0, this simplifies into

(7.4) f_{1} ∘ m_{3}^{H^{∗}A} = m_{2}^{A} ∘ (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) − f_{2} ∘ (m_{2}^{H^{∗}A} ⊗ id − id ⊗m_{2}^{H^{∗}A}) + m_{1}^{A} ∘ f_{3}.

Now one checks that the map

(7.5)  Φ_{3} = m_{2}^{A} ∘ (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) − f_{2} ∘ (m_{2}^{H^{∗}A} ⊗ id − id ⊗m_{2}^{H^{∗}A})

has its image in the cycles Z^{∗}A of A. So we define

(7.6)  m_{3}^{H^{∗}A} = π ∘ Φ_{3},

where π denotes the quotient map Z^{∗}A → H^{∗}A. Then

f_{1} ∘ m_{3}^{H^{∗}A} − Φ_{3} = (f_{1} ∘ π − id)Φ_{3}

has its image in the coboundaries and thus we can indeed choose a k-linear map

f_{3} : H^{∗}A^{⊗3} → A

of degree −2 such that

f_{1} ∘ m_{3}^{H^{∗}A} − Φ_{3} = m_{1}^{A} ∘ f_{3}

as desired.

This construction depends on some choices and the secondary multiplication is not uniquely determined. However, it determines a Hochschild class which is independent of all choices:

Proposition 7.5. [5, Prop. 5.4] Let A be a dg algebra over a field k. Then the secondary multiplication m_{3}^{H^{∗}A} of the A_{∞}-algebra H^{∗}A is a (3, −1)-Hochschild cocycle. Moreover, its Hochschild class is independent of all choices in defining the maps f_{1} and f_{2}.

The Hochschild class of any choice of m_{3}^{H^{∗}A} is denoted by μ_{A} ∈ HH^{3,−1}(H^{∗}A). We are particularly interested in this Hochschild class since it determines a global obstruction for realisability ([5], see Section 10.3). Because of this special property it is also referred to as canonical class. In Chapter 13.2 we will compute the secondary multiplication and its Hochschild class in some examples.

The following proposition has applications in the Chapters 12 and 13.

Proposition 7.6. [5, Cor. 5.7] Let α: A → B be a morphism of dg algebras and H^{∗}α: H^{∗}A → H^{∗}B the induced morphism in cohomology.

(1) In the Hochschild group HH^{3,−1}(H^{∗}A, H^{∗}B), it holds

H^{∗}α ∘ μ_{A} = μ_{B} ∘ (H^{∗}α)^{⊗3},

where H^{∗}B is a (H^{∗}A, H^{∗}A)-bimodule through H^{∗}α.

(2) If α is a quasi-isomorphism, then the class μ_{A} is mapped to μ_{B} under the induced isomorphism between the Hochschild cohomology of H^{∗}A and H^{∗}B.

Hence for any choice of secondary multiplications m_{3}^{H^{∗}A} and m_{3}^{H^{∗}B}, the difference

H^{∗}α ∘ m_{3}^{H^{∗}A} − m_{3}^{H^{∗}B} ∘ (H^{∗}α)^{⊗3}

is a (3, −1)-Hochschild coboundary. If we assume in addition that the algebra map H^{∗}α: H^{∗}A → H^{∗}B is a monomorphism, we can obtain equality of H^{∗}α ∘ m_{3}^{H^{∗}A} and m_{3}^{H^{∗}B} ∘ (H^{∗}α)^{⊗3} not only on the level of Hochschild classes, but even on the level of k-linear maps:

Proposition 7.7. Let α: A → B be a morphism of dg algebras and assume that H^{∗}α: H^{∗}A → H^{∗}B is a monomorphism. Given the choices in defining m_{3}^{H^{∗}A}, we can define m_{3}^{H^{∗}B} such that

H^{∗}α ∘ m_{3}^{H^{∗}A} = m_{3}^{H^{∗}B} ∘ (H^{∗}α)^{⊗3} in Hom_{k}^{−1}(H^{∗}A^{⊗3}, H^{∗}B).

Proof. Let f_{1}^{A}: H^{∗}A → A and f_{2} : H^{∗}A ⊗ H^{∗}A → A be any choices of the first two components of a quasi-isomorphism f: H^{∗}A → A lifting the identity. We define a graded degree zero map f_{1}^{B}: H^{∗}B → B by

f_{1}^{B} = α ∘ f_{1}^{A} ∘ (H^{∗}α)^{−1}

on the image of the monomorphism H^{∗}α and extend this map k-linearly to a graded map inducing the identity in cohomology. This is indeed possible because we are working over a field and since

H^{∗}f_{1}^{B}(H^{∗}α(x)) = H^{∗}α ∘ f_{1}^{A}(x) = H^{∗}α(x).

Now we define f_{2}^{B}. On the image of H^{∗}α ⊗ H^{∗}α we set

f_{2}^{B} = α ∘ f_{2} ∘ (H^{∗}α ⊗ H^{∗}α)^{−1}.

We then extend f_{2}^{B} k-linearly to a degree −1 map satisfying

df_{2}^{B}(x, y) = f_{2}^{B}(x)f_{2}^{B}(y) − f_{2}^{B}(xy)

for all x, y ∈ H^{∗}B. Our choices for f_{1}^{B} and f_{2}^{B} then automatically yield

H^{∗}α ∘ m_{3}^{H^{∗}A} = m_{3}^{H^{∗}B} ∘ (H^{∗}α)^{⊗3}.  □

Remark 7.8. [5, Exm. 7.7] Let A and B dg algebras over a field k and m_{3}^{H^{∗}A} resp. m_{3}^{H^{∗}B} secondary multiplications of their cohomology. Then under the Künneth isomorphism H^{∗}(A ⊗_{k} B) ≅ H^{∗}A ⊗_{k} H^{∗}B, the canonical class μ_{A⊗B} is represented by the cocycle

m(x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}, x_{3} ⊗ y_{3}) =

(−1)^{|x_{3}||y_{1}|+|x_{3}||y_{2}|+|x_{2}||y_{1}|}m_{3}^{H^{∗}A}(x_{1}, x_{2}, x_{3}) ⊗ y_{1}y_{2}y_{3} + x_{1}x_{2}x_{3} ⊗ m_{3}^{H^{∗}B}(y_{1}, y_{2}, y_{3}).

8. Localisation in triangulated categories

The classical localisation S^{−1}R of a commutative ring R with respect to a multiplicatively closed subset S ⊆ R gives rise to the functor − ⊗_{R} S^{−1}R: Mod R → Mod S^{−1}R. It assigns to an R-module M the S^{−1}R-module S^{−1}M, whose elements are fractions \frac{m}{s}, m ∈ M, s ∈ S. The tensor functor −⊗_{R} S^{−1}R is right adjoint to Hom_{S^{−1}R}(S^{−1}R, −), and it is well-known that the latter functor is fully faithful.

This calculus of fractions has been generalised by Gabriel and Zisman [20] to arbitrary categories. In his thèse [54], Verdier applied this to introduce localisation of triangulated categories. In particular, he invented the Verdier quotient which is a quotient category T /B of a triangulated category T by a triangulated subcategory B.

In the first two sections we recall categories of fractions and localisation functors for arbitrary categories. Localisation of triangulated categories, in particular the Verdier quotient, will be introduced in Section 8.3. Localisation functors of triangulated categories give rise to localisation sequences, which we define in Section 8.4, and those localisation sequences which are at the same time co-localisation sequences, the recollements, are considered in Section 8.5. In the last section of this chapter we study cohomological localisations. These localisations are a key tool to prove our results stated in the Chapters 9 and 11.

8.1. Categories of fractions. A functor F: C → D is said to invert a morphism σ of C if F σ is invertible. For a category C and any class of morphisms Σ of C there exists (after taking the necessary set-theoretic precautions) the category of fractions C[Σ^{−1}], together with a canonical functor

Q_{Σ} : C → C[Σ^{−1}]

having the following properties:

(Q1) Q_{Σ} makes the morphisms in Σ invertible.

(Q2) If a functor F: C → D makes all morphisms in Σ invertible, then there is a unique functor G: C[Σ^{−1}] → D such that F = G ∘ Q_{Σ}.

An explicit construction of the category C[Σ^{−1}] can be found in the book of Gabriel and Zisman [20].

Let C, D be categories and

\xymatrix{C \ar[r]_{F} & D \ar[l]_{G}}

an adjoint pair of functors, that is, F: C → D and G: D → C are a pair of functors such that G is right adjoint to F. By η: id_{C} → G ∘ F we denote the unit, and by ε: F ∘ G → id_{D} the counit of the adjunction.

Lemma 8.1. [20, Ch. I, Prop. 1.3] Let Σ be the set of morphisms σ of C such that F σ is invertible. The following are equivalent:

(1) The functor G is fully faithful.

(2) The counit of the adjunction ε: F ∘ G → id_{D} is invertible.

(3) The functor F̄: C[Σ^{−1}] → D satisfying F = F̄ ∘ Q_{Σ} is an equivalence.

8.2. Localisation functors. Let F: C → D and G: D → C be an adjoint pair of functors satisfying the equivalent conditions of Lemma 8.1, and set L = G ∘ F. The following well-known lemma shows that the pair (F, G) can be reconstructed from L and the adjunction unit Ψ: id_{C} → L.

Lemma 8.2. Let L: C → C be a functor and Ψ: id_{C} → L a natural transformation. The following are equivalent:

(1) The map LΨ: L → L^{2} is invertible and LΨ = ΨL.

(2) There exists a pair of functors F: C → D and G: D → C such that F is left adjoint to G, G is fully faithful, L = G ∘ F, and Ψ: id_{C} → G ∘ F is the unit of the adjunction.

For a proof we refer to [36]. However, we sketch how the pair (F, G) can be constructed from the functor L: Given L: C → C, we define D to be the full subcategory of C formed by those objects X such that ΨX: X → LX is invertible. The functor F is given by F: C → D, F X = LX, and G: D → C is defined to be the inclusion. Note also that Ψ equals the adjunction unit η: id_{C} → G ∘ F.

Definition 8.3. Let C be an additive category.

(1) We call a pair (L: C → C, Ψ: id_{C} → L) a localisation functor if it satisfies the conditions of Lemma 8.2.

(2) An object X ∈ C is called L-acyclic if L(X) = 0, and the full subcategory of L-acyclic objects is denoted by Ker L.

(3) An object X ∈ C is called L-local if X ≅ LX′ for some X′ ∈ C. The full subcategory of L-local objects is denoted by C_{L}.

Remark that an object X ∈ C is L-local if and only if ΨX is invertible, see [36, Lemma 1.5]. Thus the category D constructed in Lemma 8.2 equals C_{L}.

Justified by Lemma 8.2, a functor F: C → D admitting a fully faithful right adjoint G is also called localisation functor.

8.3. Quotient categories. If the category C admits an abelian (resp. triangulated) structure and B is a Serre (resp. triangulated) subcategory, then we can form a quotient category by inverting a special class of morphisms. We present this construction for triangulated categories.

Let C be triangulated and B a triangulated subcategory. We define Σ_{B} to be the class of all morphisms σ ∈ C such that there exists an exact triangle

X → Y ⟶^{σ} Z → X[1],

with X ∈ B.

The category of fractions C[Σ_{B}^{−1}] is called the Verdier Quotient and denoted by C/B. The following properties of the Verdier Quotient are well-known.

Lemma 8.4. Let Q_{B} denote the canonical functor Q_{Σ_{B}}: C → C/B.

(1) C/B carries a unique triangulated structure such that the functor Q_{B}: C → C/B is exact.

(2) The kernel of Q_{B} consists of the thick closure of B, i.e. those objects X ∈ C such that there exists Y ∈ C with X ⨿ Y ∈ B.

(3) For any exact functor F: C → D satisfying F (B) = 0, there exists a unique functor F̄: C/B → D such that F = F̄ ∘ Q_{B}.

A pair (L: T → T , Ψ: id_{C} → L) is a localisation functor of triangulated categories if L is an exact localisation functor and Ψ commutes with the suspension functor in the sense that Ψ ∘ [1]_{C} ≅ [1]_{C} ∘ Ψ.

Then Ker L and T_{L} are triangulated subcategories of T . Moreover, the functor T → T_{L}, X ↦ LX, is exact and induces an equivalence

T / Ker L ⟶^{≃} T_{L}.

It follows from Lemma 8.2 that we have a bijection between localisation functors (L: T → T , Ψ: id_{T} → L) and quotient functors Q: T → T /B having a fully faithful right adjoint R (which are also called localisation functors). Observe that the adjunction unit η: id_{T} → RQ satisfies η ∘ [1]_{T} ≅ [1]_{T} ∘ η because R is fully faithful. Definition 8.5. A localisation functor (L: T → T , Ψ: id_{C} → L) is called smashing if L commutes with arbitrary direct sums.

A localising subcategory B of T is called smashing if Q: T → T /B admits a fully faithful right adjoint R which commutes with arbitrary direct sums. Then the quotient functor Q: T → T /B is also called smashing localisation.

Note that the composition R ∘ Q commutes with arbitrary direct sums if and only if R does. Hence we have a bijection between the smashing localisation functors in the two different senses.

8.4. Localisation sequences. A sequence of exact functors

A ⟶^{F} B ⟶^{G} C

between triangulated categories is called localisation sequence if the following conditions hold:

(L1) The functor F has a right adjoint F_{ρ}: B → A satisfying F_{ρ} ∘ F ≅ id_{A}.

(L2) The functor G has a right adjoint G_{ρ}: C → B satisfying G ∘ G_{ρ} ≅ id_{C}, i.e. G is

a localisation functor. (L3) Let X be an object in B. Then GX = 0 if and only if X ≅ F X′ for some

X′ ∈ A′. The sequence (F, G) is called colocalisation sequence if the sequence (F^{op}, G^{op}) of opposite functors is a localisation sequence.

We recall the basic properties of a localisation sequence

Lemma 8.6 (Verdier [54], see also [35]). Let A ⟶^{F} B ⟶^{G} C be a localisation sequence. Identify A = Im F and C = Im G_{ρ}.

(1) The functors F and G_{ρ} are fully faithful.

(2) For given objects X, Y ∈ B, we have

X ∈ A ⟺ Hom_{B}(X, C) = 0,
Y ∈ C ⟺ Hom_{B}(A, Y) = 0.

(3) The functor G induces an equivalence B/A ≃ C. Hence every triangulated functor G′: B → C′ satisfying G′ ∘ F = 0 factors over G.

(4) For each X ∈ T there is an exact triangle

(F ∘ F_{ρ})(X) → X → (G_{ρ} ∘ G)X → Σ((F ∘ F_{ρ})X)

which is functorial in X.

(5) The sequence

C ⟶^{G_{ρ}} B ⟶^{F_{ρ}} A

is a colocalisation sequence.

8.5. Recollements. We say that a sequence

A ⟶^{F} B ⟶^{G} C

of exact functors between triangulated categories induces a recollement

\xymatrix{A \ar[r] & B \ar[l]^{F_{λ}} \ar[l]_{F_{ρ}} \ar[r] & C \ar[l]^{G_{λ}} \ar[l]_{G_{ρ}}}

if it is at the same time a localisation and a colocalisation sequence.

This means that the functors F and G admit left adjoints F_{λ} and G_{λ} as well as right adjoints F_{ρ} and G_{ρ} such that the adjunction morphisms

F_{λ} ∘ F ⟶^{≅} id_{A} ⟶^{≅} F_{ρ} ∘ F and G ∘ G_{ρ} ⟶^{≅} id_{B} ⟶^{≅} G ∘ G_{λ}

are isomorphisms.

Let Λ be a Noetherian ring. We denote by

I: K_{ac}(Inj Λ) → K(Inj Λ)

the inclusion functor, and by Q the canonical functor given by the composition

K(Inj Λ) ⟶^{inc} K(Mod Λ) ⟶^{can} D(Mod Λ).

Theorem 8.7. [35, Cor. 4.3] The sequence

K_{ac}(Inj Λ) ⟶^{I} K(Inj Λ) ⟶^{Q} D(Mod Λ)

induces a recollement

(8.1)  \xymatrix{K_{ac}(Inj Λ) \ar[r] & K(Inj Λ) \ar[l]_{I_{ρ}} \ar[l]^{I_{λ}} \ar[r] & D(Mod Λ). \ar[l]^{Q_{λ}} \ar[l]_{Q_{ρ}}}

Remark 8.8. [6, Sect. 5] The recollement (8.1) provides two embeddings of D(Mod Λ) into K(Inj kG): The fully faithful functor Q_{ρ} assigns to X ∈ D(Mod Λ) its homotopically injective resolution iX introduced in Chapter 6.3. The other embedding is given by the functor Q_{λ}, which identifies D(Mod Λ) with the localising subcategory of K(Inj kG) generated by iΛ.

Assume now that Λ is self-injective. Then Q_{λ} maps a complex X of Λ-modules to its homotopically projective resolution pX. Furthermore, for every Λ-module M, the canonical triangle

(8.2)  pM → iM → tM → Σ(pM)

is isomorphic to the triangle

(8.3)  (Q_{λ} ∘ Q)(M̄) → M̄ → (I ∘ I_{λ})M̄ → Σ(Q_{λ} ∘ Q)M̄,

where M̄ = Q_{ρ}M.

Krause proved Theorem 8.7 more generally for a locally Noetherian Grothendieck category A such that D(A) is compactly generated. In order to prove that (I, Q) induces a localisation sequence, the essential point is the following proposition which is also stated more generally in [35].

Proposition 8.9. [35, Prop. 2.3] The triangulated category K(Inj Λ) is compactly generated by the injective resolutions iM of the Noetherian modules M. Moreover, the canonical functor K(Mod Λ) → D(Mod Λ) induces an equivalence

K^{c}(Inj Λ) ⟶^{≃} D^{b}(mod Λ),

where K^{c}(Inj Λ) denotes the full subcategory of compact objects in K(Inj Λ).

The following lemma is well-known and easy to check.

Lemma 8.10. Let Λ be a Noetherian self-injective ring. The functor

Z^{0} : K_{ac}(Inj Λ) → \underline{Mod} Λ

is an equivalence with quasi-inverse M ↦ tM, where tM denotes a Tate resolution of any representative of M ∈ \underline{Mod} Λ.

Over a finite dimensional cocommutative Hopf algebra, the adjoints in the recollement can be written down explicitly:

Proposition 8.11. [6] Let H be a finite dimensional cocommutative Hopf algebra. Then the adjoints in the recollement (8.1) take the form

\xymatrix{\underline{Mod} H ≃ K_{ac}(Inj H) \ar[r] & K(Inj H) \ar[l]^{−⊗_{k}tk} \ar[l]_{Hom_{k}(tk,−)} \ar[r] & D(Mod H). \ar[l]_{Hom_{k}(pk,−)} \ar[l]^{−⊗_{k}pk}}

In this case, the triangles (8.2) and (8.3) are isomorphic to

(8.4)  M ⊗_{k} pk → M ⊗_{k} ik → M ⊗_{k} tk → Σ(M ⊗_{k} pk).

8.6. Cohomological localisation. Cohomological localisations were first studied by Bousfield [7]. Hovey, Palmieri and Strickland [25, Thm. 3.3.7] applied them in the context of axiomatic stable homotopy theory. We refer to a paper of Krause [36] for a more algebraic and detailed approach.

Throughout this section let T be a triangulated category which admits arbitrary direct sums and is generated by a set of compact elements. We fix a compact object A ∈ T and write H^{∗} for the functor T (A, −)^{∗}.

Theorem 8.12. [36] Every exact localisation functor (L, Ψ) on Mod_{gr} Γ extends to an exact localisation functor (L̂, Ψ̂) on T such that the diagram

\xymatrix{T \ar[d]_{L̂} \ar[r]^{H^{∗}} & Mod_{gr} Γ \ar[d]^{L} \\ T \ar[r]^{H^{∗}} & Mod_{gr} Γ}

commutes up to a natural isomorphism. More precisely, it holds (1) The morphisms LH^{∗}Ψ̂, ΨH^{∗}L̂ and

\xymatrix{LH^{∗} \ar[r]^{LH^{∗}Ψ̂} & LH^{∗}L̂ \ar[r]^{(ΨH^{∗}L̂)^{−1}} & H^{∗}L̂}

are invertible.

(2) An object X in T is L̂-acyclic if and only if H^{∗}X is L-acyclic.

(3) If X ∈ T is L̂-local, then H^{∗}X is L-local.

If A is a generator of T and L preserves arbitrary direct sums, then L̂ preserves arbitrary direct sums.

Remark 8.13. [36, Rem. 2.4] Let L: Mod_{gr} Γ → Mod_{gr} Γ be an exact localisation functor and denote by L̂: T → T the exact localisation functor which exists by Theorem 8.12. Write C for the L̂-acyclic objects. By Lemma 8.2, L̂ and L give rise to adjoint pairs of functors

\xymatrix{T \ar[r]_{Q} & T /C \ar[l]_{R} & and & Mod_{gr} Γ \ar[r]_{F} & (Mod_{gr} Γ)_{L} \ar[l]_{G}}

satisfying L̂ = R ∘ Q and L = G ∘ F. The diagram below commutes up to natural isomorphism.

\xymatrix{T \ar[d]_{Q} \ar[r]^{T (A,−)^{∗}} & Mod_{gr} Γ \ar[d]^{F} \\ T /C \ar[d]_{R} \ar[r]^{T /C(QA,−)^{∗}} & (Mod_{gr} Γ)_{L} \ar[d]^{G} \\ T \ar[r]^{T (A,−)^{∗}} & Mod_{gr} Γ}

Now assume that Γ is graded-commutative and consider the localisation functor L: Mod_{gr} Γ → Mod_{gr} Γ given by localisation with respect to a multiplicatively closed subset S of Γ. See Chapter 3.1.2 for localisation of graded-commutative rings. The following results in this section are joint work with K. Brüning [8].

Proposition 8.14. Suppose that the ring T (A, A)^{∗} is graded-commutative and let L: Mod_{gr} T (A, A)^{∗} → Mod_{gr} T (A, A)^{∗} be localisation with respect to a multiplicatively closed subset of homogeneous elements S ⊆ T (A, A)^{∗}. Denoting C = Ker L̂, the diagram

\xymatrix{T \ar[d]_{Q} \ar[r]^{T (A,−)^{∗}} & Mod_{gr} T (A, A)^{∗} \ar[d]^{can} \\ T /C \ar[r]^{T /C(QA,−)^{∗}} & Mod_{gr} S^{−1}T (A, A)^{∗}}

commutes up to natural isomorphism. Furthermore, T /C(QA, QA)^{∗} and S^{−1}T (A, A)^{∗} are isomorphic not only as graded T (A, A)^{∗}-modules, but also as graded rings.

Proof. The diagram commutes by Remark 8.13. Writing again H^{∗} for T (A, −)^{∗}, the naturality of H^{∗}Ψ yields a commutative square

\xymatrix{H^{∗}A \ar[r]^{ΨH^{∗}A} \ar[d]_{H^{∗}Ψ̂A} & LH^{∗}A \ar[d]^{LH^{∗}Ψ̂A}_{≅} \\ H^{∗}L̂A \ar[r]^{ΨH^{∗}L̂A}_{≅} & LH^{∗}L̂A}

in which the lower and the right hand side morphism are bijective by Theorem 8.12. Now note that H^{∗}Ψ̂A is up to isomorphism given by the canonical map

Q: T (A, A)^{∗} → T /C(QA, QA)^{∗}, f ↦ Qf,

and that ΨH^{∗}A equals up to isomorphism the canonical ring homomorphism

can: T (A, A)^{∗} → S^{−1}T (A, A)^{∗}.

Since Q: T /(A, A)^{∗} → T /C(QA, QA)^{∗} is a multiplicative map inverting all elements in S, we obtain a ring homomorphism r: S^{−1}T (A, A)^{∗} → T /C(QA, QA)^{∗} which makes the upper triangle in the modified diagram

\xymatrix{T (A, A)^{∗} \ar[r]^{can} \ar[d]_{Q} & S^{−1}T (A, A)^{∗} \ar[d]^{S^{−1}Q}_{≅} \ar@{.>}[dl]_{r} \\ T /C(QA, QA)^{∗} \ar[r]^{ν}_{≅} & S^{−1}T /C(QA, QA)^{∗}}

commute. We now show that r is bijective by proving the commutativity of the lower triangle.

Since both the maps ν ∘r and S^{−1}Q make the following diagram of T (A, A)^{∗}-modules

\xymatrix{T (A, A)^{∗} \ar[r]^{μ} \ar[d]_{ν∘Q} & S^{−1}T (A, A)^{∗} \ar[dl]_{ν∘r} \ar[dl]^{S^{−1}Q} \\ S^{−1}T /C(QA, QA)^{∗}}

commute, the universal property of localisation of modules yields ν ∘ r = S^{−1}Q and hence the claim. □

Proposition 8.15. Suppose that the ring T (A, A)^{∗} is graded-commutative and let L: Mod_{gr} T (A, A)^{∗} → Mod_{gr} T^{∗}(A, A)^{∗} be localisation with respect to a multiplicatively closed subset of homogeneous elements S ⊆ T (A, A)^{∗}. If the compact object A ∈ T is a generator, then the category C = Ker L̂ is generated by compact objects of T .

Proof. We show that C is generated by {Cone(σ) | σ: A → A[n] ∈ S}. Let M be any object of C. Using Lemma 2.1, it is enough to show that T (Cone(σ), M)^{∗} = 0 for all σ ∈ S implies M = 0. By Lemma 2.3, every triangle

A ⟶^{σ} A[n] → Cone(σ) → A[1]

gives rise to an exact sequence

T (Cone(σ), M)^{∗} → T (A[n], M)^{∗} ⟶^{T (σ,M)^{∗}} T (A, M)^{∗} → T (Cone(σ)[−1], M)^{∗}.

By assumption, we have T (Cone(σ)[−1], M)^{∗} = 0 = T (Cone(σ), M)^{∗}. Hence the map

T (A[n], M)^{∗} ⟶^{T (σ,M)^{∗}} T (A, M)^{∗}

is an isomorphism for all σ ∈ S and thus, T (A, M)^{∗} is S-local. On the other hand, T (A, M)^{∗} is S-acylic and so we conclude that T (A, M)^{∗} = 0. It follows that M = 0 because A is a compact generator. □

8.6.1. Cohomological 𝔭-Localisation. Let A be a dg algebra such that H^{∗}A is gradedcommutative and let 𝔭 be a graded prime ideal of H^{∗}A. Denote by C_{𝔭} the full subcategory of objects X in D(A) such that (H^{∗}X)_{𝔭} = 0. In other words, C_{𝔭} is the kernel of the cohomological functor

(− ⊗_{H^{∗}A} (H^{∗}A)_{𝔭}) ∘ D(A)(A, −)^{∗}.

From the previous discussion we obtain

Corollary 8.16. The localisation

\xymatrix{D(A) \ar[r]_{Q} & D(A)/C_{𝔭} \ar[l]_{R}}

is smashing, and there is an isomorphism r: D(A)(A, A)_{𝔭}^{∗} ⟶^{≅} D(A)/C_{𝔭}(QA, QA)^{∗} of graded rings making the diagram

\xymatrix{D(A)(A, A)^{∗} \ar[r]^{can} \ar[d]_{Q} & D(A)(A, A)_{𝔭}^{∗} \ar[dl]_{≅}^{r} \\ D(A)/C_{𝔭}(QA, QA)^{∗}}

commutative. Furthermore, the squares

\xymatrix{D(A) \ar[d]_{Q} \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \ar[d]^{−⊗_{H^{∗}A}(H^{∗}A)_{𝔭}} & D(A) \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \\ D(A)/C_{𝔭} \ar[r]^{D(A)/C_{𝔭}(QA,−)^{∗}} & Mod_{gr} H^{∗}A_{𝔭} & D(A)/C_{𝔭} \ar[u]_{R} \ar[r]^{D(A)/C_{𝔭}(QA,−)^{∗}} & Mod_{gr} H^{∗}A_{𝔭} \ar[u]^{inc}}

commute up to natural isomorphism. □

9. Realising smashing localisations by morphisms of dg algebras

The results in this chapter are joint work with K. Brüning [8], with substantial contributions of Bernhard Keller. We show that every smashing localisation on a derived category of a dg algebra can be realised by a morphism of dg algebras. More precisely, if A is a dg algebra and L: D(A) → D(A) a smashing localisation, we prove the existence of a dg algebra A_{L} with the property D(A)/ Ker L ≃ D(A_{L}), a dg algebra A′ quasi-isomorphic to A and a zigzag of dg algebra morphisms

A ⟵^{∼} A′ → A_{L}

which identifies in cohomology with the algebra map L: D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}. If the dg algebra A is cofibrant, then the algebra map L identifies with the cohomology of a morphism A → A_{L}, and the quotient functor is naturally isomorphic to the left derived functor − ⊗_{A}^{L} A_{L}.

As an application, we consider in Section 9.2 dg algebras with graded-commutative cohomology ring. For such a dg algebra A, we introduce the localisation of A at a prime 𝔭 in cohomology and denote this dg algebra by A_{𝔭}. It has the property H^{∗}(A_{𝔭}) ≅ (H^{∗}A)_{𝔭}. Moreover, we show that with this identification of graded algebras, the canonical morphism H^{∗}A → (H^{∗}A)_{𝔭} is induced by a zigzag of dg algebra morphisms.

9.1. Construction of a dg algebra morphism. Let A be a differential graded algebra over some commutative ring k and let

L: D(A) → D(A)

be a smashing localisation. Denoting by C the category of L-acyclic objects, we have an adjoint pair of functors

\xymatrix{D(A) \ar[r]_{Q} & D(A)/C \ar[l]_{R}}

satisfying R ∘ Q = L. The right adjoint R is fully faithful and commutes with arbitrary direct sums.

Our first aim is to write the quotient category D(A)/C as derived category of a differential graded algebra A_{L}. Then we construct a zigzag of dg algebra morphisms A ⟵^{∼} A′ → A_{L} which induces the algebra morphism

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}, f ↦ Lf,

in cohomology. For this purpose, we identify throughout this chapter the functors H^{∗}: D(A) → Mod_{gr} H^{∗}A and D(A)(A, −)^{∗}: D(A) → Mod_{gr} H^{∗}A (see Lemma 6.2).

The following lemma which we learned from Dave Benson is the key to our construction.

Lemma 9.1. Let A, B dg algebras and M a dg (B, A)-bimodule. Let α: A → M and β: B → M be maps of dg modules which satisfy α(1) = β(1). Then

X = {(a, b) ∈ A × B | α(a) = β(b)}

is a dg algebra with differential d_{X} = (d_{A}, d_{B}) and the projections p_{1}, p_{2} in the pullback diagram

\xymatrix{X \ar[r]^{p_{2}} \ar[d]_{p_{1}} & B \ar[d]^{β} \\ A \ar[r]_{α} & M}

are dg algebra morphisms. If β is a surjective quasi-isomorphism, then the diagram induces a pullback diagram in cohomology.

Proof. The first assertions are immediately checked. For the last one we show that H^{∗}X = \{(\overline{a}, \overline{b}) ∈ H^{∗}A × H^{∗}B | H^{∗}α(\overline{a}) = H^{∗}β(\overline{b})\}.

A pair (\overline{a}, \overline{b}) ∈ H^{∗}X trivially satisfies the property H^{∗}α(\overline{a}) = H^{∗}β(\overline{b}) and consequently, the inclusion ⊆ is always fulfilled.

For the other inclusion we need to assume that β is a surjective quasi-isomorphism. Let (\overline{a}, \overline{b}) ∈ H^{∗}A × H^{∗}B such that H^{∗}α(\overline{a}) = H^{∗}β(\overline{b}). We choose representing cocycles a of \overline{a} and b of \overline{b}. Then α(a) − β(b) = m for some coboundary m ∈ M. Since β is a surjective quasi-isomorphism, there is a coboundary b′ ∈ B such that β(b′) = m. Hence the tuple (a, b + b′) satisfies α(a) = β(b + b′) and thus (\overline{a}, \overline{b}) = (\overline{a}, \overline{b + b′}) ∈ H^{∗}X. □

The following lemma ensures that the cohomology of the dg algebra A_{L} which we construct below is independent of all choices that we will make.

Lemma 9.2. Let X, Y be dg A-modules and let ν: X → Y be an isomorphism in K(A).

(1) Denote by X → I(X) the injective hull of X in the Frobenius category Mod_{dg} A. There exists a dg algebra S and a zigzag of quasi-isomorphisms of dg algebras

End_{A}(X) ⟵^{∼} S ⟶^{∼} End_{A}(Y ⊕ I(X)).

(2) Let I be any injective module in the Frobenius category Mod_{dg} A. There is a dg algebra T and a zigzag of quasi-isomorphisms of dg algebras

End_{A}(Y) ⟵^{∼} T ⟶^{∼} End_{A}(Y ⊕ I).

(3) There exists a zigzag of quasi-isomorphisms of dg algebras from End_{A}(X) to End_{A}(Y).

Proof. (1) By Lemma 6.4, we can choose a representing dg A-module map

ν̄: X → Y ⊕ I(X)

of ν ∈ K(A)(X, Y) which is split as map of graded A-modules. Hence the map

ν̄^{∗}: End_{A}(Y ⊕ I(X)) → Hom_{A}(X, Y ⊕ I(X)), f ↦ f ∘ ν̄,

is surjective. Applying Lemma 9.1, the pullback

\xymatrix{S \ar[r]^{p_{2}} \ar[d]_{p_{1}} & End_{A}(Y ⊕ I(X)) \ar@{->>}[d] \ar[d]^{∼}_{ν̄^{∗}} \\ End_{A}(X) \ar[r]^{ν̄_{∗}}_{∼} & Hom_{A}(X, Y ⊕ I(X))}

gives rise to a pullback diagram in cohomology, and the object S is actually a dg algebra. In particular, we obtain quasi-isomorphisms of dg algebras

\xymatrix{End_{A}(X) & S \ar[l]^{p_{1}}_{∼} \ar[r]^{∼}_{p_{2}} & End_{A}(Y ⊕ I(X)).}

(2) The dg A-module map ι: Y ⟶^{[ id 0 ]} Y ⊕ I is obviously a split monomorphism inducing id_{Y} in the homotopy category. Hence we obtain a pullback diagram

\xymatrix{T \ar[r]^{p_{2}}_{∼} \ar[d]_{p_{1}}^{∼} & End_{A}(Y ⊕ I) \ar@{->>}[d] \ar[d]^{∼}_{ι^{∗}} \\ End_{A}(Y) \ar[r]^{ι_{∗}}_{∼} & Hom_{A}(Y, Y ⊕ I)}

yielding the claim.

(3) is a trivial consequence of (1) and (2). □

The proof of the following lemma is immediate.

Lemma 9.3. The object QA is a compact generator of D(A)/C. □

Fix a homotopically projective replacement of RQA ∈ D(A). By abuse of notation we denote the replacement also by RQA.

Proposition 9.4. The functor R Hom(RQA, R−): D(A)/C → D(End_{A}(RQA)) is an equivalence of triangulated categories.

Proof. We use Proposition 2.2 and first note that R Hom(RQA, R−) preserves arbitrary direct sums because for any family (X_{i})_{i∈I} in D(A)/C, the map

∐_{i∈I} R Hom(RQA, RX_{i}) → R Hom(RQA, R ∐_{i∈I} X_{i})

identifies in cohomology with the isomorphism

∐_{i∈I} D(A)(RQA, RX_{i}) ≅ ∐_{i∈I} D(A)/C(QA, X_{i}) ≅ D(A)/C(QA, ∐_{i∈I} X_{i}) ≅ D(A)(RQA, R ∐_{i∈I} X_{i}).

Moreover, the functor R Hom(RQA, R−) maps the compact generator QA of D(A)/C to End_{A}(RQA) which compactly generates D(End_{A}(RQA)). Finally, the map

D(A)/C(QA, QA[n]) ⟶^{R Hom(RQA,R−)} D(End_{A}(RQA))(End_{A}(RQA), End_{A}(RQA)[n])

is an isomorphism for all n ∈ ℤ since RQA being homotopically projective implies that the diagram

\xymatrix{D(A)/C(QA, QA[n]) \ar[d]_{R}^{≅} \ar[r]^{R Hom(RQA,R−)} & D(End_{A}(RQA))(End_{A}(RQA), End_{A}(RQA)[n]) \\ D(A)(RQA, RQA[n]) \ar[r]_{≅} & H^{n}(End_{A}(RQA)) \ar[u]^{≅}}

is commutative. □

Hence we have shown that the quotient category D(A)/C is equivalent to the derived category of the dg algebra End_{A}(LA), where LA was chosen to be homotopically projective. Note that Lemma 9.2 provides a zigzag of quasi-isomorphisms between the endomorphism dg algebras of two different homotopically projective replacements of an object in D(A).

In order to construct a zigzag A ⟵^{∼} A′ → End_{A}(LA) of dg algebra morphisms inducing

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}

in cohomology, we need to make another choice on the dg A-module representing LA. Let η: id → RQ be the unit and ε: QR → id the counit of the adjunction

\xymatrix{D(A) \ar[r]_{Q} & D(A)/C. \ar[l]_{R}}

Since A is homotopically projective, we have

η_{A} ∈ D(A)(A, RQA) ≅ K(A)(A, RQA).

Lemma 9.5. For any map η̄_{A} in Mod_{dg} A that represents η_{A} ∈ K(A)(A, RQA) and any dg A-module M, the map

η̄_{A}^{∗}: Hom_{A}(RQA, RQM) → Hom_{A}(A, RQM), f ↦ f ∘ η̄_{A},

is a quasi-isomorphism.

Proof. Since R is fully faithful, the usual adjunction isomorphism (see [38, Ch. IV.1]) gives rise to the mutually inverse maps

H^{n}(η̄_{A}^{∗}): D(A)(RQA, RQM[n]) → D(A)(A, RQM[n]), f ↦ f ∘ η_{A},

and

D(A)(A, RQM[n]) → D(A)(RQA, RQM[n]), g ↦ R(ε_{QA}) ∘ RQ(g).  □

Remark 9.6. By Lemma 6.4, we may represent η_{A}: A → LA by a monomorphism of dg A-modules

η̄_{A}: A → L̂A,

which is a split as map of graded A-modules. Remember that L̂A = LA ⊕ I(A), where A → I(A) is the injective hull of A in the Frobenius category Mod_{dg} A, and LA was already chosen to be homotopically projective. By Lemma 9.2, we have a zigzag of quasi-isomorphisms

End_{A}(LA) ⟵^{∼} T ⟶^{∼} End_{A}(L̂A).

We define the dg algebra A_{L} to be End_{A}(L̂A). By abuse of notation we write A_{L} = End_{A}(LA). Note that from Lemma 9.2 and Proposition 9.4, it follows that

D(A_{L}) ≃ D(A)/C.

Theorem 9.7. The algebra map

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}, f ↦ L(f),

is induced by a zigzag of dg algebra maps

A ⟵^{∼} A′ ⟶^{φ} A_{L}.

That is, there exists a dg algebra A′ quasi-isomorphic to A, a morphism of dg algebras φ: A′ → A_{L} and in cohomology, we have the commutative diagram

\xymatrix{H^{∗}A′ \ar[d]_{≅} \ar[dr]^{H^{∗}φ} \\ D(A)(A, A)^{∗} \ar[r]^{L} & D(A)(LA, LA)^{∗}}

Proof. We identify the dg algebras End_{A}(A) and A through the isomorphism given by evaluation at 1. Let

\xymatrix{A′ \ar[r]^{p_{2}} \ar[d]_{p_{1}}^{∼} & A_{L} \ar@{->>}[d] \ar[d]_{∼}^{η̄_{A}^{∗}} \\ End_{A}(A) \ar[r]^{η̄_{A∗}} & Hom_{A}(A, LA)}

be a pullback diagram.

The map η̄_{A∗} is a quasi-isomorphism (Lemma 9.5), and surjective since η̄_{A} is a split monomorphism of graded A-modules (Remark 9.6). We infer from Lemma 9.1 that A′ is a dg algebra quasi-isomorphic to A, and we set φ = p_{2}.

In cohomology, we obtain the commutative diagram

\xymatrix{H^{∗}A′ \ar[r]^{H^{∗}(p_{2})} \ar[d]_{H^{∗}(p_{1})}^{≅} & H^{∗} End_{A}(LA) \ar[d]_{≅}^{H^{∗}(η̄_{A}^{∗})} \\ H^{∗} End_{A}(A) \ar[r]^{H^{∗}(η̄_{A∗})} & H^{∗}(Hom_{A}(A, LA))}

and thus it remains to show that the composition

H^{∗}(η̄_{A}^{∗})^{−1} ∘ H^{∗}(η̄_{A∗})

identifies with the map

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}, f ↦ L(f).

In fact, for f ∈ D(A)(A, A)^{∗} we have

(H^{∗}(η̄_{A}^{∗})^{−1} ∘ H^{∗}(η̄_{A∗}))(f) = R(ε_{QA}) ∘ RQ(η_{A}) ∘ RQ(f) ∈ D(A)(RQA, RQA).

But it is well-known that ε_{QA} ∘ Q(η_{A}) ≅ id_{QA} (see [38, Ch. IV.1]) and hence the claim follows. □

Observe that the map φ: A′ → A_{L} is a monomorphism: Since η̄_{A} is split as map of graded A-modules, η̄_{A∗} is injective and so is φ = p_{2}.

If we assume in addition that A is a cofibrant dg algebra (see Section 6.5), then the algebra map L: D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗} is not only induced by a zigzag of dg algebra maps, but by a morphism A → A_{L}. Corollary 9.8. Let A be a cofibrant dg algebra. The algebra morphism

D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}, f ↦ L(f),

lifts to a dg algebra morphism ψ: A → A_{L}. Proof. Since A is cofibrant, the map p_{1} : A′ → A in the pullback diagram

\xymatrix{A′ \ar[r]^{p_{2}} \ar@{->>}[d] \ar[d]_{p_{1}}^{∼} & A_{L} \ar@{->>}[d] \ar[d]_{∼}^{η̄_{A}^{∗}} \\ A \ar[r]^{η̄_{A∗}} & Hom_{A}(A, LA)}

splits: There is a morphism of dg algebras s: A → A′ such that p_{1} ∘ s = id_{A}. We define ψ to be the composition

p_{2} ∘ s: A → A_{L}.

Then H^{∗}ψ identifies with the canonical map L: D(A)(A, A)^{∗} → D(A)(LA, LA)^{∗}. □

Now our aim is to show that if A is cofibrant, then we can identify the functors Q: D(A) → D(A)/C ≃ D(A_{L}) and −⊗_{A}^{L} A_{L}: D(A) → D(A_{L}), where A_{L} is a dg (A, A_{L})- bimodule through the morphism ψ: A → A_{L}. Lemma 9.9. There exists a natural transformation

λ: R Hom_{A}(A, −) → R Hom_{A}(LA, L−)

in D(A) which commutes with the suspension functor. For every M ∈ D(A), λ_{M} induces the map

D(A)(A, M) → D(A)(LA, LM), f ↦ Lf,

in cohomology.

Proof. By Lemma 9.5, the adjunction unit η_{A}: A → LA induces a natural isomorphism R Hom_{A}(η_{A}, LM). Therefore we can define the morphism λ_{M} to be the composition

\xymatrix{R Hom_{A}(A, M) \ar[r]^{R Hom(A,η_{M})} & R Hom_{A}(A, LM) \ar[r]^{R Hom(η_{A},LM)^{−1}}_{≅} & R Hom_{A}(LA, LM),}

which obviously induces L: D(A)(A, M)^{∗} → D(A)(LA, LM) in cohomology. The naturality of λ_{M} follows from the naturality of R Hom(A, η_{M}) and R Hom(η_{A}, LM).

The unit η of the adjoint pair (Q, R) commutes with the suspension functor [1], hence so does R Hom(A, η_{M}). Since R Hom(η_{A}, LM) commutes with [1], we conclude that λ ∘ [1] ≅ [1] ∘ λ. □

Note that if A is cofibrant, then λ_{A} equals the dg algebra morphism ψ: A → A_{L} constructed in Corollary 9.8. In addition, R Hom_{A}(LA, LM) becomes an object in D(A) through the dg algebra morphism ψ.

Proposition 9.10. Suppose that A is a cofibrant dg algebra. Then the diagram

\xymatrix{D(A) \ar[d]_{Q} \ar[r]^{−⊗_{A}^{L}A_{L}} & D(A_{L}) \\ D(A)/C \ar[ur]_{R Hom_{A}(RQA,R−)}^{≃}}

commutes up to natural isomorphism.

Proof. We show that the functors R Hom_{A}(LA, L−) and −⊗_{A}^{L} A_{L} are naturally isomorphic. A natural transformation

τ: − ⊗_{A}^{L}A_{L} ⟶ R Hom_{A}(LA, L−)

is given as composition of the three natural maps in the diagram

\xymatrix{M ⊗_{A}^{L} A_{L} \ar@{.>}[r]^{τ_{M}} \ar[d]_{can_{M} ⊗_{A}^{L}A_{L}}^{≅} & R Hom_{A}(LA, LM) \\ R Hom_{A}(A, M) ⊗_{A}^{L} A_{L} \ar[r]^{λ_{M} ⊗_{A}^{L}A_{L}} & R Hom_{A}(LA, LM) ⊗_{A}^{L} A_{L} \ar[u]_{ν_{M}}}

where can_{M} is the canonical identification and ν_{M} is defined by

ν_{M}: R Hom_{A}(LA, LM) ⊗_{A}^{L} End_{A}(LA) → R Hom_{A}(LA, LM).
f ⊗ g ↦ f ∘ g

Note that τ commutes with the suspension functor since this holds for λ by Lemma 9.9, and obviously for can and ν. In order to prove that τ is an isomorphism, it suffices to show that the full subcategory A = {M ∈ D(A) | τ_{M} is an isomorphism} of D(A) is a localising subcategory containing A.

First we point out that A is closed under triangles by the Five-Lemma for triangulated categories, and that it is easy to check that A ∈ A.

Furthermore, A is closed under taking shifts because τ commutes with the suspension functor.

Finally, we show that A is closed under taking arbitrary direct sums. To that end, recall that the functor R Hom_{A}(LA, R−) commutes with arbitrary direct sums (Proposition 9.4) and hence, so does R Hom_{A}(LA, L−). Since − ⊗_{A}^{L} A_{L} obviously commutes with arbitrary direct sums, the claim follows. □

Remark 9.11. Let T be an algebraic triangulated category in the sense of Keller. We refer to [34, Sect. 6.5] for a definition. If T is generated by a single compact object, then there is a dg algebra A such that T ≃ D(A), see [34, Sect. 6.5]. Since we can choose the dg algebra A to be cofibrant (see Lemma 6.10), every smashing localisation L: T → T is induced by a morphism of dg algebras A → A_{L}.

9.2. The 𝔭-localisation of a dg algebra. Let A be a dg algebra over a commutative ring k and assume throughout this section that the cohomology algebra H^{∗}A is gradedcommutative. We fix a graded prime ideal 𝔭 of H^{∗}A. By C_{𝔭} we denote the full subcategory of objects M in D(A) such that (H^{∗}M)_{𝔭} = 0. The localisation L_{𝔭}: D(A) → D(A), given by the adjoint pair

\xymatrix{D(A) \ar[r]_{Q} & D(A)/C_{𝔭}, \ar[l]_{R}}

is smashing by Corollary 8.16. Now we apply the results of Section 9.1 to this special case. We define

A_{𝔭} = A_{L_{𝔭}},

and we call A_{𝔭} localisation of A at a prime 𝔭 in cohomology.

From Lemma 9.2 and Proposition 9.4, we infer that D(A)/C_{𝔭} ≃ D(A_{𝔭}). For this special smashing localisation, we have

Theorem 9.12. Let A be a dg algebra over a commutative ring k such that H^{∗}A is graded-commutative and let 𝔭 be a graded prime ideal of H^{∗}A. The dg algebra A_{𝔭} has the property H^{∗}(A_{𝔭}) ≅ (H^{∗}A)_{𝔭}. Moreover, with this identification of graded algebras, the canonical map

can: H^{∗}A → (H^{∗}A)_{𝔭}

is induced by a zigzag of dg algebra maps

A ⟵^{∼} A′ ⟶^{φ} A_{𝔭}.

That is, we have a commutative diagram

\xymatrix{H^{∗}A′ \ar[d]_{≅} \ar[drr]^{H^{∗}φ} \\ H^{∗}A \ar[r]_{can} & (H^{∗}A)_{𝔭} \ar[r]_{≅} & H^{∗}(A_{𝔭})}

Proof. Since D(A)(A, A)_{𝔭} ≅ D(A)(L_{𝔭}A, L_{𝔭}A) by Corollary 8.16, the dg algebra A_{𝔭} satisfies H^{∗}(A_{𝔭}) ≅ (H^{∗}A)_{𝔭}. Theorem 9.7 shows that the zigzag A ⟵^{∼} A′ ⟶^{φ} A_{𝔭} induces the map

L_{𝔭}: D(A)(A, A)^{∗} → D(A)(L_{𝔭}A, L_{𝔭}A)^{∗}, f ↦ L_{𝔭}(f)

in cohomology. But we may identify the algebra maps can and L_{𝔭} by Corollary 8.16. □

The following result is an immediate consequence of Corollary 9.8 and Theorem 9.12.

Corollary 9.13. Let A be a cofibrant dg algebra such that H^{∗}A is graded-commutative and let 𝔭 be a graded prime ideal of H^{∗}A. Then the canonical algebra morphism can: H^{∗}A → (H^{∗}A)_{𝔭} lifts to a dg algebra morphism

\xymatrix{ψ: A \ar[r] & A_{𝔭}. & □}

A class of cofibrant dg algebras with graded-commutative cohomology are the Sullivan algebras (ΛV, d) introduced Section 6.5.

Remark 9.14. The smashing localisation L_{𝔭}: D(A) → D(A) can be interpreted as 𝔭-localisation on the derived category. It satisfies a local-global principle:

For M ∈ D(A) and a graded prime 𝔭 of H^{∗}A, we define

M_{𝔭} = R Hom_{A}(L_{𝔭}A, L_{𝔭}M) ∈ D(A_{𝔭})

and call M_{𝔭} localisation of M at a graded prime ideal 𝔭 of H^{∗}A. If A is a cofibrant dg algebra, then we have M_{𝔭} = M ⊗_{A}^{L} A_{𝔭} by Proposition 9.10. Since (H^{∗}M)_{𝔭} ≅ H^{∗}(M_{𝔭}), the following conditions are equivalent for M ∈ D(A):

(1) M = 0.

(2) M_{𝔭} = 0 for all graded prime ideals 𝔭.

(3) M_{𝔪} = 0 for all graded maximal ideals 𝔪.

9.2.1. A universal property of A_{𝔭}. Let A be a dg algebra over a commutative ring k such that H^{∗}A is graded-commutative and 𝔭 ∈ Spec_{gr}(H^{∗}A).

The cohomology of the dg algebra A_{𝔭} satisfies a universal property since H^{∗}(A_{𝔭}) is isomorphic to the ring of fractions S^{−1}(H^{∗}A) = (H^{∗}A)_{𝔭}, where S is the subset of homogeneous elements in H^{∗}A \ 𝔭. If β: A → B is a morphism of dg algebras such that H^{∗}β makes S invertible, then H^{∗}β factors uniquely over the canonical morphism can: H^{∗}A → (H^{∗}A)_{𝔭}.

Without loss of generality, we assume from now on that A is cofibrant. Then can is induced by a morphism of dg algebras ψ: A → A_{𝔭} and the universal property yields a unique algebra morphism g: H^{∗}(A_{𝔭}) → H^{∗}B making the following diagram commute:

\xymatrix{H^{∗}A \ar[r]^{H^{∗}β} \ar[d]_{H^{∗}ψ} & H^{∗}B \\ H^{∗}(A_{𝔭}) \ar@{.>}[ur]_{g}}

The dg algebra morphisms β: A → B and ψ: A → A_{𝔭} give rise to functors

F_{β}: D(A) ⟶^{−⊗_{A}^{L}B} D(B) and F_{ψ}: D(A) ⟶^{−⊗_{A}^{L}A_{𝔭}} D(A_{𝔭}).

Now we prove a universal property on the level of derived categories.

Proposition 9.15. There is a unique functor G: D(A_{𝔭}) → D(B) making the following diagram commute:

\xymatrix{D(A) \ar[r]^{F_{β}} \ar[d]_{F_{ψ}} & D(B) \\ D(A_{𝔭}) \ar@{.>}[ur]_{G}}

Proof. We first note that by Proposition 9.10, the functor F_{ψ} is nothing but the quotient functor Q: D(A) → D(A)/C_{𝔭} composed with the equivalence D(A)/C_{𝔭} ≃ D(A_{𝔭}). Thus we can use the universal property of Q and only need to show that F_{β}(C_{𝔭}) = 0.

In Proposition 8.15 we have shown that

M = {Cone(σ) | σ: A → A[n] ∈ S}

is a set of compact generators of C_{𝔭} and thus it suffices to check that F_{β} vanishes on M. Any element of M fits into an exact triangle

A ⟶^{x·} A[n] → Cone(x·) → A[1]

in D(A), where x· denotes multiplication with an element x ∈ A whose cohomology H^{∗}x belongs to S. Applying the functor F_{β} to this triangle, we obtain a triangle in D(B) naturally isomorphic to

B ⟶^{β(x)·} B[n] → F_{β}(Cone(x·)) → B[1].

Since H^{∗}β(x) is invertible, we infer that F_{β}(Cone(x·)) is contractible and consequently, the object F_{β}(Cone(x·)) is zero in D(B). □

Since C_{𝔭} is generated by compact elements, the quotient functor Q: D(A) → D(A)/C_{𝔭} gives rise to a quotient functor D^{per}(A) → D^{per}(A)/C_{𝔭}^{per}, where C_{𝔭}^{per} = C_{𝔭} ∩ D^{per}(A). Furthermore, this quotient functor identifies with the functor

D^{per}(A) ⟶^{−⊗_{A}^{L}A_{𝔭}} D^{per}(A_{𝔭}).

This proves

Corollary 9.16. There is a unique functor G: D^{per}(A_{𝔭}) → D^{per}(B) which makes the following diagram commute:

\xymatrix{D^{per}(A) \ar[r]^{F_{β}} \ar[d]_{F_{ψ}} & D^{per}(B) \\ D^{per}(A_{𝔭}) \ar@{.>}[ur]_{G} &  & □}

Remark. The discussion above raises the question whether the functor G: D(A_{𝔭}) → D(B) and with it the algebra map g: H^{∗}(A_{𝔭}) → H^{∗}B can be lifted to a zigzag of dg algebra morphisms. Our construction in Section 9.1 does not apply since in general, we cannot expect that G is a smashing localisation. It remains to enlighten the relation of our construction with DG quotients, which have a universal property and were introduced by Drinfeld [16].

There is also a construction by Toën [53] (see also in [31]) which seems to be related: Let dgcat_{k} be the category of small dg categories over a commutative ring k. The localisation of dgcat_{k} with respect to the quasi-equivalences is denoted by Hqe. If A is a small dg category and S a set of morphisms in H^{0}(A), then a morphism F: A → B in Hqe is said to make S invertible if the induced functor H^{0}(A) → H^{0}(B) takes each s ∈ S to an isomorphism. Toën constructs a morphism Q: A → A[S^{−1}] in Hqe which makes S invertible. This morphism has a universal property: Each morphism in Hqe making S invertible factors uniquely through Q.

However, if A is a dg algebra, viewed as dg category with a single object, then the object A[S^{−1}] is in general not a dg algebra, but a dg category with more than one object.

10. Realisability

In this chapter we introduce the concept of realisability as considered in a paper of Benson, Krause and Schwede [5]. They are concerned with deciding whether a graded module over the Tate cohomology ring Ĥ^{∗}(G, k), where G is a finite group and k a field, is isomorphic to Ĥ^{∗}(G, M) for some kG-module M.

More generally, they consider a triangulated category T admitting arbitrary direct sums and a cohomological functor H^{∗}: T → Mod_{gr} E into the category of graded modules over a graded ring E. The functor H^{∗} is required to preserve arbitrary direct sums and products. Then realisability deals with deciding whether a graded E-module is isomorphic to a module in the image of this cohomological functor.

Benson, Krause and Schwede [5] have stated a local obstruction for realisability up to direct summands, and a criterion for realisability which is given by an infinite sequence of obstructions.

If A is a dg algebra over a field k, then the functor in question is the cohomology functor H^{∗}: D(A) → Mod_{gr} H^{∗}A. In this setting, Benson, Krause and Schwede also prove the existence of a global obstruction for realisability up to direct summands.

In the first section we introduce the general setup of Benson, Krause and Schwede [5] and recall the construction of the local obstruction. After focusing on realisability in the setting of dg algebras in the second section, we study the global obstruction and its basic properties in Section 10.3.

10.1. A local obstruction for realisability. Let T be a triangulated category admitting arbitrary direct sums. We denote the suspension functor by Σ. Let N be a compact object in T and E = T (N, N)^{∗} = ∐_{i∈ℤ} T (N, Σ^{i}N) the graded endomorphism ring of N (see Example 3.1). If X is a graded E-module, then we denote by X[n] the n-fold shifted module.

If M is an object in T , then we obtain a graded E-module T (N, M)^{∗} by composition of graded maps. On the other hand, given any graded E-module X, when is X isomorphic to T (N, M)^{∗} for some M ∈ T ?

We will mainly consider this question only up to direct summands. Therefore, from now on, we use the following terminology for realisability^{2}:

Definition 10.1. Let X be a graded E-module. We call X realisable if there exists an object M ∈ T such that X is isomorphic to a direct summand of T (N, M)^{∗}. If X ≅ T (N, M)^{∗} for some M ∈ T , then we call X strictly realisable.

Remark 10.2. Note that the functor T (N, −)^{∗} occurs in a very natural way: For every graded ring Λ and every cohomological functor H^{∗}: T → Mod_{gr} Λ which preserves arbitrary direct sums and products, there exists a compact object C ∈ T such that H^{∗} is naturally isomorphic to T (C, −)^{∗} [36, Lemma 3.2].

Benson, Krause and Schwede [5] have constructed a local obstruction for realisability:

2Our terminology is different from the one in [5]. Benson, Krause and Schwede call an E-module realisable if X ≅ T (N, M)^{∗} for some M ∈ T .

Theorem 10.3. [5, Thm. 3.7] Let T be a triangulated category with arbitrary direct sums, N ∈ T a compact object and E = T (N, N)^{∗} the graded endomorphism algebra of N. For each graded E-module X, there exists an element

κ(X) ∈ Ext_{E}^{3,−1}(X, X)

determining the realisability: X is realisable if and only if κ(X) is trivial.

For the proof of Theorem 10.3 and more details we refer to [5]. However, for our purposes we sketch the construction of the local obstruction κ(X):

Construction 10.1. A T -presentation of a graded E-module X consists of a distinguished triangle

(10.2)  Σ^{−1}B ⟶^{δ} R_{1} ⟶^{α} R_{0} ⟶^{π} B

in T together with an epimorphism of graded E-modules ε: T (N, R_{0})^{∗} → X such that the sequence

T (N, R_{1})^{∗} ⟶^{α_{∗}} T (N, R_{0})^{∗} ⟶^{ε} X → 0

is exact. If the objects R_{0} and R_{1} are assumed to be N-free, that is, isomorphic to a direct sum of shifted copies of N, then we refer to an N-special T -presentation.

Given an N-special T -presentation

(Σ^{−1}B ⟶^{δ} R_{1} ⟶^{α} R_{0} ⟶^{π} B, ε: T (N, R_{0})^{∗} → X),

we obtain an exact sequence of graded E-modules

(10.3) 0 → X[−1] ⟶^{η[−1]} T (N, B)^{∗}[−1] ⟶^{δ_{∗}} T (N, R_{1})^{∗} ⟶^{α_{∗}} T (N, R_{0})^{∗} ⟶^{ε} X → 0

by applying the functor T (N, −)^{∗} to the triangle. The monomorphism η: X → T (N, B)^{∗} is determined by η ∘ ε = π_{∗}.

This sequence is called associated extension of the N-special T -presentation.

In [5, Prop. 3.4] it is shown that there exists an N-special T -presentation for each graded E-module X, and that the Yoneda-class of the associated extension, denoted by κ(X) ∈ Ext_{E}^{3,−1}(X, X), is independent of the choice of the N-special T -presentation.

Since T (N, R_{0})^{∗} and T (N, R_{1})^{∗} are free, the extension κ(X) is trivial if and only if the monomorphism

X[−1] ⟶^{η[−1]} T (N, B)^{∗}[−1]

is split. Thus κ(X) determines, in fact, the realisability of X.

Remark 10.4. Let T be a triangulated category admitting direct sums and N ∈ T a compact object. Denote by E = T (N, N)^{∗} the graded endomorphism algebra of N. Benson, Krause and Schwede extend their theory by an infinite sequence of obstructions which decides whether a graded E-module X is strictly realisable.

They show that if there exists an infinite sequence of obstructions

κ_{n}(X) ∈ Ext_{E}^{n,2−n}(X, X), n ≥ 3,

where the class κ_{n}(X) is defined provided that the previous one κ_{n−1}(X) vanishes, then it even holds X ≅ T (N, M)^{∗}. In this sequence of obstructions all but the first one depend on choices. Only κ_{3}(X) is uniquely determined and equals the local obstruction κ(X).

In view of the need for an infinite sequence of obstructions to decide whether X ≅ T (N, M)^{∗} for some M ∈ T , it is remarkable that the first obstruction of this sequence already tells whether X is a direct summand of T (N, M)^{∗}.

In our results we will only consider this first obstruction.

10.2. Realisability and dg algebras. Let A be a dg algebra over some commutative ring k. Remember that the functor D(A)(A, −)^{∗} is naturally isomorphic to the cohomology functor H^{∗} (Lemma 6.2). Hence a graded H^{∗}A-module is realisable if and only if it is a direct summand of H^{∗}M, where M ∈ D(A) is a dg A-module. Moreover, A is a compact object in D(A). Hence Theorem 10.3 applies, and we can decide whether X is a direct summand of H^{∗}M for some dg A-module M ∈ D(A).

Example 10.5. (1) Let k be a Noetherian ring and G a finite group. Remember that the group cohomology ring H^{∗}(G, k) is actually a graded endomorphism ring of an object in a triangulated category: We have H^{∗}(G, k) ≅ K(Inj kG)(ik, ik)^{∗}, where ik denotes an injective resolution of k. Moreover, K(Inj kG) admits arbitrary direct sums since Inj kG does. Since ik is compact in K(Inj kG) by Proposition 8.9, the assumptions of Theorem 10.3 are satisfied.

On the other hand, we know from Remark 6.1(2) that K(Inj kG)(ik, ik)^{∗} is the cohomology of the endomorphism dg algebra End(ik) of the complex ik,

H^{∗} End(ik) ≅ H^{∗}(G, k),

and we can also consider realisability in the setting of dg algebras. Now note that we have a commutative square

\xymatrix{K(Inj kG) \ar@{->>}[d]_{Hom(ik,−)} \ar[r]^{K(Inj kG)(ik,−)^{∗}} & Mod_{gr} H^{∗}(G, k) \ar[d]^{≅} \\ D(End(ik)) \ar[r]^{H^{∗}} & Mod_{gr} H^{∗} End(ik)}

Since the exact functor Hom(ik, −): K(Inj kG) → D(End(ik)) commutes with arbitrary direct sums and is dense (see for example [6, Prop. 3.1]), a graded H^{∗}(G, k)-module is realisable by a complex C ∈ K(Inj kG) if and only if it is realisable by a dg End(ik)- module M.

Observe that H^{∗}(G, k) is also isomorphic to the graded endomorphism ring D(kG)(k, k)^{∗}. However, except in trivial cases, the stalk complex k is not a compact object in D(kG) and so the assumptions for Theorem 10.3 are not fulfilled.

(2) Let k be a field and G be a finite group. The Tate cohomology ring Ĥ^{∗}(G, k) is the graded endomorphism ring \underline{Hom}_{kG}(k, k)^{∗} and k is compact in \underline{Mod} kG. With the equivalence Z^{0} : K_{ac}(Inj kG) ⟶^{≃} \underline{Mod} kG (Lemma 8.10), we can write Ĥ^{∗}(G, k) also as graded endomorphism ring K_{ac}(Inj kG)(tk, tk)^{∗} and conclude that

H^{∗} End(tk) ≅ Ĥ^{∗}(G, k).

Similarly as in (1), we have a commutative square

\xymatrix{K_{ac}(Inj kG) \ar@{->>}[d]_{Hom(tk,−)} \ar[r]^{K_{ac}(Inj kG)(tk,−)^{∗}} & Mod_{gr} Ĥ^{∗}(G, k) \ar[d]^{≅} \\ D(End(tk)) \ar[r]^{H^{∗}} & Mod_{gr} H^{∗} End(tk)}

and Hom(tk, −) is exact, dense and preserves arbitrary direct sums. Hence a graded Ĥ^{∗}(G, k)-module is realisable by an acyclic complex (or equivalently, by an object of \underline{Mod} kG) if and only if it is realisable by an object of D(End(tk)).

Note that if k is a field of characteristic p > 0 and G a p-group, then the functors Hom(ik, −) and Hom(tk, −) are even equivalences [6].

Considering realisability in the setting of dg algebras has a striking advantage, as the following section indicates.

10.3. A global obstruction for realisability. Let A be a dg algebra over a field k. We have seen in Chapter 7 that H^{∗}A is an A_{∞}-algebra whose secondary multiplication determines a Hochschild class μ_{A} ∈ HH^{3,−1}(H^{∗}A), called the canonical class. Now the importance of this result for the realisability theory becomes evident:

Theorem 10.6. [5, Thm. 6.2] Let X be a graded H^{∗}A-module. The realisability obstruction

κ(X) ∈ Ext_{H^{∗}A}^{3,−1}(X, X)

is given by the cup product pairing

id_{X} ∪μ_{A}.

In particular, if the class μ_{A} ∈ HH^{3,−1}(H^{∗}A) is trivial, then all graded H^{∗}A-modules are realisable.

Because of the last property the class μ_{A} is referred to as global obstruction.

Remark. The converse of the last statement in Theorem 10.6 is not true in general: Benson, Krause and Schwede provide an example of a dg algebra A with the property that all H^{∗}A-modules are realisable, but with non-trivial canonical class μ_{A} [5, Exm. 5.15].

11. Realisability and 𝔭-localisation

In this chapter we study the relation between realisability of modules over gradedcommutative cohomology rings and 𝔭-localisation.

A first motivation for our use of 𝔭-localisation is the problem when a module over the graded-commutative ring H^{∗}(G, k) is isomorphic to H^{∗}(G, M) for some kG-module M. This is discussed in Section 11.1.

More generally, we consider dg algebras with graded-commutative cohomology rings in Section 11.2. In our main result of this chapter we prove that the classical local-global principle of Commutative Algebra applies for realisability.

11.1. A motivation for 𝔭-localisation. Let G be a finite group and k a field such that Char(k) divides the order of G. Let X be a graded module over the group cohomology ring H^{∗}(G, k). With Theorem 10.3 we can determine whether X is realisable by a complex in the homotopy category K(Inj kG). However, we rather want to know when X is realisable by a module, i.e. X is a direct summand of H^{∗}(G, M) with M ∈ Mod kG, or even X ≅ H^{∗}(G, M).

The category Mod kG is embedded in K(Inj kG), but with Theorem 10.3 we cannot decide whether an arbitrary realisable H^{∗}(G, k)-module can actually be realised by a module. Now we show that one can say more about realisability of 𝔭-local modules, where 𝔭 is a non-maximal prime.

Throughout this section we denote by H^{∗}(G) the group cohomology ring of G. Remember that H^{∗}(G) is graded-commutative (Section 4.1).

Let 𝔭 be a graded prime ideal of H^{∗}(G) and denote by C_{𝔭} the kernel of the cohomological functor

Hom_{K}(ik, −)_{𝔭}^{∗} = K(Inj kG)(ik, −)^{∗} ⊗_{H^{∗}(G)} H^{∗}(G)_{𝔭}: K(Inj kG) → Mod_{gr} H^{∗}(G)_{𝔭}.

From Theorem 8.12 we obtain a smashing localisation

\xymatrix{K(Inj kG) \ar[r]_{Q_{𝔭}} & K(Inj kG)/C_{𝔭}. \ar[l]_{R_{𝔭}}}

On the other hand, we have a smashing localisation

\xymatrix{K(Inj kG) \ar[r]_{Q̂} & \underline{Mod} kG, \ar[l]_{R̂}}

where Q̂(C) = Z^{0}(C ⊗_{k} tk) and R̂(M) = tM, see Chapter 8.5.

Lemma 11.1. If 𝔭 is non-maximal, then Q_{𝔭} factors over Q̂, i.e. there is a functor

Q̄: \underline{Mod} kG → K(Inj kG)/C_{𝔭}

such that Q_{𝔭} = Q̄ ∘ Q̂. Moreover, Q̄ has a right adjoint R̄ satisfying R̄Q̄ ≅ id and it holds R_{𝔭} = R̂R̄.

Proof. By Proposition 8.11, we have a localisation sequence

D(Mod kG) ⟶^{−⊗_{k}pk} K(Inj kG) ⟶^{Q̂=Z^{0}(−⊗_{k}tk)} \underline{Mod} kG.

Consequently, in order to obtain the functor Q̄, it suffices to show that the composition Q_{𝔭} ∘ (− ⊗_{k} pk) is zero (see Lemma 8.6(3)).

Now observe that Hom_{K}(ik, kG)^{∗} ≅ Hom_{kG}(H^{0}(ik), kG) ≅ k. Since 𝔭 is nonmaximal, it holds k_{𝔭} = 0 and hence kG is contained in C_{𝔭}. We infer that the composition Q_{𝔭} ∘ (− ⊗_{k} pk) vanishes on kG, and by dévissage on all objects of D(Mod kG).

The functor Q̄: \underline{Mod} kG → K(Inj kG)_{𝔭} we have obtained that way commutes with arbitrary direct sums since this holds for Q̂ and Q_{𝔭}, and because Q̂ is dense. Furthermore, the category \underline{Mod} kG is compactly generated by the finite dimensional modules (see [47]), thus Q̄ has a right adjoint R̄ by Proposition 2.4. Obviously it holds R_{𝔭} = R̂R̄ and we conclude that R̄Q̄ = id. □

Remember that a H^{∗}(G)-module X is 𝔭-local if X ≅ X_{𝔭} as H^{∗}(G)-modules.

Proposition 11.2. Let 𝔭 be a non-maximal prime and assume that X ∈ Mod_{gr} H^{∗}(G, k) is a 𝔭-local module. If X is realisable, then it can be realised by a Tate resolution tM of some kG-module M. Furthermore, X_{𝔭} is isomorphic to a direct summand of H^{∗}(G, M)_{𝔭}.

Proof. From Theorem 8.12 and Remark 8.13 we obtain diagrams

\xymatrix{K(Inj kG) \ar[d]_{Q_{𝔭}} \ar[r]^{Hom(ik,−)^{∗}} & Mod_{gr} H^{∗}(G) \ar[d]^{−⊗H^{∗}(G)_{𝔭}} & K(Inj kG) \ar[r]^{Hom(ik,−)^{∗}} & Mod_{gr} H^{∗}(G) \\ K(Inj kG)/C_{𝔭} \ar[r]^{Hom(Q_{𝔭}(ik),−)^{∗}} & Mod_{gr} H^{∗}(G)_{𝔭} & K(Inj kG)/C_{𝔭} \ar[u]_{R_{𝔭}} \ar[r]^{Hom(Q_{𝔭}(ik),−)^{∗}} & Mod_{gr} H^{∗}(G)_{𝔭} \ar[u]^{inc}}

which commute up to isomorphism. Let X be a graded 𝔭-local module over H^{∗}(G). If X is realisable by some complex C ∈ K(Inj kG), then X_{𝔭} is realisable by an object Q_{𝔭}(C) ∈ K(Inj kG)/C_{𝔭}. Hence X ≅ inc(X_{𝔭}) is realisable by R_{𝔭}Q_{𝔭}(C). We infer from Lemma 11.1 that X can be realised by R̂(M), where M denotes R̄Q_{𝔭}(C) ∈ \underline{Mod} kG. But the functor R̂ assigns to M its Tate resolution tM.

For the second claim, we show that Hom_{K}(ik, tM)_{𝔭}^{∗} ≅ H^{∗}(G, M)_{𝔭}. Applying the cohomological functor Hom_{K}(ik, −)_{𝔭} to the canonical triangle

pM → iM → tM → pM[1]

yields an exact sequence

Hom_{K}(ik, pM[−1])_{𝔭}^{∗} → Hom_{K}(ik, iM)_{𝔭}^{∗} → Hom_{K}(ik, tM)_{𝔭}^{∗} → Hom_{K}(ik, pM)_{𝔭}^{∗}

of graded H^{∗}(G)-modules. In the proof of Lemma 11.1 we have shown that pM and pM[−1] lie in the kernel of Hom_{K}(ik, −)_{𝔭}. Thus H^{∗}(G, M)_{𝔭} and Hom_{K}(ik, tM)_{𝔭}^{∗} are, in fact, isomorphic. □

Observe that the proof also shows that for a strictly realisable, graded H^{∗}(G, k)- module X which is moreover 𝔭-local, there exists a module M ∈ Mod kG such that X ≅ Hom_{K}(ik, tM) and X_{𝔭} ≅ H^{∗}(G, M)_{𝔭}.

Remark. If the module H^{∗}(G, M) in Proposition 11.2 is 𝔭-local, then X is realisable by the kG-module M. However, in general we cannot expect H^{∗}(G, M) to be 𝔭-local.

11.2. Realisability is a local property. In this section we show that the classical local-global principle applies for realisability. Let A be a differential graded algebra over a commutative ring k, and assume that the cohomology ring H^{∗}A is gradedcommutative. Under some finiteness assumptions we prove that a graded H^{∗}A-module is realisable if and only if X_{𝔭} is realisable for all graded prime ideals 𝔭 of H^{∗}A.

Lemma 11.3. Let A be a dg algebra with graded-commutative cohomology ring H^{∗}A and fix a graded prime ideal 𝔭 of H^{∗}A. If a graded H^{∗}A-module X is (strictly) realisable, then X_{𝔭} is (strictly) realisable.

Proof. We use the commutative diagrams

\xymatrix{D(A) \ar[d]_{Q} \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \ar[d]^{−⊗_{H^{∗}A}(H^{∗}A)_{𝔭}} & D(A) \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \\ D(A)/C_{𝔭} \ar[r]^{D(A)/C_{𝔭}(QA,−)^{∗}} & Mod_{gr} H^{∗}A_{𝔭} & D(A)/C_{𝔭} \ar[u]_{R} \ar[r]^{D(A)/C_{𝔭}(QA,−)^{∗}} & Mod_{gr} H^{∗}A_{𝔭} \ar[u]^{inc}}

from Corollary 8.16. If X is realisable, then the H^{∗}A_{𝔭}-module X_{𝔭} is realisable by an object of D(A)/C_{𝔭}. But then the H^{∗}A-module X_{𝔭} is realisable by an object of D(A), by the right hand diagram. □

Now our aim is to show that X_{𝔭} being realisable for all graded primes 𝔭 of H^{∗}A implies that X is realisable. On that purpose, we study the behaviour of the local obstruction κ(X) ∈ Ext_{H^{∗}A}^{3,−1}(X, X) under 𝔭-localisation. The following lemma is stated in a more generally setting, but applies to this situation.

Lemma 11.4. Let T , U be triangulated categories with arbitrary direct sums and let N ∈ T be a compact object. Assume that

F: T → U

is an exact functor such that F N is compact in U, and that

F̃: Mod_{gr} T (N, N)^{∗} → Mod_{gr} U(F N, F N)^{∗}

is an exact functor such that the diagram below commutes up to natural isomorphism.

\xymatrix{T \ar[d]_{F} \ar[r]^{T (N,−)^{∗}} & Mod_{gr} T (N, N)^{∗} \ar[d]^{F̃} \\ U \ar[r]^{U(F N,−)^{∗}} & Mod_{gr} U(F N, F N)^{∗}}

Then we have for every graded T (N, N)^{∗}-module X

F̃(κ(X)) = κ(F̃X) in Ext_{U(F N,F N)^{∗}}^{3,−1}(F̃X, F̃X).

Proof. Let (Σ^{−1}B ⟶^{δ} R_{1} ⟶^{α} R_{0} ⟶^{π} B, ε: T (N, R_{0})^{∗} → X) be an N-special T -presenta-tion with associated extension κ(X) ∈ Ext_{T (N,N)^{∗}}^{3,−1}(X, X). Since F is exact and preserves

direct sums, we obtain a triangle Σ^{−1}F B ⟶^{F δ} F R_{1} ⟶^{F α} F R_{0} ⟶^{F π} F B in U, where F R_{0} and F R_{1} are F N-free. The exactness of F̃ and the commutativity of the diagram yield an epimorphism ζ: U(F N, F R_{0})^{∗} → F̃X. Consequently

(Σ^{−1}F B ⟶^{F δ} F R_{1} ⟶^{F α} F R_{0} ⟶^{F π} F B, ζ: U(F N, F R_{0})^{∗} → F̃X)

is an F N-special U-presentation and the commutativity of the diagram

[figure: 0; (F̃X)[−1]; ≅; U(F N, Σ^{−1}F B)^{∗}; (F δ)_{∗}; ≅; U(F N, F R_{1})^{∗}; (F α)_{∗}; ≅; U(F N, F R_{0})^{∗}; ζ; ≅; F̃X; 0; 0; F̃(X[−1]); F̃(T (N, Σ^{−1}B)^{∗}); F̃(δ_{∗}); F̃(T (N, R_{1})^{∗}); F̃(α_{∗}); F̃(T (N, R_{0})^{∗}); F̃X; 0]

shows that κ(F̃X) = F̃(κ(X)) in Ext_{U(F N,F N)^{∗}}^{3,−1}(F̃X, F̃X). □

The next lemma is well-known for strictly commutative Noetherian rings (see for example [9]), and it is easy to check that it also holds true for graded-commutative coherent rings. Remember that a graded ring R is called coherent if finitely generated graded submodules of finitely presented graded R-modules are assumed to be finitely presented.

Lemma 11.5. Let R be a graded-commutative and coherent ring, and let 𝔭 be a graded prime ideal of R. Let M, N be graded R-modules and assume that M is finitely presented. Then there is a natural isomorphism

Ext_{R}^{i,∗}(M, N)_{𝔭} ≅ Ext_{R_{𝔭}}^{i,∗}(M_{𝔭}, N_{𝔭})

for all i ≥ 0. □

Theorem 11.6 (Local-global principle). Let A be a dg algebra such that H^{∗}A is gradedcommutative and coherent. The following conditions are equivalent for a finitely presented graded H^{∗}A-module X:

(1) X is realisable.

(2) X_{𝔭} is realisable for all graded prime ideals 𝔭 of H^{∗}A.

(3) X_{𝔪} is realisable for all graded maximal ideals 𝔪 of H^{∗}A.

Proof. By Lemma 11.3, it suffices to show that X is realisable if X_{𝔪} is realisable by an object in D(A)/C_{𝔪} for all graded maximal ideals 𝔪 of H^{∗}A.

The H^{∗}A-module X is realisable if and only if the class κ(X) ∈ Ext_{H^{∗}A}^{3,−1}(X, X) is trivial (Theorem 10.3). By Proposition 3.6, the latter holds true if and only if the fraction \frac{κ(X)}{1} in Ext_{H^{∗}A}^{3,−1}(X, X)_{𝔪} equals zero for all graded maximal ideals 𝔪 of H^{∗}A. But since X is finitely presented and H^{∗}A is assumed to be graded-commutative and coherent, we may apply the natural isomorphism

(Ext_{H^{∗}A}^{3,∗}(X, X))_{𝔪} ≅ Ext_{(H^{∗}A)_{𝔪}}^{3,∗}(X_{𝔪}, X_{𝔪})

from Lemma 11.5, which maps the fraction \frac{κ(X)}{1} to the extension κ(X) ⊗_{H^{∗}A} (H^{∗}A)_{𝔪}. The object QA ∈ D(A)/C_{𝔪} is compact (Lemma 9.3), hence we can apply Lemma 11.4 to the commutative diagram

\xymatrix{D(A) \ar[d]_{Q} \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \ar[d]^{−⊗_{H^{∗}A}(H^{∗}A)_{𝔪}} \\ D(A)/C_{𝔪} \ar[r]^{D(A)/C_{𝔪}(QA,−)^{∗}} & Mod_{gr} H^{∗}A_{𝔪}}

and obtain

κ(X) ⊗_{H^{∗}A} (H^{∗}A)_{𝔪} = κ(X ⊗_{H^{∗}A} (H^{∗}A)_{𝔪}).

Altogether we infer that X is realisable if and only if the class κ(X_{𝔪}) is trivial for all 𝔪, or equivalently, if X_{𝔪} is realisable by an object in D(A)/C_{𝔪} for all graded maximal ideals 𝔪 of H^{∗}A. □

A graded H^{∗}A-module is realisable if and only if all its direct summands are realisable. Since 𝔭-localisation commutes with direct sums, this shows

Corollary 11.7. Let A be a dg algebra such that H^{∗}A is graded-commutative and coherent. Let X ∈ Mod_{gr} H^{∗}A be an arbitrary direct sum of finitely presented graded H^{∗}A-modules. Then the following are equivalent:

(1) X is realisable.

(2) X_{𝔭} is realisable for all graded prime ideals 𝔭 of H^{∗}A.

(3) X_{𝔪} is realisable for all graded maximal ideals 𝔪 of H^{∗}A. □

Theorem 11.6 and Corollary 11.7 apply in particular for realisability over the group cohomology ring H^{∗}(G, k), where G is a finite group and k is a noetherian ring (see Example 10.5). The finiteness of the group is not necessary as long as H^{∗}(G, k) is still coherent and kG is noetherian. The latter is to ensure that the category K(Inj kG) is closed under taking arbitrary direct sums.

We also remark that in all results in this section, the realisability setting

D(A) ⟶^{H^{∗}} Mod_{gr} H^{∗}A

can be replaced by the more general setting

T ⟶^{T (N,−)^{∗}} Mod_{gr} T (N, N)^{∗},

where T is a triangulated category which admits arbitrary direct sums and is generated by compact objects, and N ∈ T is a compact object such that T (N, N)^{∗} is gradedcommutative and coherent.

Remark. (1) One might want to have this local-global principle for arbitrary graded H^{∗}A-modules. It is well-known that every graded module is a direct limit of finitely presented graded modules, but it is open whether a realisable finitely presented module can be written as direct limit of realisable finitely presented modules. It is not known either whether an arbitrary direct limit of realisable modules is realisable.

(2) It would be nice to have a local-global principle also for strict realisability (see Remark 10.4). Let X be a graded H^{∗}A-module, where A is a dg algebra with gradedcommutative cohomology ring. If X is strictly realisable, then so is X_{𝔭} (Lemma 11.3). If X is finitely presented and H^{∗}A coherent, does X_{𝔭} being strictly realisable for all primes 𝔭 imply that X is strictly realisable?

The infinite sequence of obstructions deciding on strict realisability

κ_{n}(X) ∈ Ext_{H^{∗}A}^{n,2−n}(X, X), n ≥ 3,

where κ_{n}(X) is defined provided that the previous one κ_{n−1}(X) vanishes, arises from an A-exact ∞-Postnikov-System which in cohomology gives rise to a map having cokernel X (see [5, App. A] for details). More precisely, an A-exact l-Postnikov system gives rise to the obstructions κ_{n}(X), 3 ≤ n ≤ l, and can be extended to an A-exact (l + 1)-Postnikov system provided that the class κ_{l}(X) is trivial. If an A-exact ∞-Postnikov system exists, then X is strictly realisable [5, Prop. A.19].

Now one might want to prove iteratively κ_{l}(X_{𝔭}) ≅ κ_{l}(X)_{𝔭} for all l ≥ 3 and use the same methods as in the proof of Theorem 11.6. The problem is that all but the first obstruction are not uniquely determined and consequently, the compatibility of the obstructions for the realisability of X and the realisability of X_{𝔭} cannot be expected in general.

12. Localising the global obstruction

Let A be a dg algebra over a field k and assume that H^{∗}A is graded-commutative. We have shown in Section 11.2 that realisability is a local property. The local-global principle we have shown applies for finitely presented modules but does not yield information on global realisability. In this chapter we develop a local-global principle for global realisability.

Applying our results from Chapter 9, we can state a global obstruction for the 𝔭local H^{∗}A-modules: Let A_{𝔭} the localisation of A at a prime 𝔭 in cohomology, defined in Section 9.2. From Corollary 8.16, Proposition 9.4 and Theorem 9.12 we conclude that the diagram

\xymatrix{D(A) \ar[d] \ar[r]^{D(A)(A,−)^{∗}} & Mod_{gr} H^{∗}A \ar[d]^{−⊗_{H^{∗}A}(H^{∗}A)_{𝔭}} \\ D(A_{𝔭}) \ar[r]^{D(A_{𝔭})(A_{𝔭},−)^{∗}} & Mod_{gr} H^{∗}(A_{𝔭})}

commutes up to isomorphism. Since in particular H^{∗}(A_{𝔭}) ≅ (H^{∗}A)_{𝔭}, we infer that the canonical class μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}A_{𝔭}) is a global obstruction for the 𝔭-local modules.

At first sight, it is not clear whether the Hochschild classes μ_{A} ∈ HH^{3,−1}(H^{∗}A) and μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}A_{𝔭}) are associated in some way. In order to relate them, we show in Section 12.1 the existence of a map of Hochschild cohomology rings

Γ: HH^{∗,∗}(H^{∗}A) ⟶ HH^{∗,∗}(H^{∗}A_{𝔭})

which has the property Γ(μ_{A}) = μ_{A_{𝔭}}.

After discussing localisation of Hochschild cohomology groups of graded-commutative algebras in Section 12.2, we are ready to prove a local-global principle for global realisability.

12.1. A map of Hochschild cohomology rings. In general, a morphism of graded algebras φ: R → T does not induce a homomorphism HH^{∗,∗}(R) → HH^{∗,∗}(T). We show in this section that such a map does exist whenever T_{R} and _{R}T are flat and φ: R → T is an epimorphism in the category of rings, i.e. for any ring T′ and morphisms α, β: T → T′ with αφ = βφ, it follows α = β. Such a map can be characterised in the following way:

Lemma 12.1. [52, Ch. XI, Prop. 1.2] The following conditions are equivalent for a morphism of graded rings φ: R → T:

(1) φ is an epimorphism in the category of rings.

(2) The map R ⊗_{T} R → R, r ⊗ r′ ↦ rr′, is an isomorphism.

(3) The restriction functor Mod_{gr} R → Mod_{gr} T is full.

We call a map of graded algebras φ: R → T a flat epimorphism if φ is an epimorphism in the category of rings and furthermore, the modules T_{R} and _{R}T are flat.

Example 12.2. If R is a graded-commutative ring and S ⊆ R a multiplicative subset of homogeneous elements, then R → S^{−1}R is a flat epimorphism: S^{−1}R is flat as both left and right R-module, and it is an epimorphism of rings by the universal property of the ring of fractions.

By 𝔹(Λ) we denote the graded Bar resolution of a graded algebra Λ as introduced in Chapter 5.2.

Lemma 12.3. Let φ: R → T be a map of graded algebras which is a flat epimorphism of rings. Then the complex T ⊗_{R} 𝔹(R) ⊗_{R} T is a T^{e}-projective resolution of T and a chain map η: T ⊗_{R} 𝔹(R) ⊗_{R} T → 𝔹(T) is given by

η_{n}: T ⊗_{R} 𝔹(R)_{n} ⊗_{R} T ⟶ 𝔹(T)_{n}
(t, r_{0}, ⋯, r_{n+1}, t′) ⟼ (tφ(r_{0}), φ(r_{1}), ⋯, φ(r_{n}), φ(r_{n+1})t′).

Proof. We first remark that T ⊗_{R} 𝔹(R) ⊗_{R} T is exact because T ⊗_{R} − and − ⊗_{R} T are exact. Since _{R}T_{T} and _{T} T_{R} are projective as T -modules and 𝔹(R)_{n} is a projective R^{e}-module for all n ≥ 0, we get indeed a T^{e}-projective resolution of T ⊗_{R} R ⊗_{R} T. But since φ: R → T is an epimorphism of rings, the map

T ⊗_{R} R ⊗_{R} T → T, (t, r, t′) ↦ tφ(r)t′,

is an isomorphism of T^{e}-modules. □

As a consequence we obtain

Proposition 12.4. The maps

Hom_{R^{e}}^{l}(𝔹(R)_{n}, R) ⟶ Hom_{T^{e}}^{l}(T ⊗_{R} 𝔹(R)_{n} ⊗_{R} T, T),
ζ ⟼ T ⊗_{R} ζ ⊗_{R} T

where l ∈ ℤ and n ≥ 0, induce a homomorphism of bigraded algebras

Γ: HH^{∗,∗}(R) ⟶ HH^{∗,∗}(T).

Proof. Γ is obviously a morphism of graded vector spaces, and it is easy to check that it commutes with Yoneda multiplication. □

Theorem 12.5. Let A, B dg algebras over a field k and suppose that ψ: A → B is a morphism of dg algebras inducing a flat epimorphism ψ^{∗}: H^{∗}A → H^{∗}B in cohomology. Then the map

Γ: HH^{∗,∗}(H^{∗}A) ⟶ HH^{∗,∗}(H^{∗}B)

satisfies Γ(μ_{A}) = μ_{B}.

Proof. We choose representing cocycles m_{A} ∈ Hom_{(H^{∗}A)^{e}}^{−1}(H^{∗}A^{⊗5}, H^{∗}A) of μ_{A} and m_{B} ∈ Hom_{(H^{∗}B)^{e}}^{−1}(H^{∗}B^{⊗5}, H^{∗}B) of μ_{B} and show that the diagram

\xymatrix{H^{∗}B ⊗_{H^{∗}A} H^{∗}A^{⊗5} ⊗_{H^{∗}A} H^{∗}B \ar[d]_{η_{3}} \ar[r]^{H^{∗}B⊗m_{A}⊗H^{∗}B} & H^{∗}B ⊗_{H^{∗}A} H^{∗}A ⊗_{H^{∗}A} H^{∗}B \ar[d]^{η_{0}}_{≅} \\ H^{∗}B^{⊗5} \ar[r]^{m_{B}} & H^{∗}B}

commutes up to coboundaries.

An element (s, x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, t) ∈ H^{∗}B ⊗ H^{∗}A^{⊗5} ⊗ H^{∗}B is sent by m_{B} ∘ η_{3} to

(12.1)  (−1)^{|s|}sm_{B}(ψ^{∗}(x_{0}), ψ^{∗}(x_{1}), ψ^{∗}(x_{2}), ψ^{∗}(x_{3}), ψ^{∗}(x_{4}))t.

On the other hand, under η_{0} ∘ (H^{∗}B ⊗ m_{A} ⊗ H^{∗}B) our element maps to

(12.2)  (−1)^{|s|}sψ^{∗}(m_{A}(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}))t.

By Lemma 7.6, we have a (2, −1)-Hochschild cocycle u ∈ Hom_{(H^{∗}A)^{e}}^{−1}(H^{∗}A^{⊗4}, H^{∗}B) such that

m_{B} ∘ (ψ^{∗})^{⊗5} − ψ^{∗} ∘ m_{A} = u ∘ d_{3},

where d_{3} : H^{∗}A^{⊗5} → H^{∗}A^{⊗4} is the differential of the Bar resolution of H^{∗}A. Hence the difference of (12.2) and (12.1) equals

(H^{∗}B ⊗ (u ∘ d_{3}) ⊗ H^{∗}B)(s, x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, t).

But now we are done since H^{∗}B ⊗ (u ∘ d_{3}) ⊗ H^{∗}B is a coboundary in the complex

Hom_{(H^{∗}B)^{e}}^{−1}(H^{∗}B ⊗_{H^{∗}A} 𝔹(H^{∗}A) ⊗_{H^{∗}A} H^{∗}B, H^{∗}B)

which computes the Hochschild cohomology HH^{∗,−1}(H^{∗}B). □

12.2. Local-global principle for the global obstruction. Throughout this section let A be a dg algebra over a field k and suppose that H^{∗}A is graded-commutative. Fix a graded prime ideal 𝔭 of H^{∗}A. In Theorem 9.12 we have shown the existence of a dg algebra A′ quasi-isomorphic to A and a zigzag of dg algebra maps

A ⟵^{∼} A′ ⟶^{φ} A_{𝔭}

which induces the canonical map can: H^{∗}A → (H^{∗}A)_{𝔭} in cohomology. Now we use this result to obtain more information about the global obstruction for the 𝔭-local modules.

Proposition 12.6. The map

Γ: HH^{∗,∗}(H^{∗}A′) ⟶ HH^{∗,∗}(H^{∗}A_{𝔭})

satisfies Γ(μ_{A′}) = μ_{A_{𝔭}}. Moreover, the composition

\xymatrix{HH^{∗,∗}(H^{∗}A) \ar[r]^{≅} & HH^{∗,∗}(H^{∗}A′) \ar[r]_{Γ} & HH^{∗,∗}(H^{∗}A_{𝔭})}

maps μ_{A} to μ_{A_{𝔭}}.

Proof. The dg algebra morphism φ: A′ → A_{𝔭} induces in cohomology the flat epimorphism of rings H^{∗}A → (H^{∗}A)_{𝔭}. The first claim then follows from Theorem 12.5. For the second just note that by Lemma 7.6, the isomorphism HH^{∗,∗}(H^{∗}A) ⟶^{≅} HH^{∗,∗}(H^{∗}A′) induced by the quasi-isomorphism A′ → A maps μ_{A} to μ_{A′}. □

Proposition 12.6 implies that if μ_{A} is trivial, then so is μ_{A_{𝔭}} for all graded primes 𝔭 of H^{∗}A. Observe that this was not clear before: if all H^{∗}A-modules are realisable, then so are in particular all 𝔭-local modules. But this does not imply that μ_{A_{𝔭}} is trivial, see Remark 10.3.

Under an additional assumption for H^{∗}A we now show that if μ_{A_{𝔭}} is trivial for all prime ideals 𝔭 of H^{∗}A, then μ_{A} is trivial.

Lemma 12.7. Let R be a graded-commutative algebra over a commutative ring k. Assume that M is a graded R^{e}-module which admits a resolution of finitely generated projective graded R^{e}-modules. Let N be any graded R^{e}-module. Then for all i ≥ 0, there is a natural isomorphism

Ext_{R^{e}}^{i,∗}(M, N) ⊗_{R^{e}} (R_{𝔭})^{e} ≅ Ext_{(R_{𝔭})^{e}}^{i,∗}(M ⊗_{R^{e}} (R_{𝔭})^{e}, N ⊗_{R^{e}} (R_{𝔭})^{e}).

Proof. Note first that R^{e} is graded-commutative and consequently, Ext_{R^{e}}^{i,∗}(M, N) is indeed a graded R^{e}-module.

The claim is immediately checked if M is a finitely generated graded free R^{e}-module and i = 0.

Now assume that M admits a resolution of finitely generated projective graded R^{e}- modules. Then M is in particular finitely presented and there exists an exact sequence

F_{1} ⟶^{f} F_{0} → M → 0,

where F_{0} and F_{1} are finitely generated graded free R^{e}-modules. The commutative diagram

\xymatrix{Hom_{R^{e}}^{∗}(F_{0}, N) ⊗_{R^{e}} (R_{𝔭})^{e} \ar[r]_{f^{∗}⊗_{R^{e}} (R_{𝔭})^{e}} \ar[d]^{≅} & Hom_{R^{e}}^{∗}(F_{1}, N) ⊗_{R^{e}} (R_{𝔭})^{e} \ar[d]^{≅} \\ Hom_{(R_{𝔭})^{e}}^{∗}(F_{0} ⊗_{R^{e}} (R_{𝔭})^{e}, N ⊗_{R^{e}} (R_{𝔭})^{e}) \ar[r]_{(f⊗_{R^{e}} (R_{𝔭})^{e})^{∗}} & Hom_{(R_{𝔭})^{e}}^{∗}(F_{1} ⊗_{R^{e}} (R_{𝔭})^{e}, N ⊗_{R^{e}} (R_{𝔭})^{e})}

and the exactness of − ⊗_{R^{e}} (R_{𝔭})^{e} ≅ R_{𝔭} ⊗_{R} − ⊗_{R} R_{𝔭} give the isomorphism

Hom_{R^{e}}^{∗}(M, N) ⊗_{R^{e}} (R_{𝔭})^{e} ≅ Hom_{(R_{𝔭})^{e}}^{∗}(M ⊗_{R^{e}} (R_{𝔭})^{e}, N ⊗_{R^{e}} (R_{𝔭})^{e}).

The claim for the Ext-groups follows from the condition that M admits a resolution of finitely generated projective R^{e}-modules. □

Remark 12.8. The assumptions of Lemma 12.7 are satisfied if R^{e} is Noetherian and M a finitely generated R^{e}-module. Note that it is not enough to assume that R is Noetherian. In general, we cannot expect a tensor product of two Noetherian algebras to be Noetherian: If F is a perfect field of characteristic p > 0 and k is an imperfect subfield of F, then the tensor product F ⊗_{k} F is not Noetherian, see [39, Sect. 1].

Before we prove the local-global principle, we establish a nice relation between the Hochschild groups of R and R_{𝔭}.

Proposition 12.9. Let R be a graded-commutative algebra over a field k. Suppose that the enveloping algebra R^{e} is Noetherian, and let 𝔭 be a graded prime ideal of R. For all n ≥ 0, we have

HH^{n,∗}(R_{𝔭}) ≅ HH^{n,∗}(R)_{𝔭}

as graded R-modules. In particular, a Hochschild group HH^{n,∗}(R) is trivial if and only if HH^{n,∗}(R_{𝔭}) is trivial for all graded prime ideals 𝔭 of R.

Proof. We first point out that

HH^{n,∗}(R_{𝔭}) ≅ R_{𝔭} ⊗_{R} HH^{n,∗}(R) ⊗_{R} R_{𝔭}

by Lemma 12.7. In order to show that R_{𝔭} ⊗_{R} HH^{n,∗}(R) ⊗_{R} R_{𝔭} is, in fact, just localisation of HH^{n,∗}(R) at 𝔭, we apply our results from Chapter 5.3. The Hochschild cohomology ring HH^{∗,∗}(R) is bigraded-commutative by Theorem 5.3 and since R is graded-commutative, Remark 5.4 implies that

HH^{0,∗}(R) = R.

Hence we have a well-defined R-linear map

ν: R_{𝔭} ⊗_{R} HH^{n,∗}(R) ⊗_{R} R_{𝔭} ⟶ HH^{n,∗}(R)_{𝔭},
\frac{r}{s} ⊗ ζ ⊗ \frac{r′}{s′} ⟼ (−1)^{|r||ζ|}ζ · \frac{rr′}{ss′}

and it is easy to check that ν is bijective by stating the obvious inverse map. □

Note that the denominators s, s′ do not need sign adjustment because we have chosen them to have even degree (see Section 3.1.2).

Now we are ready to prove Theorem 12.10 (Local-global principle). Let A be a differential graded algebra over a field k such that H^{∗}A is graded commutative. Assume that the enveloping algebra (H^{∗}A)^{e} is Noetherian. Then the following conditions are equivalent:

(1) μ_{A} ∈ HH^{3,−1}(H^{∗}A) is trivial.

(2) μ_{A_{𝔭}} ∈ HH^{3,−1}(H^{∗}A_{𝔭}) is trivial for all graded prime ideals 𝔭 of H^{∗}A.

(3) μ_{A_{𝔪}} ∈ HH^{3,−1}(H^{∗}A_{𝔪}) is trivial for all graded maximal ideals 𝔪 of H^{∗}A. In particular, all graded H^{∗}A-modules are realisable if the Hochschild class μ_{A_{𝔭}} is trivial for all graded prime ideals 𝔭 of H^{∗}A.

Proof. Fix a graded prime 𝔭 of H^{∗}A. We prove that under the isomorphism

HH^{3,∗}(H^{∗}A_{𝔭}) ≅ HH^{3,∗}(H^{∗}A)_{𝔭}

of Proposition 12.9, the class μ_{A_{𝔭}} is mapped to the fraction \frac{μ_{A}}{1} . This shows the claim.

Let A ⟵^{∼} A′ ⟶^{φ} A_{𝔭} be a zigzag of dg algebra maps which induces the canonical map H^{∗}A → (H^{∗}A)_{𝔭} in cohomology. Corollary 12.6 implies that in the Hochschild group HH^{3,−1}(H^{∗}A_{𝔭}), we have

μ_{A_{𝔭}} = [H^{∗}A_{𝔭} ⊗_{H^{∗}A′} m_{A′} ⊗_{H^{∗}A′} H^{∗}A_{𝔭}],

where m_{A′} is a representing cocycle for μ_{A′} ∈ HH^{3,−1}(H^{∗}A′). We conclude from Lemma 12.7 and Proposition 7.6 that under the composition of isomorphisms

HH^{3,∗}(H^{∗}A_{𝔭}) ≅ HH^{3,∗}(H^{∗}A_{𝔭} ⊗_{H^{∗}A′} H^{∗}A′ ⊗_{H^{∗}A′} H^{∗}A_{𝔭})
≅ H^{∗}A_{𝔭} ⊗_{H^{∗}A′} HH^{3,∗}(H^{∗}A′) ⊗_{H^{∗}A′} H^{∗}A_{𝔭}
≅ H^{∗}A_{𝔭} ⊗_{H^{∗}A} HH^{3,∗}(H^{∗}A) ⊗_{H^{∗}A} H^{∗}A_{𝔭},

the Hochschild class μ_{A_{𝔭}} maps to \frac{1}{1} ⊗ μ_{A} ⊗ \frac{1}{1}. But now we can apply the isomorphism

ν: H^{∗}A_{𝔭} ⊗_{H^{∗}A} HH^{3,∗}(H^{∗}A) ⊗_{H^{∗}A} H^{∗}A_{𝔭} ⟶ HH^{3,∗}(H^{∗}A)_{𝔭}

from the proof of Proposition 12.9. Since ν(\frac{1}{1} ⊗ μ_{A} ⊗ \frac{1}{1}) = \frac{μ_{A}}{1} , we have proved the claim. □

Remark 12.11. The reader might have noticed that the dg algebra A_{𝔭} was not shown to be uniquely determined up to quasi-isomorphism. The universal property we have proved in Section 9.2.1 only holds on the level of derived categories. Hence for another dg algebra A′_{𝔭} satisfying H^{∗}(A′_{𝔭}) ≅ (H^{∗}A)_{𝔭}, we obtain a canonical class μ_{A′_{𝔭}} which could behave differently.

However, what we have actually shown in Theorem 12.10 is that for every dg algebra A′_{𝔭} admitting a zigzag of dg algebra morphisms A ⟵^{∼} A′′ ⟶^{φ} A′_{𝔭} which induces can: H^{∗}A → (H^{∗}A)_{𝔭} in cohomology, the canonical class μ_{A′_{𝔭}} is the image of μ_{A} under the map

HH^{3,−1}(H^{∗}A) ⟶ HH^{3,−1}(H^{∗}A)_{𝔭}, ζ ⟼ \frac{ζ}{1}.

So the choice of the dg algebra inducing (H^{∗}A)_{𝔭} in cohomology is not relevant, as long as it admits such a zigzag.

Remark. One might want to have Proposition 12.9 and with it Theorem 12.10 under weaker assumptions. In Proposition 12.9 we have assumed that R^{e} is Noetherian to ensure that R admits a resolution of finitely generated R^{e}-projective modules. We do not know whether there is a way to avoid the latter condition.

13. Comparing realisability over group and Tate cohomology

Now we focus on realisability in group representation theory and compare realisability over the group cohomology ring and the Tate cohomology ring.

The group cohomology ring H^{∗}(G, k) has better properties than the Tate cohomology ring Ĥ^{∗}(G, k) which, for instance, is not Noetherian in general. However, when it comes to the source categories of realisability, the stable module category \underline{Mod} kG is more “handsome” than the homotopy category K(Inj kG). This is the reason why we are interested in studying the relation of realisability over group and Tate cohomology.

The triangulated categories K(Inj kG) and \underline{Mod} kG are related by a smashing localisation

\xymatrix{K(Inj kG) \ar[r]_{Q} & \underline{Mod} kG \ar[l]_{R}}

(see Proposition 8.11) and we are now concerned with finding a relation between realisability and this localisation of triangulated categories.

Remember that both H^{∗}(G, k) and Ĥ^{∗}(G, k) are the cohomology of a dg algebra (Example 10.5) and thus, they admit an A_{∞}-algebra structure yielding global obstructions which we now denote by μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) and μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)).

In the first section we study realisability of fixed modules. Then we focus on global realisability. The canonical class μ̂_{G} has been computed for some groups G by Benson, Krause and Schwede [5], and by Langer [37]. We consider the same groups and compute the global obstructions for the group cohomology rings. We will see in Section 13.2 that in all but one case, the Hochschild classes μ_{G} and μ̂_{G} turn out to behave surprisingly similar. As a first explication for this similarity we show in Section 13.3 that the algebra morphism H^{∗}(G, k) → Ĥ^{∗}(G, k) is induced by a zigzag of dg algebra morphisms. The main result of this chapter is stated in the last section and gives a complete explanation for the relation between μ_{G} and μ̂_{G} we observed before in examples.

We like to thank Dave Benson for discussions that helped to improve this chapter and in particular, for a contribution to the proof of Theorem 13.21.

13.1. Local realisability. Let k be a field of characteristic p > 0 and G be a finite group such that p divides the order of G. In this section we discuss ways to construct a realisable H^{∗}(G, k)-module from a realisable module over H^{∗}(G, k), and vice versa.

The following proposition will play an important role in the next section.

Proposition 13.1. Assume that the p-rank of G equals one.

(1) If X ∈ Mod_{gr} H^{∗}(G, k) is realisable, then X ⊗_{H^{∗}(G,k)} Ĥ^{∗}(G, k) is a realisable Ĥ^{∗}(G, k)-module.

(2) A graded Ĥ^{∗}(G, k)-module Y is realisable if and only if its restriction to H^{∗}(G, k) is a realisable H^{∗}(G, k)-module.

Proof. By Lemma 4.4 and Theorem 4.5, there exists a multiplicative subset S ⊆ H^{∗}(G, k) such that Ĥ^{∗}(G, k) = S^{−1}H^{∗}(G, k). Hence we may apply Theorem 8.12 and Remark 8.13 and obtain commutative diagrams

\xymatrix{K(Inj kG) \ar[d]_{Q} \ar[r]^{Hom_{K}(ik,−)^{∗}} & Mod_{gr} H^{∗}(G, k) \ar[d]^{−⊗_{H^{∗}(G,k)}Ĥ^{∗}(G,k)} & K(Inj kG) \ar[r]^{Hom_{K}(ik,−)^{∗}} & Mod_{gr} H^{∗}(G, k) \\ \underline{Mod} kG \ar[r]^{\underline{Hom}_{kG}(k,−)^{∗}} & Mod_{gr} Ĥ^{∗}(G, k) & \underline{Mod} kG \ar[u]_{R} \ar[r]^{\underline{Hom}_{kG}(k,−)^{∗}} & Mod_{gr} Ĥ^{∗}(G, k) \ar[u]^{res}}

which yield the claim. □

If the p-rank of G is at least two, then we cannot expect to obtain realisable modules from realisable modules by induction or restriction as above. The reason for this is that in general, the induction functor − ⊗_{H^{∗}(G,k)} Ĥ^{∗}(G, k) is not given by localisation with respect to a multiplicatively closed subset.

However, there is another possibility to obtain a realisable H^{∗}(G, k)-module from a realisable Ĥ^{∗}(G, k)-module. This construction also works in the general case.

Lemma 13.2. Let X ∈ Mod_{gr} Ĥ^{∗}(G, k) realisable. Then its truncation to non-negative degrees X^{⩾0} is a realisable H^{∗}(G, k)-module.

Proof. Let X be a direct summand of Ĥ^{∗}(G, M), where M is a kG-module. Without loss of generality, we may assume that M does not have any projective direct summands. Then Ĥ^{0}(G, M) ≅ H^{0}(G, M) and consequently, Ĥ^{∗}(G, M)^{⩾0} ≅ H^{∗}(G, M). It follows that X^{⩾0} is a direct summand of H^{∗}(G, M). □

13.2. Examples for the global obstruction. Let k be a field of characteristic p > 0. We study the group and Tate cohomology rings of cyclic p-groups and the Quaternion group, and focus on the global obstructions μ_{G} and μ̂_{G}.

Theorem 13.3. [5, Thm. 7.1] Let G be a cyclic group of order p^{n}, where n ≥ 1. If p^{n} = 2, then the Tate cohomology ring is a Laurent polynomial ring k[X, X^{−1}] on a 1-dimensional class X. If p^{n} ≥ 3, then the Tate cohomology ring is a truncated Laurent polynomial ring in two variables,

Ĥ^{∗}(G, k) = k[X, Y, Y^{−1}]/(X^{2}),

with deg(X) = 1 and deg(Y) = 2. The secondary multiplication m_{3} of the A_{∞}-algebra Ĥ^{∗}(G, k) and thus μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)) is trivial except when p = 3 and n = 1. In this case, the secondary multiplication is given by

m_{3}(XY^{i}, XY^{j}, XY^{l}) = Y^{i+j+l+1}, i, j, l ∈ ℤ,

and vanishes on all other tensor products of monomials. Furthermore, its Hochschild class μ̂_{ℤ/(3)} is non-trivial.

Note that strictly speaking, one can make choices to obtain m_{3} in the shape as stated above. With other choices of f_{1} and f_{2} in Construction 7.1 we could obtain a different map. However, the Hochschild class μ̂_{G} of m_{3} is independent of all choices.

Now we consider the group cohomology ring H^{∗}(G, k) of a cyclic group G as above. It identifies with the subring of non-negative degrees of Ĥ^{∗}(G, k). The global obstruction μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) turns out to behave very similarly:

Proposition 13.4. Let G be a cyclic group of order p^{n}, where n ≥ 1. If p^{n} = 2, then the group cohomology ring is a polynomial ring k[X], where X has degree 1. If p^{n} ≥ 3, then H^{∗}(G, k) is a truncated polynomial ring in two variables,

H^{∗}(G, k) = k[X, Y]/(X^{2}),

with deg(X) = 1 and deg(Y) = 2. The secondary multiplication m_{3} of the A_{∞}-algebra H^{∗}(G, k) and thus μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) is trivial except when p = 3 and n = 1. In this case, it satisfies

m_{3}(XY^{i}, XY^{j}, XY^{l}) = Y^{i+j+l+1}, i, j, l ≥ 0,

and vanishes on all other tensor products of monomials. Its Hochschild class μ_{ℤ_{3}} is non-trivial.

Proof. Since the characteristic of k is positive, we may identify kG with the truncated polynomial ring K[T]/(T^{r}), where r = p^{n}. An injective resolution of k which is moreover 2-periodic is given by

\xymatrix{ik: & ⋯ \ar[r] & 0 \ar[r] & 0 \ar[r] & I_{0} \ar[r]^{T} & I_{1} \ar[r]^{−T^{r−1}} & I_{2} \ar[r]^{T} & I_{3} \ar[r]^{−T^{r−1}} & ⋯}

Let x: ik → Σik be the degree-one chain map

\xymatrix{ik \ar[d]^{x} & ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \ar[r] \ar[d] & I_{0} \ar[r]^{T} \ar[d]^{1} & I_{1} \ar[r]^{−T^{r−1}} \ar[d]^{T^{r−2}} & I_{2} \ar[r]^{T} \ar[d]^{1} & I_{3} \ar[d]^{T^{r−2}} \ar[r]^{−T^{r−1}} & ⋯ \\ Σik & ⋯ \ar[r] & 0 \ar[r] & I_{0} \ar[r]_{−T} & I_{1} \ar[r]_{T^{r−1}} & I_{2} \ar[r]_{−T} & I_{3} \ar[r]_{T^{r−1}} & I_{4} \ar[r]_{−T} & ⋯}

and y: ik → Σ^{2}ik the degree-two chain map

\xymatrix{ik \ar[d]^{y} & ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \ar[r] \ar[d] & I_{0} \ar[r]^{T} \ar[d]^{1} & I_{1} \ar[r]^{−T^{r−1}} \ar[d]^{1} & I_{2} \ar[r]^{T} \ar[d]^{1} & ⋯ \\ Σ^{2}ik & ⋯ \ar[r] & I_{0} \ar[r]_{T} & I_{1} \ar[r]_{−T^{r−1}} & I_{2} \ar[r]_{T} & I_{3} \ar[r]_{−T^{r−1}} & I_{4} \ar[r]_{T} & ⋯}

One easily checks that xy = yx. If r ≥ 3, then x^{2} is nullhomotopic by the homotopy q, given by multiplication with T^{r−3} in odd degrees, and the zero map in even degrees. If r = 2, then obviously x^{2} = y. We infer that x and y are cycles of End(ik) representing the classes X ∈ H^{1}(G, k) and Y ∈ H^{2}(G, k), respectively.

In order to compute the secondary multiplication, we define a cycle selection homomorphism

f_{1} : H^{∗} End(ik) → End(ik)

on the k-basis {X^{ϵ}Y^{i} | ϵ ∈ {0, 1}, i ≥ 0} of H^{∗}(G, k), given by

f_{1}(X^{ϵ}Y^{i}) = x^{ϵ}y^{i}.

If r = 2, then this map is multiplicative and it follows that f_{2} and with it m_{3} can be chosen to be trivial. If r ≥ 3, then f_{1} is multiplicative except on two odd dimensional classes. The product f_{1}(XY^{i})f_{1}(XY^{j}) = x^{2}y^{i+j} is only nullhomotopic, by the homotopy qy^{i+j}.

We define

f_{2} : H^{∗} End(ik) ⊗ H^{∗} End(ik) → End(ik)

to be trivial except on two odd dimensional classes: In this case, we set

f_{2}(XY^{i}, XY^{j}) = qy^{i+j}.

Since m_{3} maps (A, B, C) to the cohomology class of the expression

(13.1)  (−1)^{|A|}f_{1}(A)f_{2}(B, C) − f_{2}(A, B)f_{1}(C) − f_{2}(AB, C) + f_{2}(A, BC)

(see (7.6)), we infer that m_{3} vanishes on all tensor product of monomials with at least one monomial having even degree. Using the fact that the homotopy q commutes with y, one checks that the tensor product of three odd degree monomials (XY^{i}, XY^{j}, XY^{l}) is mapped under (13.1) to

(qx + xq)y^{i+j+l}.

The chain map qx + xq: ik → Σ^{2}ik is given by multiplication with T^{r−3} in each degree. Now if r > 3, then qx + xq is nullhomotopic via the homotopy given by the zero map in even degrees, and by multiplication with T^{r−4} in odd degrees. Thus m_{3} = 0. But if r = 3, then qx + xq = y and we conclude

m_{3}(XY^{i}, XY^{j}, XY^{l}) = Y^{i+j+l+1}.

It remains to show that also the Hochschild class μ_{ℤ_{3}} is non-trivial. But this is is a consequence of the following remark. □

Remark 13.5. Using Massey Products, Benson, Krause and Schwede proved that Ĥ^{∗}(ℤ_{3}, k)/(X) is a non-realisable Ĥ^{∗}(ℤ_{3}, k)-module [5, Exm. 7.6]. From Proposition 13.1 we conclude that Ĥ^{∗}(ℤ_{3}, k)/(X), viewed as H^{∗}(ℤ_{3}, k)-module, is not realisable either.

Theorem 13.6. [37, Satz 2.10] Denote by Q_{8} be the quaternion group and let k be a field of characteristic two. Then the Tate cohomology ring Ĥ^{∗}(Q_{8}, k) is given by

k[X, Y, S^{±1}]/(X^{2} + Y^{2} = XY, X^{3} = Y^{3} = 0, X^{2}Y = XY^{2}),

with |X| = |Y | = 1, |S| = 4. The canonical class μ̂_{Q_{8}} is non-trivial.

Furthermore, Langer shows the existence of a non-realisable module [37, Lemma 2.23]: Write Ĥ for Ĥ^{∗}(Q_{8}, k). The cokernel of the map

g: Ĥ[−1] ⊕ Ĥ[−1] \underrightarrow{[ \begin{matrix}Y & X+Y \\ X & Y \end{matrix} ]} Ĥ ⊕ Ĥ

is not a realisable module.

Since r_{2}(Q_{8}) = 1 (see Theorem 4.6), it follows from Proposition 13.1 that Coker g, viewed as module over H^{∗}(Q_{8}, k), is not realisable. We obtain

Corollary 13.7. Denote by Q_{8} be the quaternion group and let k be a field of characteristic two. Then

H^{∗}(Q_{8}, k) = k[X, Y, S]/(X^{2} + Y^{2} = XY, X^{3} = Y^{3} = 0, X^{2}Y = XY^{2}),

with |X| = |Y | = 1, |S| = 4. The canonical class μ_{Q_{8}} is non-trivial. □

One might wonder whether it always holds true that μ_{G} is trivial if and only if μ̂_{G} is trivial. But in general, this is not the case. For any finite abelian 2-group G, the class μ_{G} is trivial by Proposition 13.4 and Remark 7.8. However, the situation for Tate cohomology is entirely different when it comes to the Klein four group:

Theorem 13.8. [5, Exm. 7.7] [37, Thm. 3.1] Let G = ℤ_{2} × ℤ_{2} and k be a field with Char(k) = 2. Then μ̂_{ℤ_{2}} is non-trivial. For any other finite abelian 2-group, the canonical class is trivial.

13.2.1. Reduction to Sylow subgroups. Let k be a field of characteristic p > 0 and G be a finite group such that p divides the order of G. Let P a p-Sylow subgroup of G. Benson, Krause and Schwede [5] have shown that the canonical class μ̂_{G} is already determined by μ̂_{P}. In order to compare μ_{G} and μ̂_{G} in more cases, we now briefly explain why the same holds true for μ_{G} and μ_{P}.

Let P be a subgroup of G. If M, N are kG-modules, we define the transfer or trace map

tr: Hom_{kP} (M, N) → Hom_{kG}(M, N)

as follows:

tr_{P,G}(Φ)(m) = ∑_{i∈I} g_{i}Φ(g_{i}^{−1}m),

where {g_{i} | i ∈ I} is any choice of left coset representatives of P in G. See [2, Ch. 3.6] for details. The transfer tr induces a well-defined map

tr: Ĥ^{n}(P, k) → Ĥ^{n}(G, k)

for n ∈ ℤ, independent on the choice of the resolution [2, Lemma 3.6.16].

By considering a kG-Tate resolution of the trivial module as kP -Tate resolution of k regarded as kP -module, we obtain for n ∈ ℤ a restriction map

res: Ĥ^{n}(G, k) → Ĥ^{n}(P, k).

Lemma 13.9. [2, Ch. 3.6] The composition tr ∘ res: Ĥ^{n}(G, k) → Ĥ^{n}(G, k) is given by multiplication with [G : P].

Note that the restriction map induces morphisms of graded algebras res: H^{∗}(G, k) → H^{∗}(P, k) and res: Ĥ^{∗}(G, k) → Ĥ^{∗}(P, k). Moreover, if P is a p-Sylow subgroup of G, then these algebra morphisms are split monomorphisms by Lemma 13.9.

Theorem 13.10. [5, Thm. 8.3] Let k be a field of characteristic p > 0, G a finite group and P a p-Sylow subgroup of G. Then the canonical class μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)) is determined by the canonical class μ̂_{P} ∈ HH^{3,−1}(Ĥ^{∗}(P, k)) of the Sylow subgroup, and is given by the formula

μ̂_{G} = \frac{tr ∘μ̂_{P} ∘ res^{⊗3}}{[G : P]} .

The key point of the proof is that the restriction map in Tate cohomology is induced by a morphism of dg algebras. Since similarly, res: H^{∗}(G, k) → H^{∗}(P, k) is induced by the inclusion dg algebras

End_{kG}(ik) → End_{kP} (ik)

given by viewing the kG-injective resolution ik as injective resolution over kP, we obtain

Lemma 13.11. Let k be a field of characteristic p > 0, G a finite group and P a p-Sylow subgroup of G. Then for the canonical classes μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)) and μ_{P} ∈ HH^{3,−1}(H^{∗}(P, k)), it holds

μ_{G} = \frac{tr ∘μ_{P} ∘ res^{⊗3}}{[G : P]} .  □

Benson, Krause and Schwede concluded in particular that if G is a group whose p-Sylow group is cyclic with order different from 3, then μ̂_{G} is trivial [5, Cor. 8.4]. We infer from Lemma 13.11 and Proposition 13.4 that the same holds true for μ_{G}.

Corollary 13.12. Let k be a field of characteristic p > 0 and G be a group whose p-Sylow subgroup is cyclic of order p^{n}, n ≥ 1. Suppose that n ≥ 2 if p = 3. Then both the Hochschild classes μ_{G} and μ̂_{G} are trivial.

In order to investigate the case P = ℤ/(3), one needs to distinguish whether G is 3-nilpotent or not:

Definition 13.13. Let G be a finite group and let P be a p-Sylow subgroup. G is called p-nilpotent if there exists a normal subgroup U ⊴ G such that the composite

P ⟶^{incl} G ⟶^{proj} G/U

is an isomorphism.

Proposition 13.14. [5, Prop. 8.5] Let k be a field of characteristic p > 0 and G be a finite group with p-Sylow subgroup P. The following conditions are equivalent:

(1) res: Ĥ^{∗}(G, k) → Ĥ^{∗}(P, k) is an isomorphism.

(2) res: H^{∗}(G, k) → H^{∗}(P, k) is an isomorphism.

(3) res: H^{1}(G, k) → H^{1}(P, k) is an isomorphism.

(4) The group G is p-nilpotent.

Since both the restriction maps Ĥ^{∗}(G, k) → Ĥ^{∗}(P, k) and H^{∗}(G, k) → H^{∗}(P, k) are induced by a morphism of dg algebras, we obtain as a consequence of Proposition 7.6

Corollary 13.15. Let k be a field of characteristic 3 and G a finite, 3-nilpotent group whose 3-Sylow group is cyclic of order 3. Then both the Hochschild classes μ_{G} and μ̂_{G} are non-trivial.

For μ̂_{G}, the result above is stated in [5, Sect. 8].

Theorem 13.16. [5, Thm. 8.6] Let k be a field of characteristic 3 and G a finite group whose 3-Sylow group is cyclic of order 3. Assume that G is not 3-nilpotent. Then

Ĥ^{∗}(G, k) = k[V, W, W^{−1}]/(V^{2}),

where V is of degree 3 and W of degree 4. The canonical class μ̂_{G} is represented by the (3, −1)-cocycle m given by

m(V W^{i}, V W^{j}, V W^{l}) = W^{i+j+l+2}, i, j, l ∈ ℤ,

and vanishes on all other tensor products of monomials in V and W. The Hochschild class μ̂_{G} is non-trivial.

In [5], it is further shown that Ĥ^{∗}(G, k)/(V) is not realisable. We conclude from Proposition 13.1 that V viewed as module over H^{∗}(G, k) is not realisable either. In particular, the canonical class μ_{G} must be non-trivial. Using Lemma 13.11, we can compute a representing cocycle for the canonical class, using the same methods as in [5, Thm. 8.6].

Proposition 13.17. Let k be a field of characteristic 3 and G a finite group whose 3-Sylow group is cyclic of order 3. Assume that G is not 3-nilpotent. Then

H^{∗}(G, k) = k[V, W]/(V^{2})

where V is of degree 3 and W of degree 4. A representing (3, −1)-cocycle m for the canonical class μ_{G} is given by

m(V W^{i}, V W^{j}, V W^{l}) = W^{i+j+l+2}, i, j, l ≥ 0,

and vanishes on all other tensor products of monomials in V and W. Its Hochschild class μ̂_{G} is non-trivial. □

13.3. Lifting H^{∗}(G, k) → Ĥ^{∗}(G, k) to a morphism of dg algebras. Let G be a finite group and k a field. In the examples we have considered in the last section, the class μ_{G} always arises as restriction of μ̂_{G} to non-negative degrees. Now we give a explanation for this fact: We show that the canonical inclusion

ι: H^{∗}(G, k) → Ĥ^{∗}(G, k)

is induced by a zigzag of dg algebra morphisms. The construction is analogous to the one in Section 9. However, for the convenience of the reader we sketch it briefly. Let η: id → RQ be the unit and ε: QR → id the counit of the adjunction

\xymatrix{K(Inj kG) \ar[r]_{Q} & K_{ac}(Inj kG), \ar[l]_{R}}

where Q = − ⊗ tk and R is the inclusion. This is a smashing localisation by Proposition 8.11.

The map

η_{ik}: ik → RQ(ik)

is up to natural isomorphism just the canonical inclusion ik → tk, see Proposition 8.11. Therefore the map

K(Inj kG)(ik, ik)^{∗} → K(Inj kG)(RQ(ik), RQ(ik))^{∗}, f ↦ RQ(f)

is up to isomorphism the canonical inclusion

ι: H^{∗}(G, k) → Ĥ^{∗}(G, k).

We have K(Inj kG)(ik, RQ(ik)) ≅ H^{0}(Hom(ik, RQ(ik)). For any representing cocycle η̄_{ik} ∈ Z^{0}(Hom(ik, RQ(ik)) of η_{ik} we obtain as in Lemma 9.5 a quasi-isomorphism

η̄_{ik}^{∗}: End(RQ(ik)) → Hom(ik, RQ(ik)), f ↦ f ∘ η̄_{ik}.

We choose for η̄_{ik} the inclusion of complexes ik → tk which is a degree-wise split monomorphism of complexes. Thus η̄_{ik}^{∗} is surjective.

Theorem 13.18. There exists a dg algebra End(ik)′ quasi-isomorphic to End(ik) and a zigzag of dg algebra morphisms

\xymatrix{End(ik) & End(ik)′ \ar[l]_{ρ}^{∼} \ar[r]^{φ} & End(tk)}

inducing the canonical inclusion ι: H^{∗}(G, k) → Ĥ^{∗}(G, k) in cohomology. That is, we have diagrams

\xymatrix{End(ik)′ \ar[d]^{∼}_{ρ} \ar[dr]^{φ} &  & H^{∗}(G, k) \ar[d]^{≅}_{H^{∗}ρ} \ar[dr]^{H^{∗}φ} \\ End(ik) & End(tk) & H^{∗}(G, k) \ar[r]^{ι} & Ĥ^{∗}(G, k)}

where right hand diagram is commutative and identifies with the cohomology of the left hand diagram.

Proof. We form the pullback diagram

\xymatrix{End(ik)′ \ar[r]^{p_{2}} \ar[d]_{p_{1}} & End(RQ(ik)) \ar@{->>}[d] \ar[d]_{∼}^{η̄_{ik}^{∗}} \\ End(ik) \ar[r]^{η̄_{ik ∗}} & Hom(ik, RQ(ik))}

Since η̄_{ik}^{∗} is a surjective quasi-isomorphism, it follows from Lemma 9.1 that End(ik)′ is a dg algebra quasi-isomorphic to End(ik).

In cohomology, we obtain a commutative diagram

\xymatrix{H^{∗} End(ik)′ \ar[r]^{H^{∗}(p_{2})} \ar[d]_{H^{∗}(p_{1})}^{≅} & H^{∗} End(RQ(ik)) \ar[d]_{≅}^{H^{∗}(η̄_{ik}^{∗})} \\ H^{∗} End(ik) \ar[r]^{H^{∗}(η̄_{ik ∗})} & H^{∗}(Hom(ik, RQ(ik))}

where the composition

H^{∗}(η̄_{ik}^{∗})^{−1}H^{∗}(η̄_{ik ∗}): K(Inj kG)(ik, ik)^{∗} → K(Inj kG)(RQ(ik), RQ(ik))^{∗}

is given by applying the functor RQ to a map f ∈ K(Inj kG)(ik, ik)^{∗}. But this is up to isomorphism just the inclusion ι: H^{∗}(G, k) → Ĥ^{∗}(G, k). □

Observe that the map φ: End(ik)′ → End(tk) is a monomorphism which moreover induces a monomorphism in cohomology.

Now we show that the relation between μ_{G} and μ̂_{G} which we have observed in the examples is true in general.

Proposition 13.19. In the Hochschild group HH^{3,−1}(H^{∗}(G, k), Ĥ^{∗}(G, k)), it holds

ι ∘ μ_{G} = μ̂_{G} ∘ ι^{⊗3},

where ι: H^{∗}(G, k) → Ĥ^{∗}(G, k) is the canonical inclusion.

Proof. We use the zigzag

\xymatrix{End(ik) & End(ik)′ \ar[l]_{ρ}^{∼} \ar[r]^{φ} & End(tk)}

of Theorem 13.18. By Proposition 7.6, we have

(13.2) H^{∗}ρ ∘ μ_{End(ik)′} = μ_{End(ik)} ∘ (H^{∗}ρ)^{⊗3} in HH^{3,−1}(H^{∗} End(ik)′, H^{∗} End(ik))

and

(13.3) H^{∗}φ ∘ μ_{End(ik)′} = μ_{End(tk)} ∘ (H^{∗}φ)^{⊗3} in HH^{3,−1}(H^{∗} End(ik)′, H^{∗} End(tk)).

Hence in the Hochschild group HH^{3,−1}(H^{∗}(G, k), Ĥ^{∗}(G, k)), it holds

ι ∘ μ_{End(ik)} = μ_{End(tk)} ∘ ι^{⊗3}.  □

Remark 13.20. Even more is true: Since both ρ and φ induce monomorphisms in cohomology, there exist choices in defining the secondary multiplications of H^{∗} End(ik) and H^{∗} End(tk) to obtain the equations (13.2) and (13.3) even on the level of k-linear maps, see Proposition 7.7. With these choices, we have

ι ∘ m_{3}^{H^{∗}(G,k)} = m_{3}^{Ĥ^{∗}(G,k)} ∘ ι^{⊗3} in Hom_{k}^{−1}(H^{∗}(G, k)^{⊗3}, Ĥ^{∗}(G, k)).

The secondary multiplication m_{3}^{H^{∗}(ℤ_{3},k)} we computed in Proposition 13.4 and m_{3}^{Ĥ^{∗}(ℤ_{3},k)}

of Theorem 13.3 satisfy this equation. However, we cannot just restrict a fixed m_{3}^{Ĥ^{∗}(G,k)} to non-negative degrees to obtain m_{3}^{H^{∗}(G,k)}.

13.4. Relating the global obstructions of H^{∗}(G, k) and Ĥ^{∗}(G, k). In all but one example we have considered in the first section, the Hochschild classes μ_{G} and μ̂_{G} were either both trivial or both non-trivial. Now we are ready to give a general explanation for this fact. The main result of this chapter is

Theorem 13.21. Let G be finite group, k a field of characteristic p > 0 and assume that p divides the order of G. If the Hochschild class μ̂_{G} ∈ HH^{3,−1}(Ĥ^{∗}(G, k)) is trivial, then so is the Hochschild class μ_{G} ∈ HH^{3,−1}(H^{∗}(G, k)). If the p-rank of G equals one, then μ̂_{G} is trivial if and only if μ_{G} is trivial.

Proof. Let m: H^{∗}(G, k)^{⊗3} → H^{∗}(G, k) be any (3, −1)-cocycle representing μ_{G}. In the Hochschild group HH^{3,−1}(H^{∗}(G, k), Ĥ^{∗}(G, k)), we have the equation

ι ∘ μ_{G} = μ̂_{G} ∘ ι^{⊗3}

by Proposition 13.19. Hence if the Hochschild class μ̂_{G} is trivial, then the k-linear map ι ∘ m is a (3, −1)-Hochschild coboundary, say

ι ∘ m = δg,

where g ∈ Hom_{k}^{−1}(H^{∗}(G, k)^{⊗2}, Ĥ^{∗}(G, k)).

If g(1, 1) = 0, then we actually have that g ∈ Hom_{k}^{−1}(H^{∗}(G, k)^{⊗2}, H^{∗}(G, k)) and

m = δg.

Thus μ_{G} is trivial in this case.

If g(1, 1) ≠ 0, then we choose any map u ∈ Hom_{k}^{−1}(H^{∗}(G, k), Ĥ^{∗}(G, k)) satisfying u(1) ≠ 0. Since Ĥ^{−1}(G, k) is one-dimensional by Tate duality (see Proposition 4.2), there exists an element κ ∈ k such that g(1, 1) = κ · u(1). Setting

g′ = g − κ · δu,

we obtain

ι ∘ m = δg′,

and since g′(1, 1) = 0, we conclude that μ_{G} is trivial.

If the p-rank of G equals one, then the inclusion ι: H^{∗}(G, k) → Ĥ^{∗}(G, k) is a flat epimorphism of rings by Lemma 4.4. Moreover, it is induced by a zigzag of dg algebra morphisms by Theorem 13.18. Thus we can apply Theorem 12.5 and conclude that the map Γ: HH^{∗,∗}(H^{∗}(G, k)) → HH^{∗,∗}(Ĥ^{∗}(G, k)) satisfies Γ(μ_{G}) = μ̂_{G}. In particular, μ_{G} being trivial implies that μ̂_{G} is trivial. □

Note that the second statement of the theorem is not true if r_{p}(G) ≥ 2: If G is the Klein four group, then μ_{G} is trivial but μ̂_{G} is not, see Section 13.2.

Remark. In order to prove that μ_{G} is non-trivial in Proposition 13.4, Corollary 13.7 and Proposition 13.17, we have shown the existence of a non-realisable module over H^{∗}(G, k). But in general, one cannot expect to have a non-realisable module whenever the global obstruction is non-trivial: Benson, Krause and Schwede provide an example of a dg algebra A such that the canonical class μ_{A} ∈ HH^{3,−1}(H^{∗}A) is non-trivial, but such that all H^{∗}A-modules are realisable [5, Prop. 5.16].

However, in all known examples of non-trivial global obstructions for group or Tate cohomology, the existence of a non-realisable module could always be shown. Benson, Krause and Schwede use Massey products to show the existence of a non-realisable module over Ĥ^{∗}(ℤ_{3}, k) and Ĥ^{∗}(ℤ_{2} × ℤ_{2}, k) [5, Exm. 6.7, Exm. 7.7]. Langer has used Matrix Massey products [37, Lemma 2.23] to construct a non-realisable module over Ĥ^{∗}(Q_{8}, k).

It is an open question whether for group and Tate cohomology, one can expect to have a non-realisable module whenever the global obstruction is non-trivial.

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