DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS

PETTER ANDREAS BERGH, SRIKANTH B. IYENGAR, HENNING KRAUSE, AND STEFFEN OPPERMANN

Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.

1. Introduction

A notion of dimension for a triangulated category was introduced by Rouquier in [25]. Roughly speaking, it corresponds to the minimum number of steps needed to generate the category from one of its objects. Consideration of this invariant has been critical to some recent developments in algebra and geometry: Using the dimension of the stable category of an exterior algebra on a d-dimensional vector space, Rouquier [26] proved that the representation dimension of the exterior algebra is d+1, thereby obtaining the first example of an algebra with representation dimension more than three.

On the other hand, Bondal and Van den Bergh [11] proved that any cohomological finite functor on the bounded derived category of coherent sheaves on a smooth algebraic variety over a field is representable, by establishing that that triangulated category has finite dimension.

In this paper we establish lower bounds for the dimension of a triangulated category and discuss some applications. We make systematic use of the gradedcommutative structure of the triangulated category arising from its graded center; in particular Koszul objects are employed.

A triangulated category T is by definition an additive ℤ-category equipped with a class of exact triangles satisfying various axioms [30]. Here, ℤ-category simply means that there is a fixed equivalence Σ: T → T.

Given any additive ℤ-category T = (T, Σ), we introduce some natural finiteness condition for objects of T as follows: Let R = ⨁_{i⩾0} R^{i} be a graded-commutative ring that acts on T via a homomorphism of graded rings R → Z^{∗}(T) to the graded center of T. Thus for each pair of objects X, Y in T, the graded abelian group

Hom_{T}^{∗}(X, Y) = ⨁_{i∈ℤ} Hom_{T}(X, Σ^{i}Y)

is a graded R-module.

2000 Mathematics Subject Classification. 16E30, 16E40, 18E30.

Key words and phrases. triangulated category, Koszul object, representation dimension.

Version from June 15, 2008.

Bergh and Oppermann were supported by NFR Storforsk grant no. 167130, and Iyengar was partly supported by NSF grant, DMS 0602498. Iyengar’s work was done at the University of Paderborn, on a visit supported by a Forschungspreis awarded by the Humboldt-Stiftung.

Now fix an object X in T and suppose that for each Y ∈ T there exists an integer n such that the following properties hold:

(1) the graded R-module ⨁_{i⩾n} Hom_{T}^{i}(X, Y) is noetherian, and

(2) the R^{0}-module Hom_{T}^{i}(X, Y) is of finite length for i ≥ n. In this case, Hom_{T}^{∗}(X, Y) has finite (Krull) dimension over R^{ev}, the subring of R consisting of elements of even degree, which is a commutative ring. If X has this finiteness property also with respect to another ring S, then the dimension of Hom_{T}^{∗}(X, Y) over S coincides with that over R; see Lemma 4.3. For this reason, we denote this number by dim Hom_{T}^{∗}(X, Y). The main result in this work is as follows.

Theorem 1.1. Let T be a triangulated category and X an object with properties as above. One then has an inequality

dim T ≥ dim End_{T}^{∗}(X) − 1 .

An intriguing feature of this result is that the invariant appearing on the right hand side of the inequality involves only the additive ℤ-structure of T. Theorem 1.1 is contained in Theorem 4.2. The proof is based on a systematic use of Koszul objects and elementary observations concerning ‘eventually noetherian modules’; this is inspired by the approach in [6]. Another important ingredient is a version of the ‘Ghost Lemma’ from [10]; see Lemma 3.2.

Our principal motivation for considering dimensions of triangulated categories is that it provides a way to obtain lower bounds on the representation dimension of an Artin algebra. Indeed, for a non-semisimple Artin algebra A one has an inequality

rep. dim A ≥ dim D_{st}^{b}(A) + 2 ,

where D_{st}^{b}(A) is the stable derived category of A, in the sense of Buchweitz [13]. As one application of the preceding result, we bound the representation dimension of A by the Krull dimension of Hochschild cohomology.

Corollary 1.2. Let k be an algebraically closed field and A a finite dimensional kalgebra with radical 𝔯, where A is not semi-simple. If Ext_{A}^{∗}(A/𝔯, A/𝔯) is noetherian as a module over the Hochschild cohomology algebra HH^{∗}(A) of A over k, then

rep. dim A ≥ dim HH^{∗}(A) + 1 .

This result is a special case of Corollary 5.12. In Section 5 we present further applications of Theorem 4.2.

2. Eventually noetherian modules

Many of the arguments in this article are based on properties of ‘eventually noetherian modules’ over graded commutative rings, introduced by Avramov and Iyengar [6, §2]. In this section we collect the required results. For the benefit of the reader we provide (sketches of) proofs, although the results are well-known, and the arguments based on standard techniques in commutative algebra. For unexplained terminology the reader is referred to Bruns and Herzog [12].

Graded-commutative rings. Let R = ⨁_{i⩾0} R^{i} be a graded-commutative ring; thus R is an ℕ-graded ring with the property that rs = (−1)^{|r||s|}sr for any r, s in R. Elements in a graded object are assumed to be homogeneous.

Let M = ⨁_{i∈ℤ} M^{i} be a graded R-module. For any integer n, we set

M^{⩾n} = ⨁_{i⩾n} M^{i}  and  R^{+} = R^{⩾1} .

Note that M^{⩾n} is an R-submodule of M, and that R^{+} is an ideal in R.

As in [6, §2], we say that M is eventually noetherian if the R-module M^{⩾n} is noetherian for some integer n; we write noeth(R) for the full subcategory with objects the eventually noetherian modules. In this work, the focus is on eventually noetherian modules M that have the additional property that length_{R^{0}}(M^{n}) is finite for n ≫ 0. The corresponding full subcategory is denoted noeth^{fl}(R). It is easy to verify that both noeth(R) and noeth^{fl}(R) are abelian subcategories.

Recall that the annihilator of M, which we denote ann_{R} M, is the homogenous ideal of R consisting of elements r such that r · M = 0. The following remark is easily justified. It allows one, when considering eventually noetherian modules, to pass to a situation where the ring itself is noetherian.

Remark 2.1. Suppose that the R-module M^{⩾n} is noetherian. Set I = ann_{R}(M^{⩾n}). The ring R/I is then noetherian, and M^{⩾n} is a finitely generated and faithful R/Imodule. If in addition length_{R^{0}}(M^{i}) is finite for i ≥ n, then (R/I)^{0} is artinian.

One way to study modules over graded-commutative rings is to pass to the subring R^{ev} consisting of elements of even degree, which is then a commutative graded ring: rs = sr for any r, s in R^{ev}. In this work, this passage is facilitated by the following observation; confer the proof of [12, Theorem 1.5.5].

Lemma 2.2. Let R be a graded-commutative ring, and let M be an R-module. (1) If M is in noeth(R), then it is also in noeth(R^{ev}).

(2) If M is in noeth^{fl}(R), then it is also in noeth^{fl}(R^{ev}).

Proof. Suppose M is in noeth(R). By Remark 2.1, one can assume R is itself noetherian and M a finitely generated R-module. It then suffices to prove that the subring R^{ev} is noetherian and that R is finitely generated as a module over it. Observe that there is a decomposition R = R^{ev} ⊕ R^{odd} as R^{ev}-modules. In particular, for any ideal I ⊆ R^{ev} one has IR ∩ R^{ev} = I, and hence R noetherian implies R^{ev} noetherian. By the same token, one obtains that R^{odd}, and hence also R, is a noetherian R^{ev}-module. □

Dimension over a commutative graded ring. Let R be a commutative graded ring. We recall some basic facts concerning the dimension and support of an Rmodule M. It is convenient to employ the following notation:

Proj R = {𝔭 is a homogeneous prime in R with 𝔭 ⊉ R^{+}}
SuppR^{+} M = {𝔭 ∈ Proj R | M_{𝔭} ≠ 0}
Ass_{R}^{+} M = {𝔭 ∈ Proj R | Hom_{R_{𝔭}}(R_{𝔭}/𝔭R_{𝔭}, M_{𝔭}) ≠ 0}.

Evidently, Ass_{R}^{+} M = Ass_{R} M ∩ SuppR^{+} M, where Ass_{R} M is the set of associated primes of M. The next result is readily verified.

Lemma 2.3. Let R be a commutative graded ring and M a graded R-module. (1) For any integer n, one has equalities

SuppR^{+} (M^{⩾n}) = SuppR^{+} M and Ass_{R}^{+}(M^{⩾n}) = Ass_{R}^{+} M .

(2) If L ⊆ M is a submodule, then

SuppR^{+} L ⊆ SuppR^{+} M and Ass_{R}^{+} L ⊆ Ass_{R}^{+} M .  □

A module M is said to be eventually zero if M^{⩾n} = 0 for some integer n. The next result is part of [6, 2.2], where it is stated without proof. We give details, for the convenience of readers.

Proposition 2.4. Let R be a commutative graded ring and M an eventually noetherian R-module. The set Ass_{R}^{+} M is finite and the conditions below are equivalent:

(i) Ass_{R}^{+} M = ∅;

(ii) SuppR^{+} M = ∅;

(iii) M is eventually zero.

Proof. In view of Remark 2.1 and Lemma 2.3(1), one may assume R is noetherian and that M is a faithful R-module. In this case Ass_{R} M is a finite set and therefore Ass_{R}^{+} M is finite; see [22, Theorem 6.5].

An ideal 𝔭 ∈ Spec R belongs to Supp_{R} M if and only if there exists 𝔮 ∈ Ass_{R} M with 𝔮 ⊆ 𝔭; see [22, Theorem 6.5]. From this the implications (iii) ⟹ (ii) and (ii) ⟺ (i) are obvious consequences.

It remains to show (ii) ⟹ (iii). Since the R-module M is finitely generated and faithful, one has that Supp_{R} M = Spec R. Thus, SuppR^{+} M = ∅ implies R^{+} ⊆ 𝔭 for each 𝔭 ∈ Spec R, hence the ideal R^{+} is nilpotent. Since R is noetherian, this implies that R is eventually zero, and hence also that M is eventually zero. □

We say that an element r ∈ R^{+} is filter-regular on M if Ker(M ⟶^{r} M) is eventually zero. This notion is a minor variation on a well-worn theme in commutative algebra; confer, for instance, Schenzel, Trung, and Cuong [27, (2.1)].

Lemma 2.5. Let R be a commutative graded ring and M an eventually noetherian R-module. There then exists an element in R^{+} that is filter-regular on M.

Proof. Proposition 2.4 yields that the set Ass_{R}^{+} M is finite, so by prime avoidance [12, Lemma 1.5.10] there exists an element r in R^{+} not contained in any prime 𝔭 in Ass_{R}^{+} M. This element is filter-regular on M.

Indeed, for K = Ker(M ⟶^{r} M), one has Ass_{R}^{+} K ⊆ Ass_{R}^{+} M; see Lemma 2.3(2). However, for any 𝔭 in Ass_{R}^{+} M one has K_{𝔭} = 0, since r ∉ 𝔭, and hence Ass_{R}^{+} K = ∅. Since K is eventually noetherian, being a submodule of M, Proposition 2.4 applies and yields that K is eventually zero. □

As usual, the (Krull) dimension of a module M over R is the number

dim_{R} M = sup \{d ∈ ℕ | \begin{matrix}there exists a chain of prime ideals \\ 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{d} in Supp_{R} M \end{matrix}\} .

When M is in noeth^{fl}(R) one can compute its dimension in terms of the rate of growth of its components. To make this precise, it is convenient to introduce the complexity of a sequence of non-negative integers (a_{n}) as the number

cx(a_{n}) = inf \{d ∈ ℕ | \begin{matrix}there exists a real number c such that \\ a_{n} ≤ cn^{d−1} for n ≫ 0 \end{matrix}\} .

For basic property of this notion see, for example, [2, §2 and Appendix]. As usual, the set of prime ideals of R containing a given ideal I is denoted V(I).

Proposition 2.6. For each M in noeth^{fl}(R), the following properties hold: (1) If r_{1}, …, r_{n} are elements in R^{+} with n < dim_{R} M, then V(r)∩SuppR^{+} M ≠ ∅.

(2) One has an equality dim_{R} M = cx(length_{R^{0}}(M^{n})).

Proof. By Remark 2.1, one may assume R is noetherian, R^{0} is artinian, and M is a faithful, finitely generated R-module. Part (1) then follows from the Krull height theorem; see [22, Theorem 13.5], while (2) is contained in [22, Theorem 13.2]. □

Dimension over a graded-commutative ring. Let R be a graded-commutative ring. For each R-module M in noeth^{fl}(R), we introduce its dimension as the number

dim_{R} M = cx(length_{R^{0}}(M^{n})) .

It follows from Lemma 2.2 and Proposition 2.6(2) that this number is finite and coincides with the dimension of M as a module over R^{ev}. This remark will be used without further comment.

Proposition 2.7. Let R → S be a homomorphism of graded-commutative rings and M an S-module. If M, viewed as an R-module by restriction of scalars, is in noeth^{fl}(R), then dim_{R} M = dim_{S} M.

Proof. The module M is in noeth^{fl}(S) as well and therefore, by Remark 2.1, one can pass to a situation where S is noetherian and M is a faithful S-module that is also noetherian over R. Passing to R/I, where I is the kernel of the homomorphism R → S, one may also assume that the homomorphism is injective. Since one has injective homomorphisms of R-modules

R ↪ S ↪ Hom_{R}(M, M) ,

one thus obtains that the ring R itself is noetherian with R^{0} artinian, and that S is a finitely generated R-module. This implies that the R^{0}-module S^{0} is finitelygenerated, and hence, for any S^{0}-module N, one has inequalities

length_{S^{0}} N ≤ length_{R^{0}} N ≤ (length_{R^{0}} S^{0})(length_{S^{0}} N) .

This yields dim_{R} M = dim_{S} M, as claimed. □

3. Koszul objects

Let T be a triangulated category. For any objects X and Y in T, we set

Hom_{T}^{∗}(X, Y) = ⨁_{n∈ℤ} Hom_{T}(X, Σ^{n}Y) and End_{T}^{∗}(X) = Hom_{T}^{∗}(X, X) .

The graded center of T, that we denote Z^{∗}(T), consists in degree n of natural transformations η: id_{T} → Σ^{n} satisfying ηΣ = (−1)^{n}Ση. Composition gives Z^{∗}(T) a structure of a graded-commutative ring; see, for instance, [14, §3], especially Lemma 3.2.1, which explains the signed commutation rule, and also [21].

In what follows, we assume that a graded-commutative ring R acts centrally on T, via a homomorphism R → Z^{∗}(T). What this amounts to is specifying for each X in T a homomorphism of rings ϕ_{X}: R → End_{T}^{∗}(X) such that the induced R-module structures on Hom_{T}^{∗}(X, Y) coincide up to the usual sign rule:

η ∘ ϕ_{X}(r) = (−1)^{|r||η|}ϕ_{Y} (r) ∘ η

for any η ∈ Hom_{T}^{∗}(X, Y) and r ∈ R.

We now recall an elementary, and extremely useful, construction.

Koszul objects. Let r be a homogeneous element in R of degree d = |r|. Given an object X in T, we denote X//r any object that appears in an exact triangle

(3.1)  X ⟶^{r} Σ^{d}X ⟶ X//r ⟶ ΣX .

It is well-defined up to isomorphism; we call it a Koszul object of r on X.

Let Y be an object in T and set M = Hom_{T}^{∗}(X, Y). Applying Hom_{T}^{∗}(−, Y) to the triangle above yields an exact sequence of R-modules:

M[d + 1] ⟶^{∓r} M[1] ⟶ Hom_{T}^{∗}(X//r, Y) ⟶ M[d] ⟶^{±r} M[0] .

This gives rise to an exact sequence of graded R-modules

(3.2)  0 ⟶ (M/rM)[1] ⟶ Hom_{T}^{∗}(X//r, Y) ⟶ (0 : r)_{M}[d] ⟶ 0 ,

where (0 : r)_{M} denotes {m ∈ M | r · m = 0}.

Applying the functor Hom_{T}^{∗}(Y, −) results in a similar exact sequence.

Given a sequence of elements r = r_{1}, …, r_{n} in R, consider objects X_{i} defined by

(3.3)  X_{i} = \begin{cases}X & for i = 0, \\ X_{i−1}//r_{i} & for i ≥ 1. \end{cases}

Set X//r = X_{n}; this is a Koszul object of r on X. The result below is a straightforward consequence of (3.2) and an induction on n; see [9, Lemma 5.11(1)].

Lemma 3.1. Let n ≥ 1 be an integer and set s = 2^{n}. For any sequence of elements r = r_{1}, …, r_{n} in R^{+}, and any object X ∈ T one has that

r_{i}^{s} · Hom_{T}^{∗}(X//r, −) = 0 = r_{i}^{s} · Hom_{T}^{∗}(−, X//r) for i = 1, …, n.  □

The next construction quantifies the process of ‘building’ objects out of a given object in the triangulated category T.

Thickenings. Given an object G of T we write thick_{T}(G) for the thick subcategory of T generated by G. This subcategory has a filtration

{0} = thick_{T}^{0}(G) ⊆ thick_{T}^{1}(G) ⊆ ⋯ ⊆ ⋃_{n⩾0} thick_{T}^{n}(G) = thick_{T}(G)

where thick_{T}^{1}(G) consists of retracts of finite direct sums of suspensions of G, and thick_{T}^{n}(G) consists of retracts of n-fold extensions of thick_{T}^{1}(G). In the literature, the subcategory thick_{T}^{n}(G) has sometimes been denoted ⟨G⟩_{n}.

The next result is contained in [10, Lemma 2.1]. Similar results have appeared in Kelly [20], Carlsson [15, Proof of Theorem 16], Christensen [16, Theorem 3.5], Beligiannis [7, Corollary 5.5], Rouquier [25, Lemma 4.11], and Avramov, Buchweitz, and Iyengar [3, Proposition 2.9].

Lemma 3.2 (Ghost Lemma). Let T be a triangulated category, and let F, G be objects in T. Suppose there exist morphisms

K_{c} ⟶^{θ_{c}} K_{c−1} ⟶^{θ_{c−1}} ⋯ ⟶^{θ_{1}} K_{0}

in T such that the following conditions hold:

(1) Hom_{T}^{n}(G, θ_{i}) = 0 for n ≫ 0 and for each i = 1, …, c;

(2) Hom_{T}^{n}(F, θ_{1} ⋯ θ_{c}) ≠ 0 for infinitely many n ≥ 0.

One then has that F ∉ thick_{T}^{c}(G). □

There is also a contravariant version of the Ghost Lemma, involving Hom_{T}^{∗}(−, G).

Theorem 3.3. Let T be a triangulated category and R a graded-commutative ring acting centrally on it. Let X, Y be objects in T with the property that the R-module Hom_{T}^{∗}(X, Y) is in noeth^{fl}(R).

For any c < dim_{R} Hom_{T}^{∗}(X, Y) there exist elements r_{1}, …, r_{c} in (R^{ev})^{+} with

X//r ∉ thick_{T}^{c}(X) and Y//r ∉ thick_{T}^{c}(Y) .

Remark 3.4. In the language of levels, introduced in [4, §2.3], the conclusion of the preceding theorem reads:

level_{T}^{X}(X//r) > c and level_{T}^{Y} (Y//r) > c .

This formulation is sometimes more convenient to use in arguments.

Proof. The plan is to apply the Ghost Lemma.

By Lemma 2.2, one can assume that R = R^{ev}, and, in particular, that the graded ring R is commutative. Since the R-module Hom_{T}^{∗}(X, Y) is in noeth^{fl}(R), so are Hom_{T}^{∗}(X, Y//x) and Hom_{T}^{∗}(X//x, Y), for any finite sequence of elements x in R; this can be readily established using (3.2) and an induction on the length of the sequence x. Using Lemma 2.5, one can thus find by iteration, elements r_{1}, …, r_{c} in R^{+} such that r_{i} is filter-regular on the R-modules Hom_{T}^{∗}(X, Y//{r_{1}^{s}, …, r_{i−1}^{s}}) and Hom_{T}^{∗}(X//{r_{1}^{s}, …, r_{i−1}^{s}}, Y), where s = 2^{c}, for each i = 1, …, c.

We now verify that X//r is not in thick_{T}^{c}(X).

Set K_{0} = Y, set K_{i} = Σ^{−i}(Y//{r_{1}^{s}, …, r_{i}^{s}}) for i = 1, …, c, and let

(3.4)  K_{i} ⟶^{θ_{i}} K_{i−1} ⟶^{±r_{i}^{s}} Σ^{s|r_{i}|}K_{i−1} → ΣK_{i} ,

be the exact triangle obtained (by suitable suspension) from the one in (3.1). We claim that for each i = 1, …, c the following properties hold:

(1) Hom_{T}^{n}(X, θ_{i}) = 0 for n ≫ 0;

(2) Hom_{T}^{∗}(X//r, θ_{i}) is surjective;

(3) Hom_{T}^{n}(X//r, K_{0}) ≠ 0 for infinitely many n ≥ 0.

Indeed, for each W ∈ T the triangle (3.4) induces an exact sequence

Hom_{T}^{∗}(W, K_{i}) ⟶^{Hom_{T}^{∗}(W,θ_{i})} Hom_{T}^{∗}(W, K_{i−1}) ⟶^{±r_{i}^{s}} Hom_{T}^{∗}(W, K_{i−1})[s|r_{i}|]

of graded R-modules.

(1) With W = X in the sequence above, r_{i} is filter-regular on Hom_{T}^{∗}(X, K_{i−1}), by choice, and hence so is r_{i}^{s}. This proves the claim.

(2) Set W = X//r in the exact sequence above, and note that r_{i}^{s} annihilates Hom_{T}^{∗}(X//r, K_{i−1}), by Lemma 3.1.

(3) Recall that K_{0} = Y. It suffices to prove that one has an equality

SuppR^{+} Hom_{T}^{∗}(X//r, Y) = V(r) ∩ SuppR^{+} Hom_{T}^{∗}(X, Y) .

For then the choice of c ensures that the set above is non-empty, by Proposition 2.6(1), and hence Hom_{T}^{∗}(X//r, Y) is not eventually zero, by Proposition 2.4.

The equality above can be established as in the proof of [6, Proposition 3.10]: By induction on the length of the sequence r, it suffices to consider the case where r = r. Setting M = Hom_{T}^{∗}(X, Y), it follows from (3.2) that one has an equality

SuppR^{+} Hom_{T}^{∗}(X//r, Y) = SuppR^{+}(M/rM) ∪ SuppR^{+}(0 : r)_{M} .

It then remains to note that one has

SuppR^{+}(M/rM) = SuppR^{+} M ∩ V(r) and SuppR^{+}(0 : r)_{M} ⊆ SuppR^{+} M ∩ V(r) ,

where the equality holds because one has M/rM = M ⊗_{R}R/Rr, while the inclusion holds because (0 : r)_{M} is a submodule of M annihilated by r.

This justifies claims (1)–(3) above.

Observe that (2) and (3) imply that Hom_{T}^{∗}(X//r, θ_{1} ⋯ θ_{c}) is not eventually zero. Therefore, the Ghost Lemma yields X//r ∉ thick_{T}^{c}(X), as desired.

A similar argument, employing the contravariant version of the Ghost Lemma, establishes that Y//r is not in thick_{T}^{c}(Y). □

4. The dimension of a triangulated category

The dimension of a triangulated category T is the number

dim T = inf{n ∈ ℕ | there exists a G ∈ T with thick_{T}^{n+1}(G) = T}.

Evidently, if dim T is finite there exists an object G with thick_{T}(G) = T; we call such an object G a generator for T. The dimension of T can be infinite even if it has a generator.

Lemma 4.1. Let T be a triangulated category and R a graded-commutative ring acting centrally on it. If G is a generator for T, then for each object X in T one has equalities

Supp_{R^{ev}} Hom_{T}^{∗}(X, G) = Supp_{R^{ev}} End_{T}^{∗}(X) = Supp_{R^{ev}} Hom_{T}^{∗}(G, X) .

Proof. We may assume R = R^{ev}. For any subset U of Spec R the subcategory

{Y ∈ T | Supp_{R} Hom_{T}^{∗}(X, Y) ⊆ U}

of T is thick. Since X is in thick_{T}(G), one thus obtains an inclusion

Supp_{R} End_{T}^{∗}(X) ⊆ Supp_{R} Hom_{T}^{∗}(X, G) .

The reverse inclusion holds because R acts on Hom_{T}^{∗}(X, G) via a homomorphism of rings R → End_{T}^{∗}(X). This settles the first equality.

A similar argument gives the second one. □

Theorem 4.2. Let T be a triangulated category and R a graded-commutative ring acting centrally on it. If an object X ∈ T is such that the R-module Hom_{T}^{∗}(X, G), or Hom_{T}^{∗}(G, X), is in noeth^{fl}(R), for some generator G, then one has an inequality

dim T ≥ dim_{R} End_{T}^{∗}(X) − 1 .

Proof. Suppose that the R-module Hom_{T}^{∗}(X, G) is in noeth^{fl}(R). The full subcategory of T with objects

{Y ∈ T | Hom_{T}^{∗}(X, Y) ∈ noeth^{fl}(R)}

is thick. Since it contains G it coincides with T, so one may assume that G is an arbitrary generator for T. For c = dim_{R} Hom_{T}^{∗}(X, G) − 1, Theorem 3.3 yields a Koszul object, G//r, not contained in thick_{T}^{c}(G). This implies the inequality below:

dim T ≥ dim_{R} Hom_{T}^{∗}(X, G) − 1 = dim_{R} End_{T}^{∗}(X) − 1 ;

the equality is by Lemma 4.1. The other case is handled in the same way. □

In the theorem, the number dim_{R} End_{T}^{∗}(X) is independent of the ring R, in a sense explained in the following lemma. These results together justify Theorem 1.1.

Lemma 4.3. Let T be an additive ℤ-category and let X, Y be objects in T. Suppose that there are graded-commutative rings R and S acting centrally on T such that Hom_{T}^{∗}(X, Y) is in both noeth^{fl}(R) and noeth^{fl}(S). One then has an equality

dim_{R} Hom_{T}^{∗}(X, Y) = dim_{S} Hom_{T}^{∗}(X, Y) .

Proof. Indeed, the graded tensor product R⊗_{ℤ}S is a graded-commutative ring, and one has natural homomorphisms of graded rings R → R ⊗_{ℤ} S ← S. The central actions of R and S on T extends to one of the ring R ⊗_{ℤ} S, and Hom_{T}^{∗}(X, Y) is in noeth^{fl}(R ⊗_{ℤ} S). Proposition 2.7, applied to the preceding homomorphisms, now yields equalities

dim_{R} Hom_{T}^{∗}(X, Y) = dim_{R⊗_{ℤ}S} Hom_{T}^{∗}(X, Y) = dim_{S} Hom_{T}^{∗}(X, Y) .  □

Remark 4.4. The preceding result suggests that one should consider the full subcategory of objects X in T with the property that, for some ring R acting centrally on T and all Y ∈ T, one has Hom_{T}^{∗}(X, Y) ∈ noeth^{fl}(R); let us denote it noeth^{fl}(T). Arguing as in the proof of Lemma 4.3, it is not difficult to prove that noeth^{fl}(T) is precisely the subcategory noeth^{fl}(Z^{⩾0}(T)), where Z^{⩾0}(T) is the non-negative part of the graded center of T. This implies, for instance, that noeth^{fl}(T) is a thick subcategory of T, and also that one has an ‘intrinsic’ notion of dimension for objects in this subcategory. Thus, one could state the main results of this section without involving an ‘external’ ring R. In practice, however there are usually more convenient choices than Z^{⩾0}(T), for a ring R acting centrally on T.

Cohomological functors. There are also versions of Theorem 4.2 which apply to cohomological functors. In order to explain this, let T be a triangulated category and H: T → Ab a cohomological functor to the category of abelian groups. Let R be a graded-commutative ring that acts centrally on T. The graded abelian group

H^{∗}(Y) = ⨁_{n∈ℤ} H(Σ^{n}Y)

then has a natural structure of a graded R-module.

Assume that idempotents in T split and that there exists a generator G of T such that the R-module H^{∗}(G) is noetherian, and the R^{0}-module H^{i}(G) has finite length for each i.

One can check, as in Lemma 4.3, that in this case, for any Y ∈ T, the dimension of the R-module H^{∗}(Y) is finite, and independent of R; denote it dim H^{∗}(Y).

Theorem 4.5. If H is a cohomological functor as above, then one has an inequality:

dim T ≥ dim H^{∗}(Y) − 1 for each Y ∈ T.

Sketch of a proof. Under the hypotheses of the theorem, the functor H is representable; this can be proved by an argument similar to that for [11, Theorem 1.3] due to Bondal and Van den Bergh. The result is thus contained in Theorem 4.2. □

The following result is a variation on Theorems 4.2 and 4.5 which might be useful in some contexts. The hypothesis on T holds, for example, when it is algebraic, in the sense of Keller [19].

Theorem 4.6. Let T be a triangulated category with functorial mapping cones. If H is a cohomological functor and G a generator for T such that H^{∗}(G) is in noeth^{fl}(R), for some ring R acting centrally on T, then one has an inequality:

dim T ≥ dim H^{∗}(Y) − 1 for each Y ∈ T.

Sketch of a proof. Since T has functorial mapping cones, for each r ∈ R, the construction of the Koszul object Y//r can be made functorial, and so the assignment Y ↦ H(Σ^{−1}Y//r) defines a cohomological functor; let us denote it H//r, with a caveat that it is a desuspension of what is introduced in (3.1). This functor comes equipped with a natural transformation θ_{r}: H//r → H.

Let G be a generator for T, set c = dim H^{∗}(G) − 1 and s = 2^{c}. Arguing as in the proof of Theorem 3.3, one can pick a sequence of elements r_{1}, …, r_{c} such that r_{i+1} is filter-regular on H_{i}^{∗}(G), where H_{0} = H and H_{i} = H_{i−1}//r_{i}^{s} for i ≥ 1. One thus has natural transformations

H_{c} ⟶^{θ_{rc}} H_{c−1} ⟶^{θ_{r_{c−1}}} ⋯ ⟶^{θ_{r_{1}}} H_{0}

satisfying, for each i = 1, …, c, the following conditions:

(1) θ_{r_{i}}^{∗}(G): H_{i}^{∗}(G) → H_{i−1}^{∗}(G) is eventually zero;

(2) θ_{r_{i}}^{∗}(G//r): H_{i}^{∗}(G//r) → H_{i−1}^{∗}(G//r) is surjective;

(3) H^{∗}(G//r) is not eventually zero. It now follows from (an analogue of) the Ghost Lemma that G//r is not in thick_{T}^{c}(G). This implies the desired result. □

5. Applications

Let A be a noetherian ring. In what follows, D^{b}(A) denotes the bounded derived category of finitely generated A-modules, with the usual structure of a triangulated category. Following Buchweitz [13], the stable derived category of A is the category

D_{st}^{b}(A) = D^{b}(A)/D^{per}(A) ,

where D^{per}(A) = thick(A) denotes the category of perfect complexes. Here the quotient is taken in the sense of Verdier; see [30]. It has a structure of a triangulated category, for which the canonical functor D^{b}(A) → D_{st}^{b}(A) is exact.

Remark 5.1. The quotient functor D^{b}(A) → D_{st}^{b}(A) induces a homomorphism of graded rings Z^{∗}(D^{b}(A)) → Z^{∗}(D_{st}^{b}(A)). Thus a central action of a graded commutative ring R on D^{b}(A) induces a central action on D_{st}^{b}(A). In particular, for any pair of complexes X, Y ∈ D^{b}(A) the natural map

Ext_{A}^{∗}(X, Y) = Hom_{D^{b}(A)}^{∗}(X, Y) → Hom_{D_{st}^{b}}^{∗}(X, Y)

is one of R-modules.

Gorenstein rings. A noetherian ring A is called Gorenstein if A is of finite injective dimension both as a left module and a right module over itself. In the commutative case, this is more restrictive than the usual definition of a Gorenstein ring; however both definitions coincide if A has finite Krull dimension.

The following result is [13, Corollary 6.3.4].

Lemma 5.2. Let A be a noetherian Gorenstein ring. Then for each pair of complexes X, Y ∈ D^{b}(A) the natural map

Hom_{D^{b}(A)}^{n}(X, Y) → Hom_{D_{st}^{b}(A)}^{n}(X, Y)

induced by the quotient functor D^{b}(A) → D_{st}^{b}(A) is bijective for n ≫ 0. □

The notion of complexity of a sequence of non-negative integers was recalled in the paragraph preceding Proposition 2.6. We define the complexity of a pair X, Y of complexes of A-modules to be the number

cx_{A}(X, Y) = cx(length_{Z(A)}(Ext_{A}^{n}(X, Y)))

where Z(A) denotes the center of A.

Example 5.3. Let A be an Artin k-algebra, and let 𝔯 denote its radical. Then every finitely generated A-module M admits a minimal projective resolution

⋯ → P_{2} → P_{1} → P_{0} → M → 0

and one defines the complexity of M as

cx_{A}(M) = cx(length_{k}(P_{n})) .

It is well-known that cx_{A}(M) = cx_{A}(M, A/𝔯); see [2, A.13] or [8, 5.3] for details.

Recall that a ring A is said to be a noetherian algebra if there exists a commutative noetherian ring k such A is a k-algebra and a finitely generated k-module.

Theorem 5.4. Let A be a noetherian algebra which is Gorenstein. Let X ∈ D^{b}(A) be such that Ext_{A}^{∗}(X, Y) is in noeth^{fl}(R), for some graded-commutative ring R acting centrally on D^{b}(A), and for all Y ∈ D^{b}(A). One then has inequalities

dim D^{b}(A) ≥ dim D_{st}^{b}(A) ≥ cx_{A}(X, X) − 1 .

Proof. The inequality on the left holds because D_{st}^{b}(A) is a quotient of D^{b}(A).

The R-action on D^{b}(A) induces an action on D_{st}^{b}(A) by Remark 5.1, and the finiteness condition on X as an object of D^{b}(A) passes to the stable category D_{st}^{b}(A) because of Lemma 5.2. In particular, one has an equality

dim_{R} End_{D^{b}(A)}^{∗}(X) = dim_{R} End_{D_{st}^{b}(A)}^{∗}(X) .

Theorem 1.1 now yields the inequality below

dim D_{st}^{b}(A) ≥ dim_{R} End_{D^{b}(A)}^{∗}(X) − 1 = cx_{A}(X, X) − 1 .

The equality follows from Propositions 2.6 and 2.7, where we use that Ext_{A}^{n}(X, X) is finitely generated over Z(A). □

Artin algebras. An Artin algebra is a noetherian k-algebra where the ring k is artinian; equivalently the center Z(A) of A is an artinian commutative ring and A is finitely generated as a module over it. Over such rings the various finiteness conditions considered in this article coincide.

Lemma 5.5. Let A be an Artin algebra and X, Y objects in D^{b}(A). If the graded module Ext_{A}^{∗}(X, Y) is in noeth^{fl}(R) for some ring R acting centrally on D^{b}(A), then it is noetherian and degreewise of finite length over the ring R ⊗_{ℤ} Z(A).

Proof. It is easy to check that the Z(A)-module Ext_{A}^{i}(X, Y) has finite length for each i. The desired result is a consequence of this observation. □

For Artin algebras we are able to establish a stronger version of Theorem 5.4, where one does not have to assume beforehand that the ring is Gorenstein. This is based on the following observation.

Proposition 5.6. Let A be an Artin algebra with radical 𝔯. If Ext_{A}^{∗}(A/𝔯, A/𝔯) is noetherian over some graded-commutative ring acting centrally on D^{b}(A), then A is Gorenstein.

Proof. Observe that G = A/𝔯 is a generator for D^{b}(A). Thus, an A-module X has finite injective dimension if and only if Ext_{A}^{∗}(G, X) is eventually zero, and X has finite projective dimension if and only if Ext_{A}^{∗}(X, G) is eventually zero. Moreover, when Ext_{A}^{∗}(G, G) is eventually noetherian over some ring R acting centrally on T, then so are Ext_{A}^{∗}(X, G) and Ext_{A}^{∗}(G, X). In view of Lemma 2.2, one may assume that R = R^{ev}, so Lemma 4.1 yields an equality

Supp_{R} Ext_{A}^{∗}(X, G) = Supp_{R} Ext_{A}^{∗}(G, X) .

Applying Proposition 2.4, it follows that X has finite projective dimension if and only if it has finite injective dimension. The duality between right and left modules then implies that A is Gorenstein. □

Recall that the Loewy length of an Artin algebra A, with radical 𝔯, is the least non-negative integer n such that 𝔯^{n} = 0; we denote it ll(A).

Corollary 5.7. Let A be an Artin algebra with radical 𝔯. If Ext_{A}^{∗}(A/𝔯, A/𝔯) is noetherian as a module over some ring acting centrally on D^{b}(A), then

ll(A) ≥ dim D^{b}(A) ≥ dim D_{st}^{b}(A) ≥ cx_{A}(A/𝔯) − 1 .

Proof. The first inequality holds because thick^{ll(A)}(A/𝔯) = D^{b}(A); see [25, Lemma 7.35]. The rest are obtained by combining Proposition 5.6 and Theorem 5.4. □

Remark 5.8. In view of results of Friedlander and Suslin [18], the preceding result applies, in particular, to the case when A is a co-commutative Hopf algebra. In this case, one could choose either the Hochschild cohomology of A over k, or the k-algebra Ext_{A}^{∗}(k, k) acting on D^{b}(A) via the diagonal action, as the ring acting centrally on D^{b}(A).

One may specialize further to the case where k is a field of characteristic p and A = kG is the group algebra of a finite group G. It follows from a theorem of Quillen [24] that cx_{kG}(k) equals the p-rank of G. Thus, Corollary 5.7 yields the following inequalities

ll(kG) ≥ dim D_{st}^{b}(kG) ≥ rank_{p}(G) − 1 .

These estimates were first obtained in [23] using different methods.

Remark 5.9. We should like to note that when A is an Artin k-algebra which is also projective as a k-module, one has a natural first choice for the ring acting centrally on D^{b}(A), namely, the Hochschild cohomology HH^{∗}(A) of A over k.

Suppose that A is finite dimensional over an algebraically closed field k. In [17, §2], Erdmann et al. introduced the following finiteness condition: There is a noetherian graded subalgebra H of HH^{∗}(A), with H^{0} = HH^{0}(A), such that Ext_{A}^{∗}(A/𝔯, A/𝔯) is finitely generated over H. This condition has been investigated by various authors, in particular, in connection with the theory of support varieties.

The present work and [6] suggest that noeth^{fl}(D^{b}(A)) = D^{b}(A) is the appropriate finiteness condition on A; see Remark 4.4. In particular, the ring R acting centrally on D^{b}(A) is not essential, and the emphasis shifts rather to properties of Ext_{A}^{∗}(A/𝔯, A/𝔯) alone. While this point of view is more general, is seems also to be technically simpler and more flexible.

Complete intersections. For a commutative local ring A, with maximal ideal 𝔪 and residue field k = A/𝔪, the number edim A − dim A is called the codimension of A, and denoted codim A; here edim A is the embedding dimension of A, that is to say, the k-vector space dimension of 𝔪/𝔪^{2}.

The result below holds also without the hypothesis that A is complete; see [5]. For the definition of a complete intersection ring, see [12, §2.3].

Corollary 5.10. Let A be a commutative local ring, complete with respect to the topology induced by its maximal ideal.

When A is complete intersection, one has inequalities

dim D^{b}(A) ≥ dim D_{st}^{b}(A) ≥ codim A − 1 .

Sketch of proof. Set c = codim A. The hypotheses on A imply that there is a polynomial ring A[χ_{1}, …, χ_{c}], where the χ_{i} are indeterminates of degree 2, acting centrally on D^{b}(A) with the property that, for any pair of complexes X, Y in D^{b}(A), the graded A-module Ext_{A}^{∗}(X, Y) is finitely generated over A[χ_{1}, …, χ_{c}]; see, for instance, [6, §7.1]. Since A is complete intersection, it is Gorenstein; see [12, Proposition 3.1.20]. Hence Theorem 5.4 applies and, for the residue field k of A, yields inequalities

dim D^{b}(A) ≥ dim D_{st}^{b}(A) ≥ cx_{A}(k, k) − 1 .

It remains to note that cx_{A}(k, k) = codim A, by a result of Tate [29, Theorem 6]. □

We now apply the preceding results to obtain bounds on the representation dimension of an Artin algebra.

Representation dimension. Let A be an Artin algebra. The representation dimension of A is defined as

rep. dim A = inf \{gl. dim End_{A}(M) | \begin{matrix}M is a generator and a \\ cogenerator for mod A \end{matrix} \} .

Auslander has proved that A is semi-simple if and only if rep. dim A = 0, and that rep. dim A ≤ 2 if and only if A has finite representation type; see [1]. The connection between this invariant and dimensions for triangulated categories is that, when A is not semi-simple, one has an inequality:

rep. dim A ≥ dim D_{st}^{b}(A) + 2 .

This result is contained in [26, Proposition 3.7]. With Theorem 4.2, it yields a lower bound for the representation dimension of Artin algebras:

Theorem 5.11. Let A be an Artin algebra that is not semi-simple, and let 𝔯 be the radical of A. If Ext_{A}^{∗}(A/𝔯, A/𝔯) is noetherian as a module over some gradedcommutative ring acting centrally on D^{b}(A), then

rep. dim A ≥ cx_{A}(A/𝔯) + 1 .  □

With a further hypothesis that A is self-injective, this result was proved in [10, Theorem 3.2]. Arguing as [10, Corollary 3.5], which again required that A be selfinjective, one obtains the following result relating the representation dimension of an algebra to the Krull dimension of its Hochschild cohomology ring. The hypotheses on A/𝔯 ⊗_{k} A/𝔯 holds, for example, when k is algebraically closed.

Corollary 5.12. Let A be a finite dimensional algebra over a field k with radical 𝔯, where A is not semi-simple and A/𝔯 ⊗_{k} A/𝔯 is semi-simple. If Ext_{A}^{∗}(A/𝔯, A/𝔯) is noetherian as a module over the Hochschild cohomology algebra HH^{∗}(A) of A over k, then

rep. dim A ≥ dim HH^{∗}(A) + 1 .

Sketch of proof. Set R = HH^{∗}(A). Given Theorem 5.11 and Proposition 2.6(2), one has only to prove that

Spec R = supp_{R} Ext_{A}^{∗}(A/𝔯, A/𝔯) .

This holds because the semi-simplicity of A/𝔯 ⊗_{k} A/𝔯 implies that the kernel of the natural map R → Ext_{A}^{∗}(A/𝔯, A/𝔯) is nilpotent; see [28, Proposition 4.4]. □

The inequality in the preceding result need not hold if A/𝔯 ⊗_{k} A/𝔯 is not semisimple, for then the kernel of the homomorphism from Hochschild cohomology to the graded center of the derived category need not be nilpotent. This is illustrated by the following example.

Example 5.13. Let k be a field of characteristic p > 0. Assume that k is not perfect, so that there is an element a ∈ k that has no pth root in k. Let A = k[a^{1/p}], the extension field obtained by adjoining the pth root of a.

Since A is a field, one has rep. dim A = 0 and dim D^{b}(A) = 0.

On the other hand, it is easy to check that the Hochschild cohomology algebra of A over k is the polynomial algebra A[x], where |x| = 1, when p = 2, and the graded-commutative polynomial algebra A[x, y], where |x| = 1 and |y| = 2, when p is odd. Thus, in either case, dim HH^{∗}(A) = 1 holds.

The center of the derived category of A is readily computed:

Z^{n}(D^{b}(A)) = \begin{cases}k & when n = 0; \\ 0 & otherwise . \end{cases}

Thus the kernel of the homomorphism HH^{∗}(A) → Z^{∗}(D^{b}(A)) is not nilpotent; compare this example with [28, Proposition 4.4].

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Petter Andreas Bergh, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

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

Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln NE 68588, U.S.A.

   E-mail address: iyengar@math.unl.edu

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

   E-mail address: hkrause@math.upb.de

Steffen Oppermann, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

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