COMPLETE INTERSECTIONS AND DERIVED CATEGORIES.

D.J.BENSON AND J.P.C.GREENLEES

Abstract. We propose a definition of when a triangulated category should be considered a complete intersection. We show that for the derived category of a complete local Noetherian commutative ring R, the condition on the derived category D(R) holds precisely when R is a complete intersection in the classical sense.

1. Introduction

1.A. The context. In algebraic geometry, the best behaved varieties are complete intersections, which are subvarieties of affine space specified by the right number of equations in the sense that if they are of codimension c then only c equations are required. Considering this locally, we obtain an analogue for commutative local Noetherian rings R, where the corresponding condition is that

R = Q/(f_{1}, …, f_{c})

for a regular local ring Q by a regular sequence, f_{1}, f_{2}, …, f_{c} The smallest possible value of c (as Q and the regular sequence vary) is called the codimension of R. In commutative algebra the terminology is slightly different, so that R is a complete intersection (ci) if its completion is of the stated form.

We recall below that if R is ci of codimension c, one may construct a resolution of any finitely generated module growing like a polynomial of degree c − 1. In particular, the ring Ext_{R}^{∗}(k, k) has polynomial growth (we say that R is gci). Perhaps the most striking result about ci rings is the theorem of Gulliksen [14] which states that the ci and gci conditions are equivalent.

In fact one may go further and show that the resolutions are constructed in an eventually multi-periodic fashion. In particular, for a ci local ring Ext_{R}^{∗}(k, k) is finite as a module over a commutative Noetherian ring of operators [15, 3.1]. This in turn opens the way to the theory of support varieties for modules over a ci ring [4].

1.B. The aspiration. We are interested in extending the notion of ci rings to commutative differential graded algebras (DGAs), commutative ring spectra [12, 17] and related contexts. We have a particular interest in studying the cochains C^{∗}(BG) on the classifying space of a finite group G, partly because of the consequences for the cohomology ring H^{∗}(BG). We would like to follow the model of [10], which shows that C^{∗}(BG) is Gorenstein in a homotopical sense for all finite groups G. This structural result then establishes the existence of a local cohomology theorem for H^{∗}(BG), and in particular proves the result of Benson and Carlson [6] that if H^{∗}(BG) is Cohen-Macaulay it is also Gorenstein.

For C^{∗}(BG), the counterpart of the Ext algebra Ext_{R}^{∗}(k, k) is the loop space homology of the p-completed classifying space, H_{∗}(Ω(BG_{p}^{∧})). Of course if G is a p-group, this is simply the group ring kG in degree 0. More generally, it is known to be of polynomial growth in many cases (Chevalley groups away from the defining characteristic, A_{4} or M_{11} in characteristic 2, ....). However R. Levi [19, 20, 21, 22] has proved there is a dichotomy between small growth and large growth, and given simple examples (semidirect products (C_{p} × C_{p}) ⋊ C_{3} at p ≥ 5) where the growth is exponential. Our aim is to give a version of the definition of ci, which is homotopy invariant and applies to the module category; when it holds for C^{∗}(BG), it should imply that Ω(BG_{p}^{∧}) has polynomial growth.

The research was partially supported by EPSRC Grant number EP/E012957/1.

The purpose of the present paper is to lay foundations for these generalizations by giving a characterization of complete intersections in terms of the derived category D(R) of Rmodules. It will be apparent that the definition applies to related triangulated categories such as the homotopy category of C^{∗}(BG)-modules. We explain briefly why existing definitions are not suitable for us. Firstly, although polynomial growth rate of the endomorphism ring of k is an interesting and important property, its application is somewhat limited since it relies on having something like a Hilbert series as a measure of size. The difficulty with Avramov’s [3] characterization in terms of André-Quillen homology (Quillen’s conjecture) is rather different. To start with, it cannot be stated in terms of the (additive) derived category, but more significant for us is that it turns out that the topological André-Quillen cohomology vanishes rather generally in the case of cochains on a space [23], so that it includes far too many spaces.

The topological and representation theoretic examples will be treated elsewhere [13, 8, 7].

1.C. Organization of the paper. In Section 2 we will give a definition which applies to any triangulated category with suitable additional structure, and briefly describe some interesting classes of examples.

We then consider the particular example of the derived category of a commutative local ring, showing that our new definition does characterize ci rings. More precisely, in Section 3 we outline our results for commutative rings. Section 4 makes some observations about the endomorphisms of the identity functor of the derived category. Sections 5 and 6 describe the Avramov-Gulliksen multiperiodic resolutions in a suitable form, showing that a ci ring satisfies our condition, and in Section 7 we establish the connection with the Ext algebra having polynomial growth rate.

1.D. Grading conventions. We will have cause to discuss homological and cohomological gradings. Our experience is that this a frequent source of confusion, so we adopt the following conventions. First, we refer to lower gradings as degrees and upper gradings as codegrees. As usual, one may convert gradings to cogradings via the rule M_{n} = M^{−n}. Thus both chain complexes and cochain complexes have differentials of degree −1 (which is to say, of codegree 1). This much is standard. However, since we need to deal with both chain complexes and cochain complexes it is essential to have separate notation for homological suspensions (Σ^{i}) and cohomological suspensions (Σ_{i}): these are defined by

(Σ^{i}M)_{n} = M_{n−i} and (Σ_{i}M)^{n} = M^{n−i}.

Thus, for example, with reduced chains and cochains of a based space X, we have

C_{∗}(Σ^{i}X) = Σ^{i}C_{∗}(X) and C^{∗}(Σ^{i}X) = Σ_{i}C^{∗}(X).

2. The definition.

Before giving the definition, we recall some standard technology and background results.

2.A. Terminology for triangulated categories. Recall that an object X of a triangulated category T is called small if the natural map

⨁_{i}[X, Y_{i}] ⟶ [X, ⨁_{i} Y_{i}]

is an isomorphism for any set of objects Y_{i}.

A thick subcategory of T is a full subcategory closed under completion of triangles and taking retracts. We write thick(X) for the smallest thick subcategory containing X, and if A ∈ thick(X) we also say ‘X finitely builds A’ and write X ⊧ A.

A localizing subcategory of T is a full subcategory closed under completion of triangles and taking arbitrary coproducts. We write loc(X) for the smallest localizing subcategory containing X, and if A ∈ loc(X) we also say ‘X builds A’ and write X ⊢ A.

Following [11] we say that X is virtually small if thick(X) contains a non-trivial small object W, and we say that any such W is a witness for the fact that X is virtually small.

2.B. Regularity. The archetype of a derived characterization of a property of rings is the Auslander-Buchsbaum-Serre characterization of regularity: a Noetherian local ring R is regular if and only if the residue field has a finite resolution by finitely generated projectives; when this holds, every finitely generated module has a finite resolution by finitely generated projectives. Since small objects are precisely those equivalent to finite complexes of finitely generated projectives, this gives a characterization in terms of structure intrinsic to the triangulated category together with a small amount of additional structure.

We will rephrase these results in a style suggestive of our intentions for complete intersections. First, since finitely generated modules are not characterized in terms of the triangulated structure, we posit a triangulated subcategory FG of objects to play the corresponding role. In the case of D(R) we take the subcategory FG to consist of bounded complexes of finitely generated modules.

We may package this in three definitions. The first is ideal-theoretic, the second refers to the rate of growth of the derived endomorphism of the simple module k, and the third is in terms of the homotopy theory of modules.

Definition 2.1. (i) A local Noetherian ring R is regular if the maximal ideal is generated by a regular sequence.

(ii) A local Noetherian ring R is g-regular if Ext_{R}^{∗}(k, k) is finite dimensional.

(iii) A local Noetherian ring R is h-regular if every finitely generated module is small in D(R).

It is not hard to see that g-regularity is equivalent to h-regularity or that regularity implies g-regularity. Serre proved that g-regularity implies regularity, so the three conditions are equivalent.

We will provide a characterization of ci rings in the same style.

2.C. Work of Dwyer-Greenlees-Iyengar. We set the context by recalling the work of [11] which led us to our definition.

Definition 2.2. ([11]) The ring R is said to be a quasi-complete intersection (qci) if every object of FG is virtually small.

The reason for the terminology is that it was shown in [11] that any ci local ring is qci, and conversely that every qci ring is Gorenstein. Examples were given to show that not all Gorenstein rings are qci.

The difficulty with the qci condition is that there is no control over how a complex X builds a non-trivial small object. It is therefore hard to deduce consequences about polynomial growth for example.

2.D. The centre of a triangulated category. To start with we suppose given a triangulated category T . The centre ZT of T is defined to be the graded ring of endomorphisms of the identity functor. This is a graded commutative ring.

If T is tensor triangulated, the endomorphism ring of the unit element gives elements of ZT , but typically there are many others.

Given χ ∈ ZT of degree a, for any object X, we may form the mapping cone X/χ of χ : Σ^{a}X ⟶ X. This is well defined up to non-unique equivalence. Indeed, given a map f : X ⟶ Y, the axioms of a triangulated category give a map f : X/χ ⟶ Y/χ consistent with the defining triangles, but this is not usually unique or compatible with composition.

Now given a sequence of elements χ_{1}, χ_{2}, …, χ_{n} we may iterate this construction, and form

K(X; χ) := X/χ_{1}/χ_{2}/ ⋯ /χ_{n},

which we sometimes refer to as the Koszul complex of the sequence.

Lemma 2.3. Up to equivalence K(X; χ) only depends on the sequence, and is independent of the order of the elements χ_{i}.

Proof. For the first statement we argue by induction on n. We have already dealt with the case n = 1. Suppose then that n ≥ 2 and Y and Y′ are two models for K(X; χ_{1}, …, χ_{n−1}). By induction there is an equivalence e : Y ⟶ Y′. Since χ_{n} is in the centre we have a commutative square

Y ⟶^{χ_{n}} Y
e ↓≃  e ↓≃
Y′  ⟶^{χ_{n}} Y′,

and so the axioms provide an equivalence Y/χ_{n} ≃ Y′/χ_{n} between any choices of completions of the triangles.

For the second statement, it suffices to deal with the case n = 2, since transpositions generate the symmetric group. For the case n = 2 we apply the octahedral axiom to the square

\begin{matrix}X & ⟶^{χ_{1}} & X \\ χ_{2} ↓ & & ↓ χ_{2} \\ X & ⟶^{χ_{1}} & X. \end{matrix}  □

2.E. Examples. We consider a triangulated category T , together with a triangulated subcategy FG of objects we call finitely generated. We recommend that readers concentrate on the first example, viewing the others as generalizations to be investigated. We treat Example 2.4 in the present paper. We defer the investigations of Examples 2.5 and 2.6 to [8] and [7].

Example 2.4. Our motivating example has T = D(R) for a commutative Noetherian ring R. The small objects are those equivalent to a bounded complex of finitely generated projectives. The class FG consists of the bounded complexes of finitely generated modules.

Example 2.5. The topological example has T = Ho(C^{∗}(X)) for a space X. The small objects include the modules C^{∗}(Y) where Y is a space with a map f : Y ⟶ X and so that the homotopy fibre F (f) has finite homology.

Example 2.6. An algebraic replacement for the category Ho(C^{∗}(BG)) is given by taking T to be the k-cellularization of K(Inj kG) for a finite group G as in [9].

Example 2.7. The category T = StMod(kG) is the stable module category for a finite group G. The small objects consist of the subcategory stmod(kG) of finitely generated modules.

This suggests that all G should be considered stably regular and that FG should again consist of finitely generated modules.

Example 2.8. For a Noetherian scheme X we take the derived category T = D(qCoh(X)) of complexes of quasi-coherent sheaves on X. The small objects include vector bundles (i.e., locally free sheaves). We take FG = D^{b}(coh(X)) to be the category of bounded complexes of coherent sheaves.

2.F. The definition. Recall that if S is a commutative ring, then the Jacobson radical J(S) is defined to be the intersection of the maximal ideals. It is convenient to use the following well-known characterization.

Lemma 2.9. We have x ∈ J(S) if and only if 1 − xy is a unit for all y ∈ S. □

Finally we may give the definition.

Definition 2.10. We say that a triangulated category T with a subcategory FG is centrally ci (zci) if there are elements χ_{1}, …, χ_{c} in J(ZT ) so that for all objects X of FG, the Koszul complex K(X; χ) is small. The smallest such c is called the codimension of T .

First, we note that the definitions are arranged so that it is a tautology that if T is h-regular, it is also zci of codimension 0. We refer to zci categories of codimension 1 as z-hypersurface categories.

3. Statement of results.

For the rest of the present paper we restrict attention to the category T = D(R) for a Noetherian local ring R, and FG denotes the subcategory of bounded complexes with finitely generated homology.

3.A. Complete intersections. In commutative algebra there are three styles for a definition of a complete intersection ring: ideal theoretic, in terms of the growth of the Ext algebra and a derived version.

Definition 3.1. (i) A local Noetherian ring R is a complete intersection (ci) ring if its completion is of the form Q/(f_{1}, f_{2}, …, f_{c}) for some regular ring Q and some regular sequence f_{1}, f_{2}, …, f_{c}. The minimum such c (over all Q and regular sequences) is called the codimension of R.

(ii) A local Noetherian ring R is gci if Ext_{R}^{∗}(k, k) has polynomial growth. The g-codimension of R is one more than the degree of the growth.

(iii) A local Noetherian ring R is zci if there are elements z_{1}, z_{2}, … z_{c} ∈ J(ZD(R)) so that M/z_{1}/z_{2}/ ⋯ /z_{c} is small for all complexes M in FG. The minimum such c is called the z-codimension of R.

Remark 3.2. To avoid confusion we should comment on a trivial but awkward detail of terminology.

Since the gci condition is insensitive to completion, the completion in the ci definition is necessary for the conditions to be equivalent for incomplete rings. It is also completely standard. On the other hand, it is inconvenient that there is no simple terminology for a local ring of the form Q/(f_{1}, …, f_{c}) for a regular ring Q and a regular sequence f_{1}, …, f_{c}, and that the algebraic geometry terminology has been subverted in commutative algebra. We have chosen to let the zci terminology correspond to the algebraic geometry conventions.

The main result of the present paper is the following.

Theorem 3.3. For a complete Noetherian local ring R, the ci condition, the gci condition and the zci condition are equivalent. The notions of codimension also agree.

The fact that ci implies zci amounts to the classical Avramov-Gulliksen construction of eventually multiperiodic resolutions of modules over a ci ring. More precisely, because the construction is uniform, the construction can be lifted to Hochschild cohomology and hence provides elements of the centre of D(R). This is described in Sections 5 and 6. The fact that zci implies gci is a rather straightforward estimate of growth given in Section 7. Finally, the fact that gci implies ci is Gulliksen’s theorem.

3.B. Hypersurfaces. In commutative algebra there are three styles for a definition of a hypersurface ring: ideal theoretic, in terms of the growth of the Ext algebra and a derived version. In this case the second and third styles have two variants each.

Definition 3.4. (i) A local Noetherian ring R is a hypersurface ring if R = Q/(f) for some regular ring Q and non-zero element f.

(ii) A local Noetherian ring R is a g-hypersurface ring if Ext_{R}^{∗}(k, k) has polynomial growth of degree ≤ 1. We say that R is a p-hypersurface ring if Ext_{R}^{∗}(k, k) is eventually periodic.

(iii) A local Noetherian ring R is a z-hypersurface ring if there is an element z ∈ J(ZD(R)) so that M/z is small for all finitely generated modules M. We say R is a q-hypersurface ring if for each finitely generated module M there is a self-map f = f_{M} : Σ^{n}M ⟶ M so that M/f non-trivial and small.

Lemma 3.5. For a complete local Noetherian ring R, the conditions hypersurface, g-hypersurface, p-hypersurface and z-hypersurface are all equivalent. They imply the q-hypersurface condition.

Proof. We have

hypersurface ⇒ p-hypersurface ⇒ g-hypersurface;

the first of these is proved in Section 5 and the second is a tautology. Avramov [2, 8.1.1.3] shows that

g-hypersurface ⇒ hypersurface, so that the first three conditions are equivalent.

We have

hypersurface ⇒ z-hypersurface ⇒ q-hypersurface;

the first of these is proved in Section 5 and the second is clear from the construction since the operator is of non-zero degree.

We show in Remark 5.2 that

z-hypersurface ⇒ p-hypersurface.  □

4. The Jacobson radical of ZD(R).

We give partial information on the Jacobson radical of the centre of D(R) sufficient to let us show that the derived category of a complete intersection is indeed zci.

4.A. Degree and the Jacobson radical. There is one easily verified condition for membership of the Jacobson radical.

Proposition 4.1. If R is a commutative ring then any element of ZD(R) of non-zero degree lies in the Jacobson radical of ZD(R).

Proof. We first prove that an element χ of negative codegree is in the Jacobson radical. Start by noting that χ is zero on modules M since negative Ext groups are zero. Now we argue by induction on l that χ is nilpotent on complexes X of length l. Indeed, if X is of length l + 1 we may take Y to be the subcomplex omitting the top nonzero entry of X, and we have the short exact sequence

0 ⟶ Y ⟶ X ⟶ Σ^{d}M ⟶ 0

where the top of X is M in degree d. The map χ induces a map of the resulting triangle. It is zero on M, and χ^{n} = 0 on Y for some n by induction. Thus χ^{n} : X ⟶ X lies in a triangle with two zero maps, and is thus nilpotent.

Thus for any y ∈ ZD(R) the element 1 − χy acts as a unit on all bounded complexes. Consider it as a map of an arbitrary complex Z; since any homology class is supported on a finite subcomplex the induced map on homology is an isomorphism, and 1 − χy is invertible on Z. In more detail, to see that 1 − χy is surjective on homology, pick a homology class α ∈ H_{∗}(Z), and then a bounded subcomplex Z′ of Z with α = i_{∗}α′ where i : Z′ ⟶ Z is the inclusion. Since 1 − χy is an isomorphism on Z′, we may choose α̃′ so that α′ = (1 − χy)(α̃′). Now α = (1 − χy)(i_{∗}(α̃′)). The proof of injectivity is similar.

Now if χ is of positive codegree, and 1 − χy is homogeneous of degree 0, then y is of negative codegree and hence in the Jacobson radical by the above argument. Thus 1 − χy is a unit as required. □

4.B. Bimodules and the centre. Bimodules provide a useful source of elements of ZD(R). Indeed, if R is a flat l-algebra, and if X ⟶ Y is a map of R-bimodules over l (which is to say, of modules over R^{e} = R ⊗_{l} R), then for any R-module M we obtain a map

X ⊗_{R} M ⟶ Y ⊗_{R} M

of R-modules, natural in M.

It is convenient to package this in terms of the Hochschild cohomology ring

HH^{∗}(R|l) = Ext_{R^{e}}^{∗}(R, R).

If l = ℤ it is usual to omit it from the notation. Thus we obtain elements of the centre from Hochschild cohomology in the form of a ring homomorphism

HH^{∗}(R|l) ⟶ ZD(R).

If R is an l-algebra which is not flat, R^{e} = R ⊗_{l} R is taken in the derived sense, and similarly for HH^{∗}(R|l).

Given maps l ⟶ l′ ⟶ R, we obtain a map R ⊗^{l} R ⟶ R ⊗^{l′} R and hence a ring map HH^{∗}(R|l′) ⟶ HH^{∗}(R|l). In particular, we have maps

R = HH^{∗}(R|R) ⟶ HH^{∗}(R|l) ⟶ HH^{∗}(R|ℤ) = HH^{∗}(R).

4.C. Hochschild cohomology transcended. It seems natural to relax the role of Hochschild cohomology. If the R^{e}-module R is virtually small, this will suffice for most purposes. Indeed, we may suppose R ⊧_{R^{e}} W with W small and non-trivial. Given any R-module M we may apply the functor ⊗_{R}M to the building process, to find

M = R ⊗_{R} M ⊧_{R} W ⊗_{R} M.

On the other hand, since

R^{e} ⊧_{R^{e}} W,

we conclude

R ⊗_{l} M ≃ R^{e} ⊗_{R} M ⊧_{R} W ⊗_{R} M.

If M is small over l (for example if l is regular and M is finitely generated over l) then R ⊗_{l} M is small, and it follows that W ⊗_{R} M is small.

This suggests defining R to be bimodule ci (bci) if R is a virtually small bimodule over l for some regular ring l over which R is small.

Remark 4.2. The relationship between bci and qci deserves further investigation. If W is an R^{e}-module, we may consider the class V (W) consisting of R-modules M so that M ⊗_{R}W ≃ 0. This is a localizing subcategory of D(R). For any virtually small R^{e}-module X we may then consider the collection I(X) of all localizing subcategories of the form V (W) as W runs through small R^{e}-modules finitely built by X. If I(X) contains a localizing subcategory V (W) intersecting FG in 0, then bci implies qci.

5. Hypersurfaces.

In this section we suppose R is a hypersurface ring, so that we have a short exact sequence

0 ⟶ Q ⟶^{f} Q ⟶ R ⟶ 0

of Q-modules for a regular local ring Q. There are two basic constructions that we need to generalize.

5.A. The degree 2 operator. We describe a construction of a cohomological operator due to Gulliksen [15, 2].

Given an R-module M we may apply (·) ⊗_{Q} M to the defining sequence to obtain the short exact sequence

0 ⟶ Tor_{1}^{Q}(R, M) ⟶ M ⟶^{f} M ⟶ R ⊗_{Q} M ⟶ 0.

Since f = 0 in any R-module, we conclude

Tor_{1}^{Q}(R, M) ≅ M, R ⊗_{Q} M ≅ M,

and

Tor_{i}^{Q}(R, M) = 0 for i ≥ 2.

Next, choose a free Q-module F with an epimorphism to M, giving a short exact sequence

0 ⟶ K ⟶ F ⟶ M ⟶ 0

of Q-modules. Applying (·) ⊗_{Q} R, we obtain

0 ⟶ Tor_{1}^{Q}(R, M) ⟶ \overline{K} ⟶ \overline{F} ⟶ M ⟶ 0.

Since Tor_{1}^{Q}(R, M) ≅ M, this gives an element

χ_{f} ∈ Ext_{R}^{2}(M, M),

or a map

χ_{f} : M ⟶ Σ^{2}M

in the derived category of R-modules.

We will see in Subsection 5.D that this construction lifts to give an element

χ_{f} ∈ HH^{2}(R|Q),

and hence in particular that it gives an element of the centre ZD(R) of D(R).

5.B. The eventually periodic resolution. Continuing with the above case, we may show that all modules M have free resolutions over R which are eventually periodic of period 2.

Indeed, M has a finite free Q-resolution

0 ⟶ F_{n} ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0.

Adding an extra zero term if necessary, we suppose for convenience that n is even. Now apply (·) ⊗_{Q} R to obtain a complex

0 ⟶ \overline{F}_{n} ⟶ \overline{F}_{n−1} ⟶ ⋯ ⟶ \overline{F}_{1} ⟶ \overline{F}_{0} ⟶ M ⟶ 0.

Since Tor_{i}^{Q}(R, M) = 0 for i ≥ 2, this is exact except in homological degree 1, where it is Tor_{1}^{Q}(R, M) ≅ M. Splicing in a second copy of the resolution, we obtain a complex

0 ⟶ \overline{F}_{n} ⟶ \overline{F}_{n−1} ⟶ ⋯ ⟶ \overline{F}_{2} ⟶ \overline{F}_{1} ⟶ \overline{F}_{0} ⟶ M ⟶ 0
⊕ ⊕ ⊕ ⊕ ↗
\overline{F}_{n−1} ⟶ \overline{F}_{n−2} ⟶ \overline{F}_{n−3} ⟶ ⋯ ⟶ \overline{F}_{0}

which is exact except in the second row, in homological degree 4, where the homology is again M. We may repeatedly splice in additional rows to obtain a free resolution

… ⟶ G_{3} ⟶ G_{2} ⟶ G_{1} ⟶ G_{0} ⟶ M ⟶ 0

over R. Remembering the convention that n is even, provided the degree is at least n, the modules in the resolution are

G_{2i} = \overline{F}_{n} ⊕ \overline{F}_{n−2} ⊕ ⋯ ⊕ \overline{F}_{2} ⊕ \overline{F}_{0}

in even degrees and

G_{2i+1} = \overline{F}_{n−1} ⊕ \overline{F}_{n−3} ⊕ ⋯ ⊕ \overline{F}_{3} ⊕ \overline{F}_{1}

in odd degrees.

5.C. Smallness. We may reformulate the eventual periodicity of the previous subsection in homotopy invariant terms.

Lemma 5.1. If R is a hypersurface R = Q/(f) and M is a finitely generated R-module then the mapping cone of χ_{f} : M ⟶ Σ^{2}M is small.

Proof. From the Yoneda interpretation, we notice that

χ_{f} : M ⟶ Σ^{2}M

is realized by the quotient map factoring out the first row subcomplex

0 ⟶ \overline{F}_{n} ⟶ \overline{F}_{n−1} ⟶ ⋯ ⟶ \overline{F}_{1} ⟶ \overline{F}_{0} ⟶ 0.

Thus the short exact sequence

0 ⟶ \overline{F}_{•} ⟶ G_{•} ⟶ Σ^{2}G_{•} ⟶ 0

of R-free chain complexes realizes the triangle

Σ^{1}M/χ ⟶ M ⟶ Σ^{2}M.  □

Remark 5.2. Note that if there is a map χ : M ⟶ Σ^{2}M whose fibre F is small, then we may form an eventually 2-periodic resolution of M. Indeed, we may suppose F is a finite complex of finitely generated projectives, and then realize the composite ΣF ⟶ ΣM ⟶ F by a map δ of complexes. The double complex formed from ⨁_{n≥0} Σ^{n}F by using δ to relate the successive copies of F is equivalent to M. It is 2-periodic once n is larger than the span of F.

5.D. Lifting to Hochschild cohomology. We now explain how the degree 2 operators can be lifted to Hochschild cohomology classes, working here in the underived context. We assume that Q and R are flat l-algebras, and use the standard notation Q^{e} = Q ⊗_{l} Q, R^{e} = R ⊗_{l} R, and the formula

HH^{∗}(R|l) = Ext_{R^{e}}^{∗}(R, R)

for the Hochschild cohomology.

Next, note that there is an exact sequence

0 ⟶ Q ⊗_{l} R ⟶^{f⊗1} Q ⊗_{l} R ⟶ R^{e} ⟶ 0

so that

Tor_{1}^{Q⊗_{l}R}(M, R^{e}) ≅ M.

Now define J by the exact sequence

(1)  0 ⟶ J ⟶ Q ⊗_{l} R ⟶ R ⟶ 0

of (Q, R)-bimodules. We obtain the required relationship by forming

R ⊗_{Q} (1) ⊗_{R} M

in two different ways.

First we apply R ⊗_{Q} (·) to obtain an exact sequence

(2)  0 ⟶ R ⟶ \overline{J} ⟶ R^{e} ⟶ R ⟶ 0

of R^{e}-modules: this is the element χ_{f} ∈ HH^{2}(R).

To see that χ_{f} gives the required endomorphism of the identity functor on D(R) we proceed as follows. For any R-module M, we can apply (·) ⊗_{R} M to Equation (2) to obtain an exact sequence

0 ⟶ M ⟶ \overline{\overline{J}} ⟶ R ⊗_{l} M ⟶ M ⟶ 0.

It remains to observe that this can also be obtained from an exact sequence of Q-modules by tensoring down. However for this we need only apply (·) ⊗_{R} M to Equation (1) to obtain

0 ⟶ \overline{J} ⊗_{R} M ⟶ Q ⊗_{l} M ⟶ M ⟶ 0,

which qualifies as a sequence

0 ⟶ K ⟶ F ⟶ M ⟶ 0

as in Subsection 5.A.

6. General complete intersections.

Building on the hypersurface case, we can deal with general complete intersections. In the general case, we suppose R = Q/(f_{1}, …, f_{c}) where Q is a regular local ring and f_{1}, f_{2}, …, f_{c} is a regular sequence.

As a matter of notation, let Q = Q_{0}, and Q_{i} = Q/(f_{1}, …, f_{i}), so that Q_{c} = R and we have a sequence of rings

Q = Q_{0} ⟶ Q_{1} ⟶ ⋯ ⟶ Q_{c} = R.

The contents of Subsection 5.A only required f to be regular and made no requirement on Q. We deduce that for any Q_{s+1}-module M_{s+1} we have

Tor_{i}^{Q_{s}}(Q_{s+1}, M_{s+1}) = \{ \begin{matrix}M_{s+1} if i = 0 \\ M_{s+1} if i = 1 \\ 0 if i ≥ 2 \end{matrix} .

We have an operator χ_{j} = χ_{f_{j}} on Q_{j}-modules.

Proposition 6.1. For any finitely generated R-module M the iterated mapping cone

M/χ_{c}/χ_{c−1}/ ⋯ /χ_{1}

is finitely built from M and small over R.

Proof. In the hypersurface case, we constructed an explicit eventually periodic resolution. That is to say, in large degrees the resolution is obtained by splicing together a fixed resolution. In general the analogue of a periodic resolution is a multigraded resolution obtained by stacking boxes. For complete intersections, one may build an explicit, resolution which is multigraded and eventually obtained by stacking boxes.

Suppose M is an R-module. We begin with the construction of Subsection 5.B. This gives a Q_{0}-resolution

0 ⟶ F_{n} ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0

of M, which we tensor down to give a Q_{1}-complex

0 ⟶ \overline{F}_{n} ⟶ \overline{F}_{n−1} ⟶ ⋯ ⟶ \overline{F}_{1} ⟶ \overline{F}_{0} ⟶ M ⟶ 0.

This is not exact, so we splice together copies until we obtain a free Q_{1}-resolution

⋯ ⟶ G_{2} ⟶ G_{1} ⟶ G_{0} ⟶ M ⟶ 0,

and there is a short exact sequence of Q_{1}-chain complexes

\overline{F}_{•} ⟶ G_{•} ⟶ Σ^{2}G_{•}

realizing

Σ^{1}M/χ_{1} ⟶ M ⟶ Σ^{2}M.

Thus

M/χ_{1} ≃ Σ^{−1}\overline{F}_{•}

is Q_{1}-small.

We may repeat the process, applying (·) ⊗_{Q_{1}} Q_{2} to give a Q_{2}-complex

⋯ ⟶ \overline{G}_{2} ⟶ \overline{G}_{1} ⟶ \overline{G}_{0} ⟶ M ⟶ 0.

Since Tor_{i}^{Q_{1}}(Q_{2}, M) = 0 for i ≥ 2 it is exact except in degree 2, where it is Tor_{1}^{Q_{1}}(Q_{2}, M) ≅ M, and we may splice together complexes to give a free Q_{2}-resolution

⋯ ⟶ H_{2} ⟶ H_{1} ⟶ H_{0} ⟶ M ⟶ 0.

There is a short exact sequence of chain complexes

\overline{G}_{•} ⟶ H_{•} ⟶ Σ^{2}H_{•}

realizing

Σ^{1}M/χ_{2} ⟶ M ⟶ Σ^{2}M.

Thus

M/χ_{2} ≃ Σ^{−1}\overline{G}_{•},

and therefore

M/χ_{2}/χ_{1} ≃ Σ^{−2}\overline{\overline{F}}_{•}

is Q_{2}-small.

Repeating this process c times, we obtain the desired conclusion. □

7. Growth conditions.

Throughout algebra and topology it is common to use the rate of growth of homology groups as a measurement of complexity. In this section we show that if R satisfies the zci condition then its Ext algebra has polynomial growth (i.e., R satisfies the gci condition). Gulliksen’s theorem then completes the proof of equivalence of the characterizations of complete intersections. Because of later applications we will prove results in greater generality than we require here.

7.A. Polynomial growth. Often, one knows that the homology groups will be zero in negative degrees, so the growth condition will be on the positive homology groups. Similarly, if homology groups are zero in positive degrees, the growth condition will be on the negative homology groups. For us it is convenient to treat positive and negative degrees symmetrically.

To continue, suppose that H is a Noetherian graded ring and that M is an H-module (we have in mind the case the H is the homology of a DGA, and M is the homology of a DG-module over it). If H_{0} is local, we write len(M_{i}) for the length of M_{i}. To extend the notion to more general rings we impose a condition to give a reasonable notion of length.

Definition 7.1. (i) An H-module M is locally finite if M is a Noetherian H-module and len((M_{i})_{𝔪_{0}}) is finite for all i and all maximal ideals 𝔪_{0} of H_{0}.

(ii) If M is locally finite then since it is Noetherian, it has finitely many associated primes ℘, and we may write

len(M_{i}) = ∑_{℘∈Ass(M)} len((M_{i})_{℘_{0}}).

Assembling these we obtain a notion of growth.

Definition 7.2. We say that a locally finite module M has polynomial growth of degree ≤ d and write growth(M) ≤ d if there is a polynomial p(x) of degree d with

len(X_{n}) ≤ p(|n|)

for almost all n.

Remark 7.3. A bounded module has growth ≤ −1. For modules with growth d ≥ 0, by adding a constant to the polynomial, we may insist that the bound applies for all n ≥ 0.

7.B. Degree 0 elements are redundant. We show that degree 0 elements of the Jacobian do not reduce the growth rate.

Lemma 7.4. If x ∈ J(H) and |x| = 0 then for any locally finite H-module M

growth(M/xM) = growth(M).

Proof. Since M is Noetherian it has finitely many associated primes ℘. If ℘ is an associated prime, the degree 0 part ℘_{0} is maximal in H_{0}, since len(M_{i}) < ∞. Altogether, we suppose 𝔪_{0}^{1}, …, 𝔪_{0}^{S} are the finitely many maximal primes of H_{0} occurring in this way. We thus have

len(M_{i}) = ∑_{s} len((M_{i})_{𝔪_{0}^{s}}),

and hence

growth(M_{i}) = max{growth((M)_{𝔪_{0}^{s}}) | s = 1, …, S}

In short, we may assume that R_{0} is local, and we write 𝔪_{0} for its maximal ideal.

Next, since ann(M) is finitely generated, ann(M)_{0} is 𝔪_{0}-primary, and therefore there is an N so that 𝔪_{0}^{N} · M_{i} = 0 for all i.

Now

len(M_{i}) ≤ len(𝔪_{0}/𝔪_{0}^{N}) · len(M_{i}/𝔪_{0}M_{i}),

and hence

len(M_{i}/xM_{i}) ≥ len(M_{i}/𝔪_{0}M_{i}) ≥ \frac{len(M_{i})}{len(𝔪_{0}/𝔪_{0}^{N})}.

This shows growth(M/xM) ≥ growth(M) and completes the proof. □

7.C. Mapping cones and growth rate. We now turn to the special case when we have a DGA R, and a DG-module X over R, and we take H = H_{∗}(R) and consider the growth of M = H_{∗}(X). Of course, we are concentrating on the special case when R is itself concentrated in a single degree, so that this means H = R and X is nothing but a chain complex.

Definition 7.5. We say that X is locally finite over R if H_{∗}(X) is locally finite over H_{∗}(R), and that X has growth ≤ d if this is true for H_{∗}(X).

The key link between the zci condition and growth is the following result.

Lemma 7.6. Given bounded below locally finite modules X, Y in a triangle

Σ^{n}X ⟶^{χ} X ⟶ Y

with χ ∈ J(ZD(R))

growth(X) ≤ growth(Y) + 1.

Proof. By the arguments of the previous subsection, we may assume n ≠ 0. Indeed if χ has degree 0 we have an exact sequence

0 ⟶ H_{∗}(X)/χ ⟶ H_{∗}(Y) ⟶ H_{∗}(X),

so that Lemma 7.4 shows that

growth(X) = growth(Y).

Next, since X and Y are locally finite, the homology modules H_{∗}(X) and H_{∗}(Y) have only finitely many associated primes and it suffices to work with one maximal ideal at a time. Accordingly we may localize and it suffices to deal with the case that H_{0}(R) is a local ring.

The homology long exact sequence of the triangle includes

⋯ ⟶ H_{i−n}(X) ⟶^{χ} H_{i}(X) ⟶ H_{i}(Y) ⟶ · · · .

This shows

len(H_{i}(X)) ≤ len(H_{i}(Y)) + len(χH_{i−n}(X)).

Iterating s times, we find

len(H_{i}(X)) ≤ len(H_{i}(Y)) + len(χH_{i−n}(Y)) + ⋯ + len(χ^{s−1}H_{i−(s−1)n}(Y)) + len(χ^{s}H_{i−sn}(X)).

First suppose that n > 0. The case that H_{∗}(X) is bounded below is familiar. If h_{X}(t) is the Hilbert series of H_{∗}(X) then we have

h_{X}(t) ≤ h_{Y} (t)(1 + t^{n+1} + t^{2n+2} + ⋯) = \frac{h_{Y} (t)}{1 − t^{n+1}} ,

giving the required growth estimate.

To extend to the general case, we argue that χ^{s}H_{i−sn}(X) = 0 for s >> 0. Indeed, we have a decreasing chain of subgroups of the finite length module H_{i}(X), which therefore stabilizes. The stable value must be zero by Nakayama’s lemma. Thus there is a j = j(i) so that if i−sn ≤ j(i) then χ^{s} is zero from degree i−sn. Now taking j = min{j(0), j(1), …, j(n−1)}, we see that for k ≤ j powers of χ from H_{k}(X) have zero image in positive degrees. This means that the number j serves as a lower bound for estimates in positive degrees, and we can use the familiar argument.

For estimates in negative degrees we argue similarly, starting with the estimate

len(H_{j}(X)) ≤ len(H_{j+n+1}(Y)) + len(χH_{j}(X)).

This gives the bounded above case in the usual way. Using Nakayama’s lemma we find as s so that χ^{s}H_{j}(X) = 0, and deduce that there is a k so that no power of χ from negative degrees has non-zero image in degrees ≥ k. The general case then follows like the bounded above case.

The argument if n < 0 is precisely similar. □

7.D. Operators and growth. It is convenient to introduce further terminology in which the letter ‘g’ stands for growth.

Definition 7.7. If R is a commutative local ring we say that it is gci of codimension c if Ext_{R}^{∗}(k, k) has polynomial growth of degree c − 1.

Theorem 7.8. (Gulliksen [14, 16]) A Noetherian local ring is a complete intersection if and only if it is gci. □

We may connect this to our module theoretic condition.

Theorem 7.9. If R is zci of codimension c then it is also gci of codimension c.

Proof. By hypothesis there are elements χ_{1}, χ_{2}, …, χ_{c} in J(ZD(R)) so that k/χ_{c}/ … /χ_{1} is small.

We may apply Hom_{R}(·, k) (i.e., derived Hom) to the resulting triangles in the derived category. For brevity, we write

X_{i} = Hom_{R}(k/χ_{c}/ ⋯ /χ_{i}, k).

Now by hypothesis, since k/χ_{c}/ … /χ_{1} is small, so X_{1} is bounded, and thus of growth −1, and

H_{∗}(X_{c+1}) = H_{∗}(Hom_{R}(k, k)) = Ext_{R}^{∗}(k, k),

so that the theorem states growth(X_{c+1}) ≤ c.

Applying the lemma to the cofibre sequence

Σ^{−|χ_{i−1}|}X_{i} ⟵ X_{i} ⟵ X_{i−1}

we find

growth(X_{i}) ≤ growth(X_{i−1}) + 1

and hence, repeating c + 1 times, we find

growth(X_{c+1}) ≤ c

as required. □

Remark 7.10. In commutative algebra one says that a locally finite complex M of Rmodules has complexity n if Ext_{R}^{∗}(M, k) has polynomial growth of degree n − 1. In these terms, the proof proceeds by showing for i = 1, 2, …, c+1 that k/χ_{c}/ ⋯ /χ_{i+1} has complexity i.

Corollary 7.11. The three conditions ci, zci and gci on Noetherian local rings are equivalent.

Proof. We showed in Section 6 that ci implies zci, and in Section 7 that zci implies gci. Gulliksen’s Theorem (7.8) shows gci implies ci. □

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Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK

E-mail address: bensondj@maths.abdn.ac.uk

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK

E-mail address: j.greenlees@sheffield.ac.uk