Levels in triangulated categories

Levels in triangulated categories Srikanth Iyengar University of Nebraska, Lincoln Leeds, 18th August 2006

The goal My aim is to make a case that the invariants that I call levels are useful and interesting invariants. Towards this end, I discuss the proof of the following result. Theorem Let R be a commutative noetherian local ring. For every finite free complex F the following inequality holds: Loewy R H n (F ) 1 + conormal free-rank of R. n Z For now, it is not relevant what conormal free-rank of R is. What is relevant is that the statement makes no mention of levels, or triangulated categories, or... Applied to group algebras of elementary abelian p-groups, this recovers results of G. Carlsson, and C. Allday and V. Puppe.

Outline Thickenings and levels DG modules over DG algebras A New Intersection Theorem for DG modules Homology of perfect complexes The dimension of the stable derived category of a local ring

Joint work with (subsets of) L. L. Avramov and C. Miller R.-O. Buchweitz Based on the following articles: Class and rank for differential modules (on the arxiv.) Homology of perfect complexes (will be on the arxiv before long.)

Thick subcategories Let C be a non-empty class of objects in triangulated category T. Let Thick T (C) be the smallest thick subcategory containing C. Its objects may be thought as being finitely built out of C. Example Let R be a ring. Write Thick R (C) for Thick D(R) (C). Thick R (R) is the category of perfect complexes: complexes quasi-isomorphic to one of the form 0 F t F s 0 where each F i is a finitely generated projective R-module. If R is semi-local, with Jacobson radical m, then Thick R (R/m) = {M D(R) length R H(M) <.}

Thickenings We consider subcategories {thick n T (C)} n 0 of Thick T (C): thick 0 T (C) = {0}. thick 1 T (C) = C closed up under shifts, finite direct sums, retracts. thick n T (C) is the subcategory with objects { } L M N ΣL is an exact triangle M with L thick n 1 T (C) and N thick1 T (C) closed up under retracts. Note that thick n T (C) is closed under shifts and direct sums, but not under triangles. We call thick n T (C) the nth thickening of C in T. It consists of (n 1)-fold extensions of objects in thick 1 T (C).

These subcategories provide a filtration {0} thick 1 T (C) thick2 T (C) n 0 thick n T (C) = Thick T(C). This filtration appears in the work of Bondal and Van den Bergh: Generators and representability... Dan Christensen: Ideals in triangulated categories... Rouquier: Dimension of triangulated categories. Representation dimension of exterior algebras. The focus in these works is on global aspects of T. Here we use the filtration to obtains invariants of objects in T. Another pertinent reference: Dwyer, Greenlees, I.: Finiteness in derived categories...

Levels Let M be an object in T. The C-level of M is the number level C T (M) = inf{n 0 M thickn T (C)}. Evidently, level C T (M) is finite if and only if M is in Thick T(C). This invariant has good formal properties. For example: If L M N ΣL is an exact triangle, then level C T (M) levelc T (L) + levelc T (N). Thus, levels are sub-additive. If f : T S is an exact functor between triangulated categories, then level C T (M) levelf(c) f(t) (f(m)).

Why levels? This is what this talk is about. By varying the class C one can model various invariants of interest: projective dimension, Loewy length, regularity, e.t.c. Most ring-theoretic invariants, and certainly those in the preceding list, do not behave well under change of categories. Levels do, and provide a versatile tool for studying them. I will now discuss the case where T = D(A), the derived category of a DG (=Differential Graded) algebra A. I will focus on level with respect to A. This models classical projective dimension for modules over rings. It is convenient to write level A A ( ) instead of levela D(A) ( ),

DG modules over DG algebras Let A be a DG algebra and M a DG A-module. Theorem One has level A A (M) d if and only if M is a retract of a DG A-module F admitting a filtration 0 F 0 F d 1 = F where F n /F n 1 is isomorphic to a direct sum of shifts of A. One direction is clear: A filtration as above induces exact triangles F n 1 F n F n /F n 1 ΣF n 1 for 0 n d 1, so level A A (M) d, by sub-additivity. Note: when M is in Thick A (A), it is quasi-isomorphic to a DG module whose underlying graded module is projective over A. The converse holds under additional hypotheses on A.

Example Let R be a ring and F a finite free complex: F = 0 F t F s 0. Then F n = 0 F s+n F s 0 gives a filtration of F, so level R R (F ) card{n F n 0}. Often the inequality is strict: F can be built more efficiently. Definition Given elements x 1,..., x n in a commutative ring R, the complex (0 R x 1 R 0) R R (0 R xn R 0) is the Koszul complex on x.

Example Let R = k[x, y], a polynomial ring with x = 0 = y. Pick d 1. Let K be the Koszul complex on x d, x d 1 y,..., xy d 1, y d. Thus K n = R ( d+1 n ) therefore level R R (K) d + 2. However, level R R (K) = 3. (This calls for an explanation!) Remark In the last example card{n K n 0} level R R (K) = d 1; in particular, the difference can be made arbitrarily large. The next few slides discuss bounds on level A A (M). Remark Upper bounds on levels are easier to obtain than lower bounds.

Let A be a DG algebra with A = 0, left coherent as a graded ring. Proposition If the A-module H(M) is finitely presented, then level A A (M) proj dim A H(M) + 1 gl dim A + 1. Remark: To get a better result one should consider levels with respect to projectives. One way to prove the proposition is to pick a projective resolution 0 H(M) P 0 ΣP 1 Σ 2 P 2 and construct an Adams resolution: M = M 0 ΣM 1 Σ 2 X 2 +1 P 0 ΣP 1 +1 Σ 2 P 2

A New Intersection Theorem for DG modules Theorem Let A be a DG algebra with A = 0 and M a DG A-module. When A is a commutative noetherian algebra over a field one has level A A (M) codim H(M) + 1. Recall: codim H(M) = height Ann A H(M). Observe that this number depends only the support of H(M). Corollary Let R be a commutative noetherian algebra over a field. If F = 0 F d F 0 0 be a finite free complex, then d + 1 card{n F n 0} level A A (F ) codim H(F ) + 1. In particular, d codim H(F ). This is the classical New Intersection Theorem.

Example Let R = k[x 1,..., x n ] (or any regular local ring) If F is a finite free complex with length R H(F ) 0,, then n + 1 = gl dim R + 1 level R R (F ) codim H(F ) + 1 = n + 1. Therefore, level R R (F ) = n + 1. This calculation applies, in particular, when: R = k[x, y] F = K, the Koszul complex on x d, x d 1 y,..., xy d 1, y d. Thus, level R R (K) = 3.

Remarks The New Intersection Theorem for algebras over a field was proved by Peskine and Szpiro, Hochster, and P. Roberts. Using intersection theory, P. Roberts proved it for all commutative noetherian rings. Even for rings, we have not been able to deduce our theorem from Roberts result. Let me remind you that the inequality is typically strict. card{n F n 0} level A A (F ) Our proof uses local cohomology and big Cohen-Macaulay modules. Hochster has constructed them for algebras over fields, hence the restriction on A.

The result is deduced from an analogous statement for differential modules, proved in: Class and rank for differential modules (Avramov, Buchweitz, I.). The idea is to construct a sequence of complexes X (d+1) θ(d+1) X (d) θ(d) θ(1) X (0) where d = codim H(M), with the following properties: (a) H(θ(n)) = 0 for each n; (b) H(M A θ) 0, where θ = θ(1) θ(d + 1). Much of what I have said so far applies to differential modules. This is important for some applications. That is the beginning of a different story, and is work in progress with L. Avramov, R.-O. Buchweitz, Lars Christensen, and Greg Piepmeyer.

General strategy to estimate levels Suppose C is an object in a triangulated category T. We wish to estimate level C T (X ), for some object X in T. One strategy is as follows: Find a (commutative noetherian) DG algebra A with A = 0 and an exact functor f : T D(A) such that f(c) is a finitely generated projective A-module. Then f(c) thick 1 A (A), so one obtains an estimate level C T (X ) levelf(c) A (f(x )) levela A (f(x )) codim f(x ).

Free summands of the conormal module Let k be a field and R = k[x ]/I where k[x ] is a polynomial over k in variables X = {x 1,..., x e }; I is a homogeneous ideal in (X ) 2. The R-module I /I 2 is the conormal module of R. It is independent of the presentation R = k[x ]/I as above. The conormal free rank of R is the number { } R n is a free direct summand cf-rank R = sup n of the conormal module of R It is a measure of the singularity of R.

Example Let R = k[x 1,..., x c ]/(x n 1 1,..., x nc c ), with each n i 2. Then the conormal module of R is I /I 2, where I = (x n 1 1,..., x nc c ). It is easy to check that I /I 2 = R c, so cf-rank R = c. Special case: n i = p, with p 2, covers group algebras of elementary abelian groups. Free summands of conormal modules arise in the following cases: When R has embedded deformations: if R = Q/xQ, where (Q, q) is a local ring and x a non-zero divisor in q 2. When R is the closed fibre of a flat homomorphism ϕ: (P, p) (Q, q) such that ϕ(p) q 2. Aside: there is a notion of conormal free rank for any (commutative noetherian) local ring.

Homology of perfect complexes The Loewy length of an R-module M equals the number Loewy R M = inf{n 0 m n M = 0}, where m is the maximal ideal of R. When length R M is finite, so is Loewy R M; the converse holds when M is finitely generated. Loewy length better reflects the structure of M than length does. Theorem If F is a finite free complex of R-modules with H(F ) 0, then Loewy R H n (F ) cf-rank R + 1. n Thus, the singularity of R imposes lower bounds on the size of homology of finite free complexes.

Group algebras of elementary abelian groups Let p a prime, and R the group algebra over F p of a rank c elementary abelian p-group. Thus, R = F p [x 1,..., x c ]/(x p 1,..., x p c ) and I /I 2 = R c. Therefore cf-rank R = c, and the theorem yields: Loewy R H n (F ) c + 1. n In this way, the theorem specializes to results of G. Carlsson, who proved it when p = 2; C. Allday and V. Puppe, who proved it for odd primes, which has application to the study of finite group actions. Neither of their methods extends to cover the other case...

Proof of theorem The first step is convert the problem to one about levels: Lemma One has an inequality: n Loewy R H n (F ) level k R (F ). This inequality follows from general properties of levels. Thus, it suffices to prove the following inequality: level k R (F ) c + 1 where c = cf-rank R. Let K be the Koszul complex on a set of generators for m. This is a DG algebra (an exterior algebra with a differential).

Let Λ be an exterior algebra on c variables of degree 1. A crucial input in the proof is: Theorem As DG algebras K A, where A = Λ k B, and under the induced equivalence a: D(K) D(A) of derived categories, one has a(k R k) n Σ n k (c n). This is where the free summand of the conormal module comes in. The proof involves calculations with various DG algebra models for the Koszul complex. It is akin to the Jacobian criterion. Here is one consequence of the preceding theorem: Corollary level a(k Rk) A ( ) = level k A ( ).

Let S be a polynomial ring on c variables of degree 2. We view it as a DG algebra with zero differential. One has exact functors between triangulated categories D(R) t D(K) a D(A) i D(Λ) h D(S) The functors involved are as follows: t = K R a is the equivalence of categories in the last result. i is induced by the inclusion Λ (Λ k B) = A, and h is the BGG functor representing RHom Λ (k, ). In particular, h(λ) Σ c k and h(k) S.

Summing up We want to prove: If F is a finite free complex of R-modules, that is to say, if level R R (F ) is finite, then level k R (F ) c + 1 where c = cf-rank R. We will deduce this from the New Intersection Theorem for S: For any DG S-module M, one has an inequality The path from R to S is level S S (M) codim H(M) + 1. D(R) t D(K) a D(A) i D(Λ) h D(S) A rappel Levels are non-increasing under application of exact functors.

level k R (F ) \ W level t(k) K (t(f )) levelr R (F ) 0, level K K (t(f )) 0, level at(k) A (at(f )) level k A (at(f )) levela A (at(f )) 0, \ W level k Λ (iat(f )) level Λ Λ (iat(f )) 0, BGG level S S (hiat(f )) BGG level k S (hiat(f )) 0, \ W NIT dim S + 1 length S (H(hiat(F ))) 0,

The proof is better than the theorem: 1. The result can be formulated (and proved) for all local rings. 2. When R is complete intersection, the same argument yields: Theorem If M is a complex of R-modules with H(M) noetherian, then Loewy R H n (M) codim V R (M) + 1, n where V R (M) is the cohomological variety of M. Specialized to group algebras, this recovers a result of Benson and Carlson, which was proved using shifted subgroups.

Dimension of stable categories The dimension of a triangulated category T is the number { } there is an object G in T dim T = inf d 0 such that thick d+1 T (G) = T This invariant was introduced by Rouquier. Theorem Let R be a local ring and set T = D b (R)/Thick R (R). Then dim T cf-rank R 1. Thus, embedded deformations of R impose lower bounds on dim T. Note: when R is complete intersection cf-rank R = codim R. Example When R = k[x 1,..., x c ]/(x n 1 1,..., x nc c ) with n i 2, then dim stmod(r) c 1.

A partial list of references L. L. Avramov, R.-O. Buchweitz, S. Iyengar, C. Miller, Homology of perfect complexes, preprint 2006. L. L. Avramov, R.-O. Buchweitz, S. Iyengar, Class and rank of differential modules, ArXiv: math.ac/0602344. A. Bondal, M. Van den Bergh, Generators and representability of functors in commutative and non-commutative geometry, Moscow Math. J. 3 (2003), 1-36. D. J. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284 339. W. G. Dwyer, J. P. C. Greenlees, S. Iyengar, Finiteness in derived categories of local rings, Commentarii Math. Helvetici 81 (2006), 383 432. R. Rouquier, Representation dimension of exterior algebras, Invent. Math. 165 (2006), 357 367. Dimensions of triangulated categories, math.ct/0310134.