Levels in triangulated categories Srikanth Iyengar University of - - PowerPoint PPT Presentation

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:

• n∈Z

LoewyR Hn(F) ≥ 1 + conormal free-rank of R . 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,

• r 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 ThickT(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 ThickR(C)− for ThickD(R)(C)−. ThickR(R) is the category of perfect complexes: complexes quasi-isomorphic to one of the form 0 → Ft → · · · → Fs → 0 where each Fi is a finitely generated projective R-module. If R is semi-local, with Jacobson radical m, then ThickR(R/m) = {M ∈ D(R)| lengthR H(M) < ∞.}

Thickenings

We consider subcategories {thickn

T(C)}n0 of ThickT(C):

thick0

T(C) = {0}.

thick1

T(C) = C closed up under shifts, finite direct sums, retracts.

thickn

T(C) is the subcategory with objects

• M
• L → M → N → ΣL is an exact triangle

with L ∈ thickn−1

T

(C) and N ∈ thick1

T(C)

• closed up under retracts.

Note that thickn

T(C) is closed under shifts and direct sums, but not

under triangles. We call thickn

T(C) the nth thickening of C in T.

It consists of (n − 1)-fold extensions of objects in thick1

T(C).

These subcategories provide a filtration {0} ⊆ thick1

T(C) ⊆ thick2 T(C) ⊆ · · · ⊆

• n0

thickn

T(C) = ThickT(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 levelC

T(M) = inf{n ≥ 0|M ∈ thickn T(C)}.

Evidently, levelC

T(M) is finite if and only if M is in ThickT(C).

This invariant has good formal properties. For example: If L → M → N → ΣL is an exact triangle, then levelC

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 levelC

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

• f 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 levelA

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 levelA

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 levelA

A(M) ≤ d, by sub-additivity.

Note: when M is in ThickA(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 → Ft → · · · → Fs → 0 . Then F n = 0 → Fs+n → · · · → Fs → 0 gives a filtration of F, so levelR

R(F) ≤ card{n | Fn = 0} .

Often the inequality is strict: F can be built more efficiently. Definition Given elements x1, . . . , xn in a commutative ring R, the complex (0 → R

x1

− → 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 xd, xd−1y, . . . , xyd−1, yd. Thus Kn ∼ = R(d+1

n )

therefore levelR

R(K) ≤ d + 2 .

However, levelR

R(K) = 3. (This calls for an explanation!)

Remark In the last example card{n | Kn = 0} − levelR

R(K) = d − 1; in

particular, the difference can be made arbitrarily large. The next few slides discuss bounds on levelA

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 levelA

A(M) ≤ proj dimA 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) ← P0 ← ΣP1 ← Σ2P2 ← · · · and construct an Adams resolution: M = M0

ΣM1

+1

• Σ2X 2

+1

• · · · · · ·

P0

• ΣP1
• Σ2P2
• · · ·

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 levelA

A(M) ≥ codim H(M) + 1 .

Recall: codim H(M) = height AnnA 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 → Fd → · · · F0 → 0 be a finite free complex, then d + 1 ≥ card{n|Fn = 0} ≥ levelA

A(F) ≥ codim H(F) + 1 .

In particular, d ≥ codim H(F). This is the classical New Intersection Theorem.

Example Let R = k[x1, . . . , xn] (or any regular local ring) If F is a finite free complex with lengthR H(F) = 0, ∞, then n + 1 = gl dim R + 1 ≥ levelR

R(F) ≥ codim H(F) + 1 = n + 1 .

Therefore, levelR

R(F) = n + 1.

This calculation applies, in particular, when: R = k[x, y] F = K, the Koszul complex on xd, xd−1y, . . . , xyd−1, yd. Thus, levelR

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|Fn = 0} ≥ levelA

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

• f 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 levelC

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) ∈ thick1

A(A), so one obtains an estimate

levelC

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 = {x1, . . . , xe}; 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 cf-rank R = sup

• n
• Rn is a free direct summand
• f the conormal module of R
• It is a measure of the singularity of R.

Example Let R = k[x1, . . . , xc]/(xn1

1 , . . . , xnc c ), with each ni ≥ 2.

Then the conormal module of R is I/I 2, where I = (xn1

1 , . . . , xnc c ).

It is easy to check that I/I 2 ∼ = Rc, so cf-rank R = c. Special case: ni = 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 q2. When R is the closed fibre of a flat homomorphism ϕ: (P, p) → (Q, q) such that ϕ(p) ⊆ q2. 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 LoewyRM = inf{n ≥ 0 | mnM = 0} , where m is the maximal ideal of R. When lengthR M is finite, so is LoewyRM; 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

• n

LoewyR Hn(F) ≥ cf-rank R + 1 . 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 Fp of a rank c elementary abelian p-group. Thus, R ∼ = Fp[x1, . . . , xc]/(xp

1 , . . . , xp c ) and I/I 2 ∼

= Rc. Therefore cf-rank R = c, and the theorem yields:

• n

LoewyR Hn(F) ≥ c + 1 . 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 LoewyR Hn(F) ≥ levelk R(F).

This inequality follows from general properties of levels. Thus, it suffices to prove the following inequality: levelk

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

Σnk(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 levela(K⊗Rk)

A

(−) = levelk

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 RHomΛ(k, −). In particular, h(Λ) ≃ Σck 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 levelR

R(F) is finite, then

levelk

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 levelS

S(M) ≥ codim H(M) + 1 .

The path from R to S is 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.

levelk

R(F)

levelR

R(F) = 0, ∞

levelt(k)

K

(t(F))

\W

levelK

K(t(F)) = 0, ∞

• levelat(k)

A

(at(F)) levelk

A(at(F))

levelA

A(at(F)) = 0, ∞

• levelk

Λ(iat(F)) \W

levelΛ

Λ(iat(F)) = 0, ∞

• levelS

S(hiat(F)) BGG

levelk

S(hiat(F)) = 0, ∞

• BGG

dim S + 1

\W NIT

• lengthS(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

• n

LoewyR Hn(M) ≥ codim VR(M) + 1 , where VR(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 dim T = inf

• d ≥ 0
• there is an object G in T

such that thickd+1

T

(G) = T

• This invariant was introduced by Rouquier.

Theorem Let R be a local ring and set T = Db(R)/ThickR(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[x1, . . . , xc]/(xn1

1 , . . . , xnc c ) with ni ≥ 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
• f 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.