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 → Σ L is an exact triangle � M � with L ∈ thick n − 1 (C) and N ∈ thick 1 � T (C) � T 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 n th 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) ⊆ thick 2 thick n T (C) ⊆ · · · ⊆ T (C) = Thick T (C) . n � 0 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 ∈ thick n 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 → Σ L is an exact triangle, then level C T ( M ) ≤ level C T ( L ) + level C T ( N ) . Thus, levels are sub-additive. If f : T → S is an exact functor between triangulated categories, then T ( M ) ≥ level f(C) level 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 level A 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 → Σ F 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 x 1 x n (0 → R − → R → 0) ⊗ R · · · ⊗ R (0 → R − → 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 = R ( d +1 n ) K n ∼ level R therefore 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 ← Σ P 1 ← Σ 2 P 2 ← · · · and construct an Adams resolution: � Σ M 1 � Σ 2 X 2 � · · · · · · M = M 0 � � � � � ��������� � � ������� � � � � � � � � � � � � � � � +1 +1 � � � � � P 0 Σ P 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 card { n | F n � = 0 } ≥ level A A ( F ) is typically strict. 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 T ( X ) ≥ level f( C ) level C (f( X )) ≥ level A A (f( X )) ≥ codim f( X ) . A
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