SUPPORT VARIETIES – AN IDEAL APPROACH

ASLAK BAKKE BUAN, HENNING KRAUSE, AND ØYVIND SOLBERG

Contents

Introduction 1 1. The prime spectrum of an ideal lattice 3 2. The prime spectrum is spectral 6 3. An adjoint of the functor Spec 7 4. Support data 9 5. Classifying support data 11 6. Thick tensor ideals 11 7. The structure sheaf of a tensor category 15 8. Applications to schemes 16 9. A projective scheme 18 10. Decompositions of ideals 23 References 24

Introduction

The spectrum of prime ideals and the support of objects like modules, sheaves, complexes etc. belong to the fundamental concepts from algebraic geometry. In fact, the use of these concepts is not restricted to algebraic geometry and similar notions exist for instance in modular representation theory. In this paper we discuss a general approach which allows us to study prime ideal spectra and supports in various settings.

A prime ideal spectrum comes naturally equipped with a topology which is usually called Zariski topology. However, there are various instances where it is more natural to consider another ‘opposite’ topology. It is one of the principal aims of this work to clarify the parallel use of two different topologies on a prime ideal spectrum. This is based on the notion of a spectral space, first introduced by Hochster [8].

We give a couple of motivating examples which illustrate the use of such different topologies. Let A be a commutative ring and denote by Spec A the set of prime ideals, together with the usual Zariski topology. We obtain another topology and write Spec^{∗} A if we take the quasi-compact Zariski open sets as a basis of closed sets. Now suppose that A is noetherian. Let mod A denote the abelian category of all finitely generated A-modules. Then the assignment

mod A ⊇ C ↦ ⋃_{M∈C} supp M

Version from August 15, 2005.

induces a bijection between all Serre subcategories of mod A and all open subsets of Spec^{∗} A; see [6]. Our second example arises from Ziegler’s work on the model theory of modules [18]. The points of the Ziegler spectrum of A are the isomorphism classes of indecomposable pure-injective A-modules and the closed subsets correspond to complete theories of modules. The indecomposable injective modules form a closed subset Inj A, and the assignment

Spec^{∗} A ∋ 𝔭 ↦ E(A/𝔭) = injective envelope of A/𝔭

induces a bijection between all closed subsets of Spec^{∗} A and all Ziegler closed subsets contained in Inj A; see [13]. We do not comment any further on the second example, but the first example is explained in some detail when we discuss the abelian category of coherent sheaves and the triangulated category of perfect complexes on a scheme.

Now let us give a brief outline of the contents of this paper. At the beginning we introduce the notion of an ideal lattice and study its prime ideal spectrum. We assign to each ideal its support and can classify ideals in terms of their support. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. Examples of ideal lattices arise from tensor categories which are abelian or triangulated. We provide a systematic treatment of such tensor categories and discuss a number of examples from algebraic geometry and modular representation theory. This is inspired by recent work of Balmer [1].

To be more specific, let C be an abelian or triangulated tensor category with a tensor identity. We consider the ideal lattice of thick tensor ideals of C and its prime ideal spectrum Spec C. This space comes naturally equipped with a sheaf of rings O_{C} and we can describe the ringed space (Spec C, O_{C}) in some interesting examples. An important application says that every quasi-compact and quasi-separated scheme X can be reconstructed from the triangulated tensor category of perfect complexes on X. This is a slight generalization of a result of Balmer [1] and based on the fundamental work of Thomason [16, 17]. On the other hand, it is the analogue – with almost identical proof – of the fact that a noetherian scheme X can be reconstructed from the abelian tensor category of coherent sheaves on X.

There are a number of interesting examples of triangulated tensor categories C where (Spec C, O_{C}) is actually a projective scheme. We provide a general criterion which explains those examples. For instance, this result establishes for a finite group G and a field k a conceptual link between the identical classifications of

– thick tensor ideals of the category of perfect complexes over the graded commutative cohomology ring H^{∗}(G, k), due to Hopkins and Neeman [9, 12], and

– thick tensor ideals of the stable category of finite dimensional k-linear represen-

tations of G, due to Benson, Carlson, and Rickard [3]. This generalizes to finite group schemes, by recent work of Friedlander and Pevtsova [5].

Our personal motivation for this project stems from the work on support varieties in non-commutative settings, for instance for modular representations of finite dimensional algebras. It turns out that most parts of our theory do not require any commutativity assumptions. However, the product formula

supp(ab) = supp(a) ∩ supp(b) = supp(ba)

for the support of two ideals a, b shows that commutativity is inherent to the subject, even though we allow ab ≠ ba.

Acknowldegements. The authors are grateful to Paul Balmer for various helpful comments on this work.

1. The prime spectrum of an ideal lattice

1.1. Ideal lattices. In this section we introduce the notion of an ideal lattice. The collection of ideals of some fixed algebraic structure is usually equipped with two additional structures. We consider the partial ordering by inclusion and the internal multiplication.

Definition 1.1. An ideal lattice is by definition a partially ordered set L = (L, ⩽), together with an associative multiplication L × L → L, such that the following holds.

(L1) The poset L is a complete lattice, that is,

⋁_{a∈A} a = sup A and ⋀_{a∈A} a = inf A

exist in L for every subset A ⊆ L. (L2) The lattice L is compactly generated, that is, every element in L is the supremum

of compact elements. (An element a ∈ L is compact, if for all A ⊆ L with a ⩽ sup A there exists some finite A′ ⊆ A with a ⩽ sup A′.)

(L3) We have for all a, b, c ∈ L

a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc.

(L4) The element 1 = sup L is compact, and 1a = a = a1 for all a ∈ L.

(L5) The product of two compact elements is again compact. A morphism ϕ: L → L′ of ideal lattices is a map satisfying

ϕ( ⋁_{a∈A} a) = ⋁_{a∈A} ϕ(a) for A ⊆ L,

ϕ(1) = 1 and ϕ(ab) = ϕ(a)ϕ(b) for a, b ∈ L.

It is useful to think of a poset L as a category L where the objects of L are the elements of L and

Hom_{L}(a, b) ≠ ∅ ⟺ card Hom_{L}(a, b) = 1 ⟺ a ⩽ b

for a, b ∈ L. Note that infimum and supremum in L correspond to product and coproduct, respectively, in L. Thus a compactly generated complete lattice is precisely a locally finitely presentable category L (in the sense of [7]) satisfying card Hom_{L}(a, b) ⩽ 1 for all a, b ∈ L. Given an ideal lattice L, the multiplication L × L → L corresponds to a tensor product L × L → L. A morphism L → L′ of ideal lattices corresponds to a functor L → L′ preserving all colimits and the tensor product.

Next observe that an ideal lattice L is essentially determined by its subset L^{c} of compact elements. To make this precise, let K be a poset and suppose that sup A exists for every finite subset A ⊆ K. A non-empty subset I ⊆ K is an ideal of K if for all a, b ∈ K

(1) a ⩽ b and b ∈ I implies a ∈ I, and

(2) a, b ∈ I implies a ∨ b ∈ I. Given a ∈ K, let I(a) = {x ∈ K | x ⩽ a} denote the principal ideal generated by a. The set K̂ of all order ideals of K is called the completion of K. This set is partially ordered by inclusion and in fact a compactly generated complete lattice. The map K → K̂ sending a ∈ K to I(a) identifies K with K̂^{c}.

Lemma 1.2. Let L be a compactly generated complete lattice. Then the map

L ⟶ L̂^{c}, a ↦ I(a) ∩ L^{c} = {x ∈ L | x ⩽ a and x compact},

is a lattice isomorphism.

Proof. The inverse map sends an ideal I ∈ L̂^{c} to sup I in L. □

Given an ideal lattice L, the multiplication L × L → L restricts to a multiplication L^{c}×L^{c} → L^{c}. It turns out that all relevant structure is determined by the multiplication of compact elements. In our applications, we always have for a, b ∈ L that ab = sup a′b′ where a′ ⩽ a and b′ ⩽ b run through all compact elements.

1.2. The prime spectrum. Let L be an ideal lattice. We define the Zariski spectrum of prime elements in L and discuss some of its basic properties. An element p ≠ 1 in L is called prime if ab ⩽ p implies a ⩽ p or b ⩽ p for all a, b ∈ L. A subset S ⊆ L is multiplicative if ab ∈ S for all a, b ∈ S. We collect some elementary facts.

Lemma 1.3. An element p ≠ 1 in L is prime if and only if ab ⩽ p implies a ⩽ p or b ⩽ p for all compact a, b ∈ L.

Proof. Use that a ⩽ p if and only if a′ ⩽ p for all compact a′ ⩽ a. □

Lemma 1.4. Let a ∈ L and S ⊆ L be a non-empty multiplicative set of compact elements. Suppose that s ⩽̸ a for all s ∈ S. Then there exists a prime p ∈ L such that a ⩽ p and s ⩽̸ p for all s ∈ S.

Proof. Consider the set A of all elements x ∈ L such that a ⩽ x and s ⩽̸ x for all s ∈ S. The set A is non-empty and for every chain B ⊆ A, we have sup B in A since the elements in S are compact. Thus A has a maximal element p, by Zorn’s lemma. We claim that p is prime. Let x, x′ ∈ L with xx′ ⩽ p. Suppose that x ⩽̸ p and x′ ⩽̸ p. Then we have s, s′ ∈ S such that s ⩽ p ∨ x and s′ ⩽ p ∨ x′, by the maximality of p. Therefore

ss′ ⩽ (p ∨ x)(p ∨ x′) = pp ∨ px′ ∨ xp ∨ xx′ ⩽ p

which contradicts the fact that p ∈ A. Thus x ⩽ p or x′ ⩽ p, and therefore p is prime. □

An element a ∈ L is semi-prime if bb ⩽ a implies b ⩽ a for all b ∈ L.

Lemma 1.5. An element a ∈ L is semi-prime if and only if a = inf V for some set V ⊆ L of prime elements.

Proof. Suppose that a is semi-prime and let V = {p ∈ L | a ⩽ p and p prime}. For any compact b ∈ L such that b ⩽̸ a, consider the multiplicative set {b^{n} | n ⩾ 1}. It follows from Lemma 1.4 that there is a prime p ∈ V such that b ⩽̸ p. Thus a = inf V. The other implication is clear. □

We denote by Spec L the set of prime elements in L and define for each a ∈ L

V (a) = {p ∈ Spec L | a ⩽ p} and D(a) = {p ∈ Spec L | a ⩽̸ p}.

The subsets of Spec L of the form V (a) are closed under forming arbitrary intersections and finite unions. More precisely,

V (⋁_{i∈Ω} a_{i}) = ⋂_{i∈Ω} V (a_{i}) and V (ab) = V (a) ∪ U(b).

Thus we obtain the Zariski topology on Spec L by declaring a subset of Spec L to be closed if it is of the form V (a) for some a ∈ L. Note that the sets of the form D(a) with compact a ∈ L form a basis of open sets. This is a consequence of the following lemma.

Lemma 1.6. For a ∈ L, we have

V (a) = ⋂_{{b⩽a \atop b compact}} V (b) and D(a) = ⋃_{{b⩽a \atop b compact}} D(b).

Proof. We have

a = ⋁_{{b⩽a \atop b compact}} b.

Given p ∈ L, it follows that a ⩽ p if and only if b ⩽ p for all compact b ⩽ a. □

Proposition 1.7. The assignments

L ∋ a ↦ V (a) = {p ∈ Spec L | a ⩽ p} and Spec L ⊇ Y ↦ inf Y

induce mutually inverse and order reversing bijections between (1) the set of all semi-prime elements in L, and

(2) the set of all closed subsets of Spec L.

Proof. Both maps are well-defined by Lemma 1.5. Given a semi-prime a ∈ L, the equality inf V (a) = a is clear since a is a join of prime elements, by Lemma 1.5. Now let Y ⊆ Spec L. The inclusion Y ⊆ V (inf Y) is purely formal. Suppose that Y is of the form Y = V (a) for some a ∈ L. If p ∈ V (inf Y), then a ⩽ inf Y ⩽ p and therefore p ∈ Y. Thus the proof is complete. □

Corollary 1.8. The assignments

L ∋ a ↦ D(a) = ⋃_{{b⩽a \atop b compact}} D(b) and Spec L ⊇ Y ↦ ⋁_{{D(b)⊆Y \atop b compact}} b

induce mutually inverse and order preserving bijections between (1) the set of all semi-prime elements in L, and

(2) the set of all open subsets of Spec L.

Proof. We apply Proposition 1.7 and need to check that for V = Spec L \ Y and a ∈ L, we have a ⩽ inf V if and only if D(a) ⊆ Y. This is clear since a ⩽ p for all p ∈ V is equivalent to a ⩽̸ q implies q ∈ Y. □

Remark 1.9. Let a ∈ L and denote by \sqrt{a} = inf V (a) the smallest semi-prime in L containing a. Then we have

\sqrt{ab} = inf(V (a) ∪ V (b)) = \sqrt{ba} for a, b ∈ L,

even though we do not assume commutativity of the multiplication in L.

2. The prime spectrum is spectral

Recall from [8] that a topological space is spectral if it is T_{0} and quasi-compact, the quasi-compact open subsets are closed under finite intersections and form an open basis, and every non-empty irreducible closed subset has a generic point. We have the following basic property of a spectral space.

Lemma 2.1. Let X be a spectral space. Endow the underlying set with a new topology by taking as open sets those of the form Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement X\Y_{i} for all i ∈ Ω, and denote the new space by X^{∗}. Then X^{∗} is spectral and (X^{∗})^{∗} = X.

Proof. See [8, Prop. 8]. □

Let L be an ideal lattice. Next we show that the space Spec L is spectral. We proceed in several steps.

Lemma 2.2. An open subset of Spec L is quasi-compact if and only if it is of the form D(c) for some compact c ∈ L.

Proof. Fix an open subset D(a) of Spec L. Suppose first that D(a) is quasi-compact. We have

D(a) = ⋃_{{b⩽a \atop b compact}} D(b)

by Lemma 1.6, and therefore D(a) = D(b) for some compact b ∈ L.

Now suppose that a ∈ L is compact and D(a) ⊆ ⋃_{b∈B} D(b) for some subset B ⊆ L. We write b̄ = ⋁_{b∈B} b and have D(a) ⊆ D(b̄). It follows from Lemma 1.4 that a^{n} ⩽ b̄ for some n ⩾ 1. Thus a^{n} ⩽ ⋁_{b∈B′} b for some finite subset B′ ⊆ B since a^{n} is compact. This implies D(a) ⊆ ⋃_{b∈B′} D(b), and therefore D(a) is quasi-compact. □

Lemma 2.3. Let p, q ∈ Spec L. Then \overline{\{p\}} = V (p). In particular, \overline{\{p\}} = \overline{\{q\}} implies p = q.

Proof. Clear. □

Lemma 2.4. Let Y ⊆ Spec L be a non-empty closed subset. If Y is irreducible, then inf Y is prime and Y = \overline{\{inf Y \}}.

Proof. First observe that we have c ⩽ inf Y for c ∈ L if and only if Y ⊆ V (c). To show that inf Y is prime, let ab ⩽ inf Y. Then

Y ⊆ V (ab) = V (a) ∪ V (b),

and we have Y ⊆ V (a) or Y ⊆ V (b) since Y is irreducible. Thus a ⩽ inf Y or b ⩽ inf Y. Let p = inf Y and Y = V (a). Then we have by construction a ⩽ p and Y ⊆ V (p). Thus Y = V (p) = \overline{\{p\}}, by Lemma 2.3. □

Proposition 2.5. The spectrum Spec L of an ideal lattice L is spectral.

Proof. The space Spec L is T_{0} by Lemma 2.3. An open subset of Spec L is quasi-compact if and only if it is of the form D(a) for some compact a ∈ L, by Lemma 2.2. Thus the definition of the topology implies that the quasi-compact open subsets form an open basis which is closed under finite intersections. Moreover, Spec L is quasi-compact since 1 = sup L is compact. If Y is a non-empty irreducible closed subset, then Y = \overline{\{p\}} for p = inf Y by Lemma 2.4. □

There is a close relation between spectral spaces and ideal lattices, and we make this more precise. Given a topological space X, we denote by L_{open}(X) the lattice of open subsets of X and consider the multiplication map

L_{open}(X) × L_{open}(X) ⟶ L_{open}(X), (U, V) ↦ UV = U ∩ V.

Note that the lattice L_{open}(X) is complete.

Lemma 2.6. Let X be a space and U ∈ L = L_{open}(X). Then (1) U is prime in L if and only if X \ U is irreducible, and

(2) U is compact in L if and only if U is quasi-compact.

Proof. Clear. □

Proposition 2.7. Let X be a spectral space. Then L_{open}(X) is an ideal lattice and every ideal in L_{open}(X) is semi-prime. Moreover, the map

X ⟶ Spec L_{open}(X), x ↦ X \ \overline{\{x\}},

is a homeomorphism.

Proof. Using Lemma 2.6, the properties of a spectral space can be translated into the defining properties of an ideal lattice. Clearly, every ideal is semi-prime, since UU = U for all U ∈ L_{open}(X). It is straightforward to check that the given map is a homeomorphism. □

Example 2.8. Let A be a commutative ring. Then the lattice L_{id}(A) of ideals of A is an ideal lattice and therefore Spec A = Spec L_{id}(A) is spectral. More generally, if X is a quasi-compact and quasi-separated scheme, then the underlying space of X is spectral.

3. An adjoint of the functor Spec

The prime spectrum of an ideal lattice satisfies a universal property which we discuss in this section. Then we view the assignment L ↦ Spec L as a functor from ideal lattices to spectral topological spaces and study its adjoint.

Definition 3.1. A spectrum of an ideal lattice L is a pair (X, δ) where X is a topological space and δ is a map which assigns to each a ∈ L an open subset δ(a) ⊆ X, such that

δ( ⋁_{a∈A} a) = ⋃_{a∈A} δ(a) for A ⊆ L, (3.1)

δ(1) = X and δ(ab) = δ(a) ∩ δ(b) for a, b ∈ L.

A morphism f: (X, δ) → (X′, δ′) of spectra is a continuous map f: X → X′ such that δ(a) = f^{−1}(δ′(a)) for all a ∈ L. Such a morphism is an isomorphism if and only if f: X → X′ is a homeomorphism.

Theorem 3.2. Let L be an ideal lattice. Then the pair (Spec L, D) is a spectrum of L. For every spectrum (X, δ) of L, there exists a unique continuous map f: X → Spec L such that δ(a) = f^{−1}(D(a)) for every a ∈ L. The map f is defined by

f(x) = ⋁_{{x∉δ(c) \atop c compact}} c for x ∈ X.

Proof. Clearly, the pair (Spec L, D) is a spectrum. Now let (X, δ) be a spectrum of L. We show that for each x ∈ X the element f(x) = ⋁_{x∉δ(c)} c is prime. First observe that f(x) ≠ 1 since δ(1) = X. Suppose that ab ⩽ f(x), and we may assume that a, b are compact. Using (3.1), observe that the compact c ∈ L with x ∉ δ(c) form a directed set. Thus ab ⩽ c for some compact c ∈ L with x ∉ δ(c). Using again (3.1), we have

δ(a) ∩ δ(b) = δ(ab) ⊆ δ(c)

and therefore x ∉ δ(a) or x ∉ δ(b). We conclude that a ⩽ f(x) or b ⩽ f(x). The definition of f implies δ(a) = f^{−1}(D(a)) for every a ∈ L, since

x ∈ δ(a) ⟺ a ⩽̸ f(x) ⟺ f(x) ∈ D(a) ⟺ x ∈ f^{−1}(D(a)).

The continuity of f follows from the fact that the sets D(a) with a ∈ L are precisely the open subsets of Spec L. Now let f_{1}, f_{2} : X → Spec L be two maps satisfying

f_{1}^{−1}(D(a)) = δ(a) = f_{2}^{−1}(D(a))

for every a ∈ L. Fix x ∈ X. Then we have

\overline{\{f_{1}(x)\}} = ⋂_{f_{1}(x)∉D(a)} V (a) = ⋂_{f_{2}(x)∉D(a)} V (a) = \overline{\{f_{2}(x)\}}.

This implies f_{1}(x) = f_{2}(x) by Lemma 2.3, and therefore the proof is complete. □

Our first application says that Spec is actually a functor into the category of topological spaces.

Lemma 3.3. A morphism of ideal lattices ϕ: L → L′ induces a unique continuous map Spec ϕ: Spec L′ → Spec L such that

D(ϕ(a)) = (Spec ϕ)^{−1}D(a) for a ∈ L.

Proof. The pair (Spec L′, D^{∘} ϕ) is a spectrum of L. Now apply Theorem 3.2. Note that we can compute more explicitly

(Spec ϕ)p = sup{a ∈ L | ϕ(a) ⩽ p} for p ∈ Spec L′.
□

The universal property of the prime spectrum yields an adjoint functor for Spec.

Theorem 3.4. We have an adjoint pair of contravariant functors

\xymatrix{Lat_{id} \ar[r]^{Spec} & Top_{sp} \ar[l]^{L_{open}}}

between the category of ideal lattices and the category of spectral spaces. More precisely, for an ideal lattice L and a spectral space X, there are mutually inverse bijections

\xymatrix{Hom_{Lat_{id}}(L, L_{open}(X)) \ar[r]^{Σ} & Hom_{Top_{sp}}(X, Spec L). \ar[l]^{Λ}}

The functor L_{open} is fully faithful, and an ideal lattice L is isomorphic to one of the form L_{open}(X) if and only if every ideal in L is semi-prime.

Proof. Both functors are well-defined by Propositions 2.5 and 2.7. The maps Σ and Λ are defined by

(Σϕ)(x) = ⋁_{{x∉ϕ(c) \atop c compact}} c and (Λf)(a) = f^{−1}(D(a)) for x ∈ X, a ∈ L.

It follows from Theorem 3.2 that both maps are mutually inverse bijections. Next observe that L_{open} is fully faithful. This follows from the fact that the adjunction morphism X → Spec L_{open}(X) is a homeomorphism; see also Proposition 2.7. It remains to describe the image of L_{open}. Clearly, every ideal in L_{open}(X) is semi-prime; see Proposition 2.7. Conversely, if every ideal in L is semi-prime, then the adjunction morphism L → L_{open}(Spec L) is an isomorphism, by Corollary 1.8. Thus the proof is complete. □

We observe that a morphism ϕ: L → L_{open}(X) is determined by its restriction to the subset L^{c} of compact elements since

ϕ(a) = ⋃_{{b⩽a \atop b compact}} ϕ(b) for a ∈ L.

In particular, we obtain an equivalent definition of a spectrum of L by taking maps δ: L^{c} → L_{open}(X) satisfying

δ(a ∨ b) = δ(a) ∪ δ(b) for a, b ∈ L,
δ(1) = X and δ(ab) = δ(a) ∩ δ(b) for a, b ∈ L.

This observation leads naturally to the concept of a support datum.

4. Support data

Let L be an ideal lattice. We have seen that the space Spec L is spectral and, in view of our applications, we consider from now on the ‘opposite’ topology on X = Spec L. To be precise, we let Spec^{∗} L = X^{∗} where the points of X^{∗} and X coincide and Y ⊆ X^{∗} is by definition open if Y = ⋃_{i∈Ω} Y_{i} with quasi-compact Zariski-open complement X \ Y_{i} for all i ∈ Ω; see Lemma 2.1. For a ∈ L, we call

supp(a) = {p ∈ Spec^{∗} L | a ⩽̸ p}

the support of a and observe that supp(a) is closed if a is compact. Let us reformulate the classification of semi-prime ideals in terms of the topology Spec^{∗} L.

Proposition 4.1. The assignments

L ∋ a ↦ supp(a) = ⋃_{{b⩽a \atop b compact}} supp(b) and Spec^{∗} L ⊇ Y ↦ ⋁_{supp(b)⊆Y ; b compact} b

induce mutually inverse and order preserving bijections between (1) the set of all semi-prime elements in L, and

(2) the set of all subsets Y ⊆ Spec^{∗} L of the form Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement Spec^{∗} L \ Y_{i} for all i ∈ Ω.

Proof. Use Corollary 1.8 and observe that the subsets Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement Spec^{∗} L \ Y_{i} are precisely the open subsets of Spec L = (Spec^{∗} L)^{∗}, by Lemma 2.1. □

Next we introduce for an ideal lattice the concept of a support datum. This is inspired by Balmer’s definition of a support datum on a triangulated tensor category [1, Defn. 3.1]. The subsequent proposition is the analogue of [1, Thm. 3.2].

Definition 4.2. A support datum on an ideal lattice L is a pair (X, σ) where X is a topological space and σ is a map which assigns to each compact a ∈ L a closed subset σ(a) ⊆ X, such that

σ(a ∨ b) = σ(a) ∪ σ(b) for a, b ∈ L, (4.1)

σ(1) = X and σ(ab) = σ(a) ∩ σ(b) for a, b ∈ L.

A morphism f: (X, σ) → (X′, σ′) of support data is a continuous map f: X → X′ such that σ(a) = f^{−1}(σ′(a)) for all compact a ∈ L. Such a morphism is an isomorphism if and only if f: X → X′ is a homeomorphism.

The following result complements the universal property of the pair (Spec L, D) of Theorem 3.2 and the proof is almost the same.

Proposition 4.3. Let L be an ideal lattice. Then the pair (Spec^{∗} L, supp) is a support datum on L. For every support datum (X, σ) on L, there exists a unique continuous map f: X → Spec^{∗} L such that σ(a) = f^{−1}(supp(a)) for every compact a ∈ L. The map f is defined by

f(x) = ⋁_{{x∉σ(c) \atop c compact}} c for x ∈ X.

Proof. Clearly, the pair (Spec^{∗} L, supp) is a support datum. Now let (X, σ) be a support datum on L. We show that for each x ∈ X the element f(x) = ⋁_{x∉σ(c)} c is prime. First observe that f(x) ≠ 1 since σ(1) = X. Suppose that ab ⩽ f(x), and we may assume that a, b are compact. Using (4.1), observe that the compact c ∈ L with x ∉ σ(c) form a directed set. Thus ab ⩽ c for some compact c ∈ L with x ∉ σ(c). Using again (4.1), we have

σ(a) ∩ σ(b) = σ(ab) ⊆ σ(c)

and therefore x ∉ σ(a) or x ∉ σ(b). We conclude that a ⩽ f(x) or b ⩽ f(x). The definition of f implies σ(a) = f^{−1}(supp(a)) for every compact a ∈ L, since

x ∈ σ(a) ⟺ a ⩽̸ f(x) ⟺ f(x) ∈ supp(a) ⟺ x ∈ f^{−1}(supp(a)).

The continuity of f follows from the fact that the sets supp(a) with compact a ∈ L form a basis of closed sets for the topology on Spec^{∗} L. Now let f_{1}, f_{2} : X → Spec^{∗} L be two maps satisfying

f_{1}^{−1}(supp(a)) = σ(a) = f_{2}^{−1}(supp(a))

for every compact a ∈ L. Fix x ∈ X. Then we have

\overline{\{f_{1}(x)\}} = ⋂_{f_{1}(x)∈supp(a)} supp(a) = ⋂_{f_{2}(x)∈supp(a)} supp(a) = \overline{\{f_{2}(x)\}}.

This implies f_{1}(x) = f_{2}(x) since the space Spec^{∗} L is T_{0}, by Proposition 2.5 and Lemma 2.1 □

5. Classifying support data

Let L be an ideal lattice. A support datum (X, σ) on L is called classifying if the space X is spectral and the assignments

L ∋ a ↦ ⋃_{{b⩽a \atop b compact}} σ(b) and X ⊇ Y ↦ ⋁_{{σ(b)⊆Y \atop b compact}} b

induce bijections between

(1) the set of all semi-prime elements in L, and

(2) the set of all subsets Y ⊆ X of the form Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement X \ Y_{i} for all i ∈ Ω.

Proposition 5.1. Let f: (X, σ) → (X′, σ′) be a morphism of support data. If both support data are classifying, then the map f: X → X′ is a homeomorphism.

Proof. Let Y ⊆ X and Y′ ⊆ X′ be subsets which are unions of subsets with quasicompact open complement, and suppose

⋁_{{σ(b)⊆Y \atop b compact}} b = a = ⋁_{{σ′(b)⊆Y′ \atop b compact}} b.

Then we have

Y = ⋃_{{b⩽a \atop b compact}} σ(b) and Y′ = ⋃_{{b⩽a \atop b compact}} σ′(b).

This implies

f^{−1}(Y′) = ⋃_{{b⩽a \atop b compact}} f^{−1}(σ′(b)) = ⋃_{{b⩽a \atop b compact}} σ(b) = Y.

It follows that the map Y ↦ f^{−1}(Y) induces an inclusion preserving bijection between the open subsets of X^{∗} and (X′)^{∗}. In fact, we use that X, X′ are spectral and apply Lemma 2.1. Thus f is a homeomorphism X^{∗} → (X′)^{∗} and therefore also a homeomorphism X → X′. □

The following consequence is the analogue of [1, Thm. 5.2]. Note that we do not assume that the support space is noetherian.

Corollary 5.2. A support datum (X, σ) on L is classifying if and only if the canonical morphism (X, σ) → (Spec^{∗} L, supp) is an isomorphism.

6. Thick tensor ideals

In this section we consider an additive category with a tensor product and study its collection of ideals. If there is an additional abelian or triangulated structure, then we consider those tensor ideals which are in addition thick subcategories.

In this paper, all categories are assumed to be small, that is, the isomorphism classes of objects form a set (in some fixed universe).

6.1. Sublattices of an ideal lattice. Let L be an ideal lattice. We fix a subset L′ ⊆ L satisfying the following conditions.

(L∧) If A ⊆ L′, then inf A ∈ L′.

(L∨) If A ⊆ L′ is directed, then sup A ∈ L′.

We consider on L′ the partial order induced from the partial order on L and define the map

π: L ⟶ L′, a ↦ ⋀_{a⩽a′∈L′} a′.

Note that we have

(6.1)  π(a) ⩽ a′ ⟺ a ⩽ a′ for a ∈ L, a′ ∈ L′.

Moreover, 1 = inf ∅ ∈ L′.

Lemma 6.1. The poset L′ is a complete and compactly generated lattice.

Proof. Let A ⊆ L′. Then we use (L∧) to compute the infimum inf A in L′ and have

sup A = inf{a′ ∈ L′ | a ⩽ a′ for all a ∈ A}.

It follows from (6.1) and (L∨) that π preserves compactness and that each element in L′ is the supremum of compact elements. Thus L′ is compactly generated. □

Given a, b ∈ L′, we define their product in L′ as

a · b = π(ab)

and use a dot to distinguish it from the product in L. We make a further assumption.

(Lπ) Given a, b ∈ L, we have π(aπ(b)) = π(ab) = π(π(a)b).

Lemma 6.2. Let a, b, c ∈ L′. Then we have

(1) (a · b) · c = a · (b · c),

(2) a · 1 = a = 1 · a,

(3) a · (b ∨ c) = (a · b) ∨ (a · c) and (a ∨ b) · c = (a · c) ∨ (b · c).

Proof. Clear. □

Proposition 6.3. Let L be an ideal lattice and L′ be a subset satisfying the conditions (L∧), (L∨), and (Lπ). Then L′ inherits from L the structure of an ideal lattice.

Proof. The lattice L′ satisfies (L1) – (L4) by Lemmas 6.1 and 6.2. To check (L5), let a′, b′ ∈ L′ be compact. Then there exist compact elements a, b ∈ L with π(a) = a′ and π(b) = b′. To see this, write a′ = ⋁_{i} a_{i} as directed union of all compact elements a_{i} ⩽ a′ in L and use that a′ = π(a′) = ⋁_{i} π(a_{i}) equals π(a_{i}) for some index i. Now we obtain

a′ · b′ = π(a′b′) = π(π(a)π(b)) = π(ab).

The element ab is compact in L, and π preserves compactness by (L∨). Thus a′ · b′ is compact in L′. □

6.2. The ideal lattice of a semi-ring. Let A = (A, +, ·) be a semi-ring, that is, A is a set together with two associative binary operations with identities (denoted by 0 and 1) such that the addition is commutative and distributivity holds. A subset I ⊆ A containing 0 is by definition an ideal if for all x, y ∈ A

(1) x ∈ I and y ∈ I implies x + y ∈ I, and

(2) x ∈ I or y ∈ I implies xy ∈ I. The ideals of A are partially ordered by inclusion and form a lattice which we denote by L_{id}(A). Given I, J ∈ L_{id}(A), we define

IJ = {∑_{i} x_{i}y_{i} | x_{i} ∈ I, y_{i} ∈ J} and I + J = {x + y | x ∈ I, y ∈ J}.

Note that I + J = I ∨ J.

Proposition 6.4. Let A be semi-ring. Then the lattice L_{id}(A) of ideals satisfies the conditions (L1) – (L4). An ideal in L_{id}(A) is compact if and only if it is finitely generated. If A is commutative, then condition (L5) is satisfied.

Proof. The proof is straightforward. To identify the compact elements, one uses that ⋁_{i} I_{i} = ⋃_{i} I_{i} for any directed set of ideals I_{i}. To show (L5), let I = ⟨I_{0}⟩ and J = ⟨J_{0}⟩ be ideals generated by subsets I_{0} and J_{0}, respectively. If A is commutative, then IJ = ⟨xy | x ∈ I_{0}, y ∈ J_{0}⟩. Therefore (L5) holds. □

Example 6.5. Let A be a (not necessarily commutative) ring and suppose A satisfies the ascending chain condition on ideals. Then the lattice L_{id}(A) of ideals is an ideal lattice. Note that we have the following weak commutativity: \sqrt{IJ} = \sqrt{JI} for any pair I, J of ideals.

6.3. Thick tensor ideals. Let C = (C, ⊗, e) be an additive category with a tensor product. To be precise, we have an additive bifunctor C × C → C and a natural isomorphism (x ⊗ y) ⊗ z ⟶^{∼} x ⊗ (y ⊗ z). In addition, we require the existence of a tensor identity e, that is, we have natural isomorphisms x ⊗ e ⟶^{∼} x and e ⊗ y ⟶^{∼} y satisfying the Pentagon Axiom and the Triangle Axiom.

Denote by C the set of isomorphism classes of objects in C, and let x + y = x ⨿ y and xy = x ⊗ y for x, y ∈ C. Then C is a semi-ring, and we shall identify C and C whenever it is convenient. A tensor ideal of C is a full additive subcategory D such that for all x, y ∈ C, we have x ⊗ y ∈ D if x ∈ D or y ∈ D. Note that the tensor ideals in C are precisely the ideals of the semi-ring C. We denote by L_{id}(C) the lattice of tensor ideals of C and define the multiplication of tensor ideals as in L_{id}(C).

Now suppose that there exists some additional exact or triangulated structure on C. A full subcategory of C is called thick if it is ‘compatible’ with this additional structure. We view the thick tensor ideals as a subset of L_{id}(C) and denote it by L_{thick}(C). An ideal D ∈ L_{thick}(C) is generated by a class D_{0} of objects and we write

D = ⟨D_{0}⟩

if D is the smallest thick tensor ideal containing D_{0}.

6.4. Abelian and triangulated tensor categories. Let C be an abelian category, or more generally, an exact category in the sense of Quillen. A full subcategory D is called thick if for every exact sequence 0 → x′ → x → x′′ → 0 in C, we have x ∈ D if and only if x′, x′′ ∈ D. Now suppose that there is a tensor product ⊗ defined on C.

Proposition 6.6. Let C be an abelian category, or more generally an exact category, with a tensor product which is exact in each variable. Suppose that either the tensor product is commutative, or that there exists an object c ∈ C such that there is no proper thick subcategory of C containing c. Then the thick tensor ideals in C form an ideal lattice. Moreover, an ideal is compact if and only if it is generated by a single object.

Proof. It is straightforward to check the conditions (L∧) and (L∨) for the subset of thick tensor ideals L_{thick}(C) in L_{id}(C). To verify (Lπ), observe that for a tensor ideal D ∈ L_{id}(C), the thick subcategory generated by D equals π(D). We deduce from Propositions 6.3 and 6.4 that L_{thick}(C) is an ideal lattice. Note that a finitely generated ideal ⟨x_{1}, …, x_{n}⟩ is generated by x_{1} ⨿ … ⨿ x_{n}. Thus compact and cyclic ideals coincide. Finally, if ⊗ is not commutative, then (L5) follows from the identity ⟨x⟩⟨y⟩ = ⟨x ⊗ c ⊗ y⟩, assuming that c generates C. □

Now let C be a triangulated category. A full subcategory D is called thick if D is a triangulated subcategory and for each x ∈ D a decomposition x = x_{1} ⨿ x_{2} implies x_{1}, x_{2} ∈ D. Now suppose that there is a tensor product ⊗ defined on C.

Proposition 6.7. Let C be a triangulated category with a tensor product which is exact in each variable. Suppose that either the tensor product is commutative, or that there exists an object c ∈ C such that there is no proper thick subcategory of C containing c. Then the thick tensor ideals in C form an ideal lattice. Moreover, an ideal is compact if and only if it is generated by a single object.

Proof. The proof is the same as that of Proposition 6.6. □

Remark 6.8. Let (C, ⊗, e) be an abelian or triangulated tensor category with exact tensor product and tensor identity e. Let E be the thick subcategory generated by e. Then we have x ⊗ y ∈ E for all x, y ∈ E and therefore (E, ⊗, e) is a category satisfying the assumptions from Propositions 6.6 or 6.7.

6.5. Support data. Let C be an abelian or triangulated tensor category. We assume from now on that the lattice of thick tensor ideals of C is an ideal lattice, for instance by imposing the assumptions from Propositions 6.6 or 6.7. We write

Spec C = Spec^{∗} L_{thick}(C)

for the spectrum of prime ideals. Note that we keep the ‘opposite’ of the Zariski topology in view of our applications. This practice is in accordance with Balmer’s notion of a spectrum in [1]. The compact ideals in L_{thick}(C) are precisely the ideals ⟨x⟩ generated by a single object x ∈ C. We write

supp(⟨x⟩) = supp(x) = {P ∈ Spec C | x ∉ P} for x ∈ C.

It is convenient to work with support data defined on objects of C instead of support data defined on ideals of C. This motivates the following definition from [1].

Definition 6.9. A support datum on C is a pair (X, τ) where X is a topological space and τ is a map which assigns to each object x ∈ C a closed subset τ(x) ⊆ X, such that for all x, y ∈ C

τ(x) = ⋃_{x′∈⟨x⟩} τ(x′), τ(x ⨿ y) = τ(x) ∪ τ(y),

τ(e) = X and τ(x ⊗ y) = τ(x) ∩ τ(y).

A morphism f: (X, τ) → (X′, τ′) of support data is a continuous map f: X → X′ such that τ(x) = f^{−1}(τ′(x)) for all x ∈ C.

Lemma 6.10. Let C be an abelian or triangulated tensor category satisfying the assumptions from Propositions 6.6 or 6.7.

(1) If (X, τ) is a support datum on C, then σ(⟨x⟩) = τ(x) defines a support datum on the lattice of thick tensor ideals of C.

(2) If (X, σ) is a support datum on the lattice of thick tensor ideals of C, then τ(x) = σ(⟨x⟩) defines a support datum on C.

Proof. We start with a support datum (X, τ) on C. The map σ on compact ideals of C is well-defined because of the condition τ(x) = ⋃_{x′∈⟨x⟩} τ(x′). Now compute for compact ideals ⟨x⟩ and ⟨y⟩

σ(⟨x⟩) ∪ σ(⟨y⟩) = τ(x) ∪ τ(y) = τ(x ⨿ y) = σ(⟨x ⨿ y⟩) = σ(⟨x⟩ ∨ ⟨y⟩),

and if the tensor product is commutative

σ(⟨x⟩) ∩ σ(⟨y⟩) = τ(x) ∩ τ(y) = τ(x ⊗ y) = σ(⟨x ⊗ y⟩) = σ(⟨x⟩⟨y⟩),

using that ⟨x⟩⟨y⟩ = ⟨x ⊗ y⟩. In the non-commutative case, we have ⟨x⟩⟨y⟩ = ⟨x ⊗ c ⊗ y⟩ for some c ∈ C with ⟨c⟩ = C. Thus

σ(⟨x⟩) ∩ σ(⟨y⟩) = τ(x) ∩ τ(y) = τ(x) ∩ τ(c) ∩ τ(y) = τ(x ⊗ c ⊗ y) = σ(⟨x ⊗ c ⊗ y⟩) = σ(⟨x⟩⟨y⟩).

We conclude that (X, σ) is support datum on the lattice of thick tensor ideals of C. The proof of the converse is analogous. □

From now on we do not distinguish between support data on the lattice of thick tensor ideals L_{thick}(C) and support data on C. We leave it to the interested reader to reformulate our general results about ideal lattices for the lattice L_{thick}(C) and its spectrum Spec C.

7. The structure sheaf of a tensor category

Let C be an abelian or triangulated tensor category with tensor identity e. Following [1], we define a structure sheaf on Spec C as follows. For an open subset U ⊆ Spec C, let

C_{U} = {x ∈ C | supp(x) ∩ U = ∅}

and observe that C_{U} is a thick tensor ideal. We obtain a presheaf of rings on Spec C by

U ↦ End_{C/C_{U}} (e).

If V ⊆ U are open subsets, then the restriction map End_{C/C_{U}} (e) → End_{C/C_{V}} (e) is induced by the quotient functor C/C_{U} → C/C_{V}. The sheafification is called the structure sheaf of C and is denoted by O_{C}. Note that the endomorphism ring of a tensor identity is commutative, if the tensor product is commutative, or if C is triangulated and the tensor product is exact in each variable; see for instance [15, Thm. 1.7]. Next observe that

O_{C,P} ≅ End_{C/P}(e) for each P ∈ Spec C.

This is an immediate consequence of the following lemma.

Lemma 7.1. Let C be an abelian or triangulated tensor category and P ∈ Spec C. Then

\underrightarrow{lim}_{P∈U} Hom_{C/C_{U}} (x, y) ⟶^{∼} Hom_{C/P}(x, y) for all x, y ∈ C,

where U runs through all (quasi-compact) open subsets containing P.

Proof. Use that P = ⋃_{P∈U} C_{U}. □

Now we discuss briefly the functoriality of the spectrum.

Lemma 7.2. Let F: C → C′ be an exact tensor functor.

(1) F induces a unique continuous map f: Spec C′ → Spec C such that

supp(F x) = f^{−1}(supp(x)) for x ∈ C.

(2) F induces a morphism

(f, f^{♯}): (Spec C′, O_{C′}) → (Spec C, O_{C})

of ringed spaces.

Proof. (1) The map sending x ∈ C to supp(F x) is a support datum on C. Now apply Proposition 4.3 to obtain a continuous map Spec C′ → Spec C. Note that f(P) = F^{−1}(P) for P ∈ Spec C′.

(2) Let U ⊆ Spec C be open. Then F maps C_{U} to C_{f^{−1}U} and induces a functor C/C_{U} → C′/C′_{f^{−1}U}. This functor induces a homomorphism End_{C/C_{U}} (e) → End_{C′/C′_{f^{−1}U}} (e′). Thus

we obtain a morphism f^{♯}: O_{C} → f_{∗}O_{C′}. □

8. Applications to schemes

8.1. Coherent sheaves on a scheme. We consider a noetherian scheme X and reconstruct it from the abelian tensor category coh X of coherent O_{X}-modules. This is based on the following well-known classification of all thick subcategories of coh X. Given x ∈ coh X, we write

supp_{X}(x) = {P ∈ X | x_{P} ≠ 0}.

Proposition 8.1. Let X be a noetherian scheme. The assignments

coh X ⊇ D ↦ ⋃_{x∈D} supp_{X}(x) and X ⊇ Y ↦ {x ∈ coh X | supp_{X}(x) ⊆ Y }

induce bijections between (1) the set of all thick subcategories of coh X, and

(2) the set of all subsets Y ⊆ X of the form Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement X \ Y_{i} for all i ∈ Ω.

Proof. See [6, Prop. VI.4]. □

Note that every open subset of a noetherian space is quasi-compact. Nonetheless, the above formulation is appropriate because it generalizes to schemes which are not necessarily noetherian.

The abelian category coh X carries a commutative tensor product ⊗_{O_{X}}, and we deduce from the classification of thick subcategories the following properties.

Proposition 8.2. Let X be a noetherian scheme and C = coh X. Then every thick subcategory of C is a tensor ideal and the thick tensor ideals of C form an ideal lattice.

Proof. We apply Proposition 8.1. The formula

supp_{X}(x ⊗_{O_{X}} y) = supp_{X}(x) ∩ supp_{X}(y)

shows that every thick subcategory is a tensor ideal. The space X is spectral because the scheme is noetherian. Thus X^{∗} is spectral and L_{open}(X^{∗}) is an ideal lattice, by Proposition 2.7. We have an isomorphism L_{open}(X^{∗}) ≅ L_{thick}(C), and therefore L_{thick}(C) is an ideal lattice. □

Theorem 8.3. Let X be a noetherian scheme and consider the abelian tensor category coh X of coherent O_{X}-modules. The pair (X, supp_{X}) is a classifying support datum on coh X and there is an induced isomorphism

(X, O_{X}) ⟶^{∼} (Spec coh X, O_{coh X})

of ringed spaces.

Proof. Let C = coh X. It follows from well-known properties of the support supp_{X}(x) that (X, supp_{X}) is a support datum on C. Thus we obtain a continuous map f: X → Spec C satisfying supp_{X}(x) = f^{−1}(supp(x)) for each x ∈ C, by Proposition 4.3. The classification of thick subcategories of C from Proposition 8.1 shows that the support datum (X, supp_{X}) is classifying. Here we use in addition that the underlying space of X is spectral. It follows from Corollary 5.2 that f is a homeomorphism.

It remains to construct an isomorphism f^{♯}: O_{C} → f_{∗}O_{X}. Observe that for each open U ⊆ Spec C, the restriction coh X → coh f^{−1}U induces an equivalence C/C_{U} ⟶^{∼} coh f^{−1}U; see [6, Prop. VI.2]. Thus we obtain for e = O_{X} an isomorphism

O_{C}(U) = End_{C/C_{U}} (e) ⟶^{∼} O_{X}(f^{−1}U)

which yields the isomorphism f^{♯}: O_{C} ⟶^{∼} f_{∗}O_{X}. □

8.2. Perfect complexes on a scheme. We consider a quasi-compact and quasi-sepa-rated scheme X and its triangulated tensor category D^{per}(X) of perfect complexes with tensor product ⊗_{O_{X}}^{L} ; see [16, Sec. 2] for a concise discussion of these concepts. For instance, every noetherian scheme is quasi-compact and quasi-separated. Let us recall Thomason’s classification of thick tensor ideals. Given x ∈ D^{per}(X), we write

supp_{X}(x) = {P ∈ X | x_{P} ≠ 0}.

Proposition 8.4. Let X be a quasi-compact and quasi-separated scheme. The assignments

D^{per}(X) ⊇ D ↦ ⋃_{x∈D} supp_{X}(x) and X ⊇ Y ↦ {x ∈ D^{per}(X) | supp_{X}(x) ⊆ Y }

induce bijections between (1) the set of all thick tensor ideals of D^{per}(X), and

(2) the set of all subsets Y ⊆ X of the form Y = ⋃_{i∈Ω} Y_{i} with quasi-compact open complement X \ Y_{i} for all i ∈ Ω.

Proof. See [17, Thm. 4.1]. □

We observe that the thick tensor ideals of D^{per}(X) form an ideal lattice by Proposition 6.7. The following result shows that a quasi-compact and quasi-separated scheme can be reconstructed from the triangulated tensor category of perfect complexes; it is a slight generalization of [1, Thm. 6.3] which assumes the scheme to be topologically noetherian.

Theorem 8.5. Let X be a quasi-compact and quasi-separated scheme and consider the triangulated tensor category D^{per}(X) of perfect complexes on X. The pair (X, supp_{X}) is a classifying support datum on D^{per}(X) and there is an induced isomorphism

(X, O_{X}) ⟶^{∼} (Spec D^{per}(X), O_{D^{per}(X)})

of ringed spaces.

Proof. The proof is essentially the same as that of Theorem 8.3, with C = coh X replaced by C = D^{per}(X). Note that the assumption on X implies that the underlying space is spectral. We use the classification of thick tensor ideals from Proposition 8.4. For the equivalence C/C_{U} ⟶^{∼} D^{per}(f^{−1}U), up to direct factors, when U ⊆ Spec C is quasicompact open, we refer to [16, Sec. 5]. □

9. A projective scheme

There are interesting examples of triangulated tensor categories C where (Spec C, O_{C}) is actually a projective scheme. Here, we present a general criterion which explains those examples. We fix a triangulated tensor category C with tensor identity e such that the tensor product is exact in each variable. Let us start with some preparation. For x, y ∈ C, we write

Hom_{C}^{∗}(x, y) = ∐_{n∈ℤ}Hom_{C}(x, Σ^{n}y)

where Σ denotes the suspension of C. The graded endomorphism ring End_{C}^{∗}(x) acts on Hom_{C}^{∗}(x, y) from the right and End_{C}^{∗}(y) acts from the left. We use the graded ring homomorphism

ϕ_{x}: End_{C}^{∗}(e) ⟶ End_{C}^{∗}(x), α ↦ α ⊗ x = x ⊗ α.

Note that End_{C}^{∗}(e) acts on Hom_{C}^{∗}(x, y) from the right via ϕ_{x} and from the left via ϕ_{y}, with

α · β = (−1)^{|α||β|}β · α

for homogeneous elements α ∈ End_{C}^{∗}(e) and β ∈ Hom_{C}^{∗}(x, y); see [15]. In particular, End_{C}^{∗}(e) is graded commutative.

9.1. Cohomological localization. We need a basic result about the localization of triangulated categories. Under appropriate assumptions, we show that first taking cohomology and then localizing is the same as first localizing and then taking cohomology. For a homogeneous element σ: Σ^{n}x → x in End_{C}^{∗}(x), we denote by x/σ its cofiber in C.

Proposition 9.1. Let C be a triangulated category and D ⊆ C a full triangulated subcategory. Let c ∈ C be an object and ϕ: H → End_{C}^{∗}(c) be a graded ring homomorphism such that H is graded commutative. Fix a subset S ⊆ H of homogeneous elements and consider for each x ∈ C the following commutative diagram of canonical homomorphisms in the category of graded H-modules.

\xymatrix{Hom_{C}^{∗}(c, x) \ar[r]^{μ} \ar[d]_{π} & S^{−1} Hom_{C}^{∗}(c, x) \ar[d]^{S^{−1}π} \\ Hom_{C/D}^{∗}(c, x) \ar[r]^{ν} & S^{−1} Hom_{C/D}^{∗}(c, x)}

(1) If {c/ϕ(σ) | σ ∈ S} ⊆ D, then ν is an isomorphism.

(2) If D ⊆ {x ∈ C | S^{−1} Hom_{C}^{∗}(c, x) = 0}, then S^{−1}π is an isomorphism.

Proof. We assume that H is graded commutative because we need in (2) that localization of graded H-modules with respect to S is an exact functor.

(1) Assume {c/ϕ(σ) | σ ∈ S} ⊆ D. Let Q: C → C/D denote the quotient functor. Then H acts on Hom_{C/D}^{∗}(c, x) via Q, and each σ ∈ S acts invertibly since Qϕ(σ) is invertible. Thus the canonical map

Hom_{C/D}^{∗}(c, x) → S^{−1} Hom_{C/D}^{∗}(c, x)

is invertible.

(2) Assume D ⊆ {x ∈ C | S^{−1} Hom_{C}^{∗}(c, x) = 0}. We embed C into the category Mod C of additive functors C^{op} → Ab via the Yoneda functor

C ⟶ Mod C, x ↦ Hom_{C}(−, x).

Note that every cohomological functor F: C → A into an abelian Grothendieck category A extends uniquely to an exact and coproduct preserving functor F̄: Mod C → A; see [11, Lem. 2.2]. Now take the composite

C ⟶^{Hom_{C}^{∗}(c,−)} Mod H ⟶^{S^{−1}} Mod H

which annihilates D by our assumption. We obtain the following commutative diagram

\xymatrix{C \ar[r]^{F =Hom_{C}^{∗}(c,−)} \ar[d]_{Q} & Mod H \ar[d]^{S^{−1}} \\ C/D \ar[r]^{G} & Mod H}

which can be extended to the following commutative diagram of exact and coproduct preserving functors.

\xymatrix{Mod C \ar[r]^{F̄} \ar[d]_{Q^{∗}} & Mod H \ar[d]^{S^{−1}} \\ Mod C/D \ar[r]^{Ḡ} & Mod H}

Note that

F̄(M) ≅ ∐_{n} M(Σ^{−n}c) ≅ ∐_{n} Hom_{C}(Hom_{C}(−, Σ^{−n}c), M)

for M in Mod C. The first isomorphism is clear for each representable functor M = Hom_{C}(−, x). Then observe that every M ∈ Mod C is a colimit of representable functors and both functors preserve colimits. The second isomorphism follows from Yoneda’s lemma. The functor Q^{∗} has a right adjoint

Q_{∗}: Mod C/D ⟶ Mod C, M ↦ M ^{∘} Q,

and the adjunction morphism Q^{∗}Q_{∗}M → M is an isomorphism for all M ∈ Mod C/D, since Q is a quotient functor. Now consider for x ∈ C the adjunction morphism

η_{x}: Hom_{C}(−, x) → Q_{∗}Q^{∗} Hom_{C}(−, x).

First observe that Q^{∗}η_{x} is an isomorphism. On the other hand, F̄η_{x} equals π up to an isomorphism, since

F̄(Q_{∗}Q^{∗} Hom_{C}(−, x)) ≅ ∐_{n} Hom_{C}(Hom_{C}(−, Σ^{−n}c), Q_{∗}Q^{∗} Hom_{C}(−, x))

≅ ∐_{n} Hom_{C/D}(Q^{∗} Hom_{C}(−, Σ^{−n}c), Q^{∗} Hom_{C}(−, x))

≅ ∐_{n} Hom_{C/D}(Hom_{C/D}(−, Σ^{−n}c), Hom_{C/D}(−, x))

≅ ∐_{n} Hom_{C/D}(Σ^{−n}c, x)

= Hom_{C/D}^{∗}(c, x).

Thus S^{−1}π ≅ S^{−1}(F̄η_{x}) = Ḡ(Q^{∗}η_{x}) is an isomorphism, and this finishes the proof. □

We formulate an immediate consequence.

Corollary 9.2. Let C be a triangulated category. Let c ∈ C be an object such that its graded endomorphism ring End_{C}^{∗}(c) is graded commutative and fix a homogeneous prime ideal 𝔭. Let

C_{𝔭} = {x ∈ C | Hom_{C}^{∗}(c, x)_{𝔭} = 0}.

Then we have a natural isomorphism Hom_{C/C_{𝔭}}^{∗}(c, x) ⟶^{∼} Hom_{C}^{∗}(c, x)_{𝔭} for all x ∈ C.

9.2. Cohomological support. We keep fixed a triangulated tensor category C with tensor identity e and suppose that H = End_{C}^{∗}(e) is concentrated in non-negative degrees. We define the cohomological support of an object x ∈ C as

supp_{H}(x) = {𝔭 ∈ Proj H | End_{C}^{∗}(x)_{𝔭} ≠ 0},

where H acts on End_{C}^{∗}(x) via the canonical ring homomorphism H → End_{C}^{∗}(x) taking an element α to α ⊗ x = x ⊗ α. It is useful to observe that for each 𝔭 ∈ Proj H

(9.1)  End_{C}^{∗}(x)_{𝔭} = 0 ⟺ Hom_{C}^{∗}(c, x)_{𝔭} = 0 for all c ∈ C.

9.3. A projective scheme. We provide a criterion for (Spec C, O_{C}) to be a projective scheme. We assume that the tensor product on C is commutative or that C is generated by a single object as a triangulated category. Thus the thick tensor ideals of C form an ideal lattice, by Proposition 6.7. The following elementary observation will be useful.

Lemma 9.3. Let C be an abelian or triangulated tensor category and (X, σ) be a support datum on C. Then C_{0} = {x ∈ C | σ(x) = ∅} is a thick tensor ideal and (X, σ) induces a support datum (X, σ′) on the quotient C/C_{0} such that σ′(x) = σ(x) for all x ∈ C.

The following is the main result.

Theorem 9.4. Let C be a triangulated tensor category with tensor identity e and suppose that H = End_{C}^{∗}(e) is concentrated in non-negative degrees. Define for each x ∈ C

supp_{H}(x) = {𝔭 ∈ Proj H | End_{C}^{∗}(x)_{𝔭} ≠ 0}

and suppose that

supp_{H}(x ⊗ y) = supp_{H}(x) ∩ supp_{H}(y) for all x, y ∈ C.

Then C_{0} = {x ∈ C | supp_{H}(x) = ∅} is a thick tensor ideal of C and (Proj H, supp_{H}) induces a support datum on the quotient C̄ = C/C_{0}. We obtain an induced morphism

f: (Proj H, O_{H}) → (Spec C̄, OC̄)

of ringed spaces which induces a ring isomorphism O_{C̄,f(𝔭)} ⟶^{∼} O_{H,𝔭} for all 𝔭 ∈ Proj H. In particular, f is an isomorphism if and only if the support datum (Proj H, supp_{H}) is classifying.

Proof. The condition supp_{H}(x⊗y) = supp_{H}(x)∩supp_{H}(y) implies that (Proj H, supp_{H}) is a support datum on C, and therefore also on C̄ by Lemma 9.3. Thus we obtain a continuous map f: Proj H → Spec C̄ satisfying

supp_{H}(x) = f^{−1}(supp(x))

for all x ∈ C̄, by Proposition 4.3.

We need to construct a morphism of sheaves f^{♯}: OC̄ → f_{∗}O_{H}. First observe that for each x ∈ C̄, the ring H acts on Hom∗ ; C̄(e, x) via the quotient functor C → C̄. In particular,

Hom_{C}^{∗}(e, x)_{𝔭} ⟶^{∼} Hom∗ ; C̄(e, x)_{𝔭} for 𝔭 ∈ Proj H

by Proposition 9.1. Now fix an open subset U ⊆ Spec C̄ and consider the composite of functors

F: C̄ ⟶^{Hom∗ ; C̄(e,−)} Mod H ⟶^{(−)}˜ Qcoh Proj H ⟶^{(−)|_{f^{−1}(U)}} Qcoh f^{−1}(U).

Here, we denote for any H-module M by M̃ its associated sheaf. Note that the stalk of M̃ at a homogeneous prime 𝔭 equals the degree zero part M_{(𝔭)} of the localized module M_{𝔭}. We claim that F annihilates C̄_{U}. In fact, x ∈ C̄_{U} implies f^{−1}(supp(x)) ∩ f^{−1}(U) = ∅ and therefore supp_{H}(x) ∩ f^{−1}(U) = ∅. Thus Hom∗ ; C̄(e, x)_{(𝔭)} = 0 for all 𝔭 ∈ f^{−1}(U) and therefore F x = 0. It follows that F factors through C̄/C̄_{U} and induces a map End_{C̄/C̄_{U}} (e) → O_{H}(f^{−1}(U)) which extends to a map OC̄(U) → O_{H}(f^{−1}(U)). This yields the morphism of sheaves f^{♯}: OC̄ → f_{∗}O_{H}.

Now fix a point 𝔭 ∈ Proj H. Then f^{♯} induces a map f_{𝔭}^{♯}: O_{C̄,f(𝔭)} → O_{H,𝔭}. We have an isomorphism O_{C̄,f(𝔭)} ≅ End_{C̄/f(𝔭)}(e) by Lemma 7.1. Next observe that

f(𝔭) = {x ∈ C̄ | End_{C}^{∗}(x)_{𝔭} = 0}.

We have

{e/σ ∈ C̄ | σ ∈ H \ 𝔭} ⊆ f(𝔭) ⊆ {x ∈ C̄ | Hom∗ ; C̄(e, x)_{𝔭} = 0}.

This follows from (9.1), and we obtain a second isomorphism

End_{C̄/f(𝔭)}(e) ≅ End∗ ; C̄(e)_{(𝔭)} ≅ End_{C}^{∗}(e)_{(𝔭)} = O_{H,𝔭}

from Proposition 9.1. We conclude that f_{𝔭}^{♯} is an isomorphism. It follows that f is an isomorphism of ringed spaces if and only if the map Proj H → Spec C̄ is a homeomorphism. This last condition is satisfied if and only if the support datum (Proj H, supp_{H}) is classifying, by Corollary 5.2. □

Note that Theorem 9.4 gives a partial answer to Balmer’s question when (Spec C, O_{C}) is a scheme [1, Rem. 6.4]. The result is best illustrated by the following example from representation theory; see [5, Thm. 7.3] for an alternative discussion.

Example 9.5. Let k be a field and let A = kG be the group algebra of a finite group G or more generally a finite group scheme. We consider the category mod A of finite dimensional A-modules and its bounded derived category D^{b}(mod A). The tensor product ⊗_{k} on mod A induces a tensor product on D^{b}(mod A) which is exact in each variable. The trivial representation k is the tensor identity and its graded endomorphism ring equals the group cohomology ring

H = H^{∗}(G, k) = Ext_{A}^{∗}(k, k).

Note that for x ∈ D^{b}(mod A), we have supp_{H}(x) = ∅ if and only if x belongs to the thick tensor ideal D^{per}(A) of perfect complexes. The composite

mod A ⟶^{inc} D^{b}(mod A) ⟶^{can} D^{b}(mod A)/D^{per}(A)

induces an equivalence \underline{mod} A ⟶^{∼} D^{b}(mod A)/D^{per}(A), where \underline{mod} A denotes the stable module category of A; see for instance [14, Thm 2.1]. The thick tensor ideals of \underline{mod} A have been classified, in case G is a finite group and k is algebraically closed by Benson, Carlson, and Rickard in [3, Thm. 3.4], and for a finite group scheme over an arbitrary field by Friedlander and Pevtsova in [5, Thm. 6.3]. The classification implies that (Proj H, supp_{H}) is a classifying support datum on \underline{mod} A, and therefore we have an isomorphism

(Proj H^{∗}(G, k), O_{H^{∗}(G,k)}) ⟶^{∼} (Spec \underline{mod} kG, O_{\underline{mod} kG})

of ringed spaces by Theorem 9.4.

The following example shows a triangulated tensor category which arises in modular representation theory. The tensor product is not necessarily commutative. This category can be used to define support varieties of representations of finite dimensional algebras, generalizing the classical case of a group algebra.

Example 9.6. Let A be a finite dimensional algebra over a field k and let A^{e} = A ⊗_{k} A^{op} be its enveloping algebra. We consider the category mod A^{e} of finite dimensional A^{e}- modules and the full subcategory B of A^{e}-modules which are projective when restricted to A or A^{op}. Note that B carries an exact structure which is induced from the natural exact structure of mod A^{e}. The inclusion B → mod A^{e} induces a fully faithful exact functor D^{b}(B) → D^{b}(mod A^{e}). The tensor product ⊗_{A} on mod A^{e} is exact in each variable when restricted to B and induces therefore an exact tensor product on D^{b}(B). The A^{e}- module A viewed as a complex concentrated in degree zero is a tensor identity of D^{b}(B). The tensor product restricts to a tensor product on the thick subcategory C ⊆ D^{b}(B) which is generated by A. We have therefore a triangulated tensor category (C, ⊗_{A}, A) and the lattice of thick tensor ideals is an ideal lattice, by Proposition 6.7. Note that the tensor product of C is not necessarily commutative. The graded endomorphism ring of the tensor identity

End_{C}^{∗}(A) = Ext_{A^{e}}^{∗}(A, A) = HH^{∗}(A)

equals the Hochschild cohomology ring of A.

10. Decompositions of ideals

10.1. Decompositions of ideals. Let L be an ideal lattice. A non-zero element a ∈ L is called indecomposable if a = a_{1} ∨ a_{2} implies a_{1} = 0 or a_{2} = 0.

Proposition 10.1. Let L be an ideal lattice and suppose that the space Spec^{∗} L is noetherian. Given a semi-prime a ∈ L, there exists a unique decomposition a = ⋁_{i∈Ω} a_{i} such that

(1) a_{i} is indecomposable and semi-prime for all i ∈ Ω, and

(2) a_{i} ∧ a_{j} = 0 for all i ≠ j in Ω.

Proof. First observe that every open subset of Spec^{∗} L is quasi-compact since Spec^{∗} L is noetherian. Thus Proposition 4.1 provides a bijection b ↦ supp(b) between all semiprimes in L and all subsets of Spec^{∗} L which are unions of closed subsets. Under this bijection, a decomposition a = ⋁_{i} a_{i} satisfying (1) and (2) corresponds to a disjoint union

supp(a) = ⋃_{i} supp(a_{i}).

Now observe that the unions of closed subsets are closed under arbitrary intersections. Thus there exists a partition supp(a) = ⋃_{i} Y_{i} into unions of closed subsets which admits no proper refinement; see Lemma 10.2 below. We obtain the decomposition a = ⋁_{i} a_{i} by taking for a_{i} the semi-prime satisfying supp(a_{i}) = Y_{i}. □

Lemma 10.2. Let X be a set and Y be a family of subsets which is closed under forming intersections. Then there exists for each Y ∈ Y a unique partition Y = ⋃_{i∈Ω} Y_{i} into non-empty subsets from Y which admits no proper refinement. More precisely, for all i, a disjoint union Y_{i} = Y_{i1} ∪ Y_{i2} with Y_{i1}, Y_{i2} ∈ Y implies Y_{i1} = ∅ or Y_{i2} = ∅.

Proof. Let (⋃_{i∈Ω_{s}} Y_{si})_{s∈Σ} be the family of all partitions of Y with Y_{si} ∈ Y for all s, i. For each x ∈ Y, let

Y_{x} = ⋂_{{s∈Σ \atop x∈Y_{si}}} Y_{si}.

Then Y = ⋃_{x∈Y} Y_{x} is a partition which admits no proper refinement. □

Remark 10.3. There are refinements of Proposition 10.1 which do not require the space Spec^{∗} L to be noetherian. For instance, if a ∈ L is compact and the space supp(a) is noetherian (with the induced topology), then we have a unique decomposition a = ⋁_{i=1}^{n} a_{i} into indecomposables such that a_{i} ∧ a_{j} = 0 for all i ≠ j.

10.2. Decompositions in tensor categories. Let C be an abelian or triangulated tensor category. We consider the lattice L_{thick}(C) of thick tensor ideals of C and recall the following definition from [10]. Given a thick tensor ideal D, a family (D_{i})_{i∈Ω} of thick tensor ideals is a decomposition of D if

(1) the objects in D are the finite coproducts of objects from the D_{i}, and

(2) D_{i} ∩ D_{j} = 0 for all i ≠ j.

Such a decomposition is denoted by D = ∐_{i∈Ω} D_{i}, and we call D indecomposable if D ≠ 0 and any decomposition D = D_{1} ⨿ D_{2} implies D_{1} = 0 or D_{2} = 0.

The decomposition D = ⋁_{i} D_{i} of a thick tensor ideal (as discussed in Proposition 10.1) amounts to a decomposition D = ∐_{i} D_{i}, provided that

D_{1} ∧ D_{2} = 0 ⟹ D_{1} ∨ D_{2} = D_{1} ⨿ D_{2}

for every pair of thick tensor ideals D_{1}, D_{2}. This property holds if C admits an appropriate internal Hom-functor, because then D_{1} ∧ D_{2} implies Hom_{C}(D_{1}, D_{2}) = 0. We do not go into details, but refer to the literature. A treatment of decompositions in the stable module category \underline{mod} kG can be found in [10], where kG denotes the group algebra of a finite group G. For further discussions, see the recent work of Balmer [2] and Chebolu [4].

References

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[16] R. W. Thomason and T. Trobaugh: Higher algebraic K-theory of schemes and of derived categories. In: The Grothendieck Festschrift, III, Birkhäuser, Progress in Mathematics 87 (1990), 247–436.

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Aslak Bakke Buan, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway.

E-mail address: aslakb@math.ntnu.no

Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany.

E-mail address: hkrause@math.uni-paderborn.de

Øyvind Solberg, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway.

E-mail address: oyvinso@math.ntnu.no