KOSZUL DUALITY AND CODERIVED CATEGORIES (AFTER K. LEFÈVRE)

BERNHARD KELLER

Abstract. This is a brief report on a part of Chapter 2 of K. Lefèvre’s thesis [5]. We sketch a framework for Koszul duality [1] where the Koszul dual algebra is replaced by a coalgebra. This allows us to free ourselves from many assumptions (e.g. finiteness assumptions) and leads to clean statements about equivalences between the derived category and a suitably defined coderived category. These results are related to work by G. Fløystad [2]. Our approach is based on classical developments in topology [6] [4] and inspired by [3].

Acknowledgment

These are notes of a 50-minute talk given at the Frobenius conference in Toruń (Poland) in September 2003. The author thanks Andrzej Skowroński and his group for their kind hospitality and their great work in organizing a wonderful conference. He is grateful to Patrick Polo and Catharina Stroppel for their remarks on previous versions of this note.

1. Koszul duality (after [1])

Let k be a field and

A = k ⊕ A_{1} ⊕ A_{2} ⊕ ⋯

a graded k-algebra with finite-dimensional components A_{i}, i ∈ N. Let Grmod A denote the category of graded (right) A-modules (with the morphisms of degree 0). For a module M ∈ Grmod A and n ∈ Z, the shifted module M⟨n⟩ is defined by

M⟨n⟩_{p} = M_{n+p} , p ∈ Z.

Assume that A is a Koszul algebra, i.e. that there is a projective resolution

(1.0.1)  … → P^{−i} → … → P^{0} → 0

of the trivial module k in Grmod A such that P^{−i} is generated in degree i. We imagine such a resolution as a bigraded object, where the differential degree is drawn horizontally and the Adams degree (=internal degree) vertically. Put

E(A) = ⨁_{i=0}^{∞} Ext_{Grmod A}^{i}(k, k⟨i⟩)

and call this algebra the Koszul dual algebra.

Theorem. [1]

a) The graded algebra E(A) is Koszul and there is a canonical isomorphism A →^{∼} E(E(A)) (cf. section 2.8, Proposition 2.9.1 and Corollary 2.3.3 of [1]). b) There is a canonical equivalence of triangulated categories

Date: October 2003, last modified on April 4, 2006.

1991 Mathematics Subject Classification. 16E40, 18E30, 16D90, 18G10.

Key words and phrases. Koszul duality, derived category, coalgebra, bar construction.

D_{gr}^{↓}(A) →^{∼} D_{gr}^{↑}(E(A)).

It sends the simple k to the projective E(A) and the graded k-dual of the free module A to the simple k (cf. Theorem 2.12.1 and Theorem 1.2.6 of [1]).

Here D_{gr}^{↓}(A) denotes the full subcategory of the unbounded derived category D(Grmod A) of the abelian category Grmod A whose objects are the complexes K such that for some N ≫ 0, we have

K_{q}^{p} ≠ 0 ⇒ (p ≥ −N or p + q ≤ N).

Analogously, D_{gr}^{↑}(E(A)) denotes the full subcategory of D(Grmod E(A)) whose objects are the complexes K such that for some N ≫ 0, we have

K_{q}^{p} ≠ 0 ⇒ (p ≤ N or p + q ≥ −N).

As an example, let V be a finite-dimensional vector space and A = SV the symmetric algebra on V. Then for the resolution 1.0.1, we can take the Koszul resolution

(1.0.2)  … → Λ^{p}V ⊗ SV ⟨−p⟩ → … → Λ^{0}V ⊗ SV → 0

with the differential given by

d(v_{1} … v_{p} ⊗ u) = ∑_{i=1}^{p}(−1)^{p−i}v_{1} … v̂_{i} … v_{p} ⊗ v_{i}u.

Then E(A) identifies with the exterior algebra Λ(DV) on the k-dual space DV of V. According to the theorem, we have an equivalence

F : D_{gr}^{↓}(SV) →^{∼} D_{gr}^{↑}(ΛDV).

Can the equivalence F be extended to an equivalence F̃ between the whole derived categories ?

\xymatrix{D_{gr}^{↓}(SV) \ar[d] \ar[r]^{F} & D_{gr}^{↑}(ΛDV) \ar[d] \\ D(Grmod SV) \ar@{.>}[r]^{F̃} & D(Grmod ΛDV)}

Suppose that such an equivalence F̃ exists. It is not hard to see that it has to take the free module SV to the doubly shifted trivial module k⟨n⟩[−n], where the braces [ ] indicate a shift of the differential degree. This is impossible since SV is compact in D(Grmod SV), i.e. the functor

Hom_{D(Grmod A)}(SV,?) : D(Grmod SV) → Mod k

commutes with infinite direct sums, but the object k is non compact in D(Grmod ΛDV).

In what follows, our aim is to present a setting where we free ourselves from the following restrictions

• A is (Adams-)graded with finite-dimensional components,

• A is a Koszul algebra,

• there is an equivalence only between certain subcategories of the derived categories.

2. Data

2.1. An algebra. Let A be a differential graded (=dg) algebra, i.e. A is an associative unital Z-graded algebra

A = ⨁_{p∈Z} A^{p}

endowed with a differential d of degree +1 such that the Leibniz rule holds: We have

d(ab) = d(a)b + (−1)^{p}ad(b)

for all a ∈ A^{p} and all b ∈ A. Note that A is differentially graded but not Adams graded. We assume that A is endowed with an augmentation, i.e. a morphism of dg algebras ε : A → k. Therefore A decomposes as

A = k ⊕ \overline{A} ,

where \overline{A} is the kernel of ε. Note that we do not require the components of A to be finite-dimensional.

2.2. A coalgebra. Let C be a dg coalgebra, i.e. C is a Z-graded coalgebra endowed with a differential d of degree +1 such that

Δ ∘ d = (d ⊗ 1 + 1 ⊗ d) ∘ Δ.

We assume that C is endowed with a co-augmentation, i.e. a morphism of dg coalgebras ε : k → C. Then C decomposes as

C = k ⊕ \overline{C} ,

where \overline{C} is the cokernel of ε. Moreover, we assume that C is cocomplete, which means that

\overline{C} = ⋃_{n≥2} ker(\overline{C} → \overline{C}^{⊗n}) ,

i.e. each element of \overline{C} is annihilated by a high enough iterate of the map induced by the comultiplication of C. Note that this implies that the k-dual algebra DC is a complete local algebra.

2.3. A twisting cochain. Let τ : C → A be a twisting cochain, i.e. τ is a k-linear homogeneous map of degree +1 such that ε_{A} ∘ τ ∘ ε_{C} = 0 and

d ∘ τ + τ ∘ d + μ ∘ (τ ⊗ τ) ∘ Δ = 0 ,

where μ is the multiplication of A. In other words, the map τ satisfies ε(τ) = 0 and d(τ) + τ ∗ τ = 0 in the dg convolution algebra Hom_{k}(C, A) of homogeneous maps from C to A.

2.4. Example. Let V be an (arbitrary) vector space and A = SV the symmetric algebra on V considered as a dg algebra concentrated in differential degree 0. The exterior algebra ΛV, graded such that V is in degree −1, admits a unique structure of super Hopf algebra such that Δ(v) = v ⊗ 1 + 1 ⊗ v for all v ∈ V. Let C be the graded coalgebra ΛV endowed with the differential d = 0. Thus, the underlying complex of C is

… → Λ^{p}V → … → Λ^{1}V → Λ^{0}V → 0 → …,

where the elements of V are of degree −1. We define the twisting cochain τ : C → A to have only one non-vanishing component and this component identifies V ⊂ ΛV with V ⊂ SV. Is is trivial to check the conditions of the preceding paragraphs.

3. Adjoint functors

We will construct a pair of adjoint functors between the category of dg modules over A and the category of certain dg comodules over C. Recall that a dg A-module is a Z-graded A-module M endowed with a differential of degree +1 such that

d(ma) = d(m)a + (−1)^{p}md(a)

for all m ∈ M^{p} and all a ∈ A. The notion of dg C-comodule is defined dually. A dg C-comodule N is cocomplete if it is the union of the kernels of the iterates

N → N ⊗ \overline{C}^{⊗n}

of the map induced by the comultiplication.

For a dg A-module M, we define the twisted tensor product M ⊗_{τ} C to be the dg C-comodule whose underlying graded comodule is the graded tensor product M ⊗_{k} C and whose differential is

d = d_{M} ⊗ 1 + 1 ⊗ d_{C} + (μ ⊗ 1)(1 ⊗ τ ⊗ 1)(1 ⊗ Δ) ,

where μ is the multiplication of M. Similarly, for a dg C-comodule N, one defines the dg A-module N ⊗_{τ} A.

In the above example 2.4, the twisted tensor product C ⊗_{τ} A is nothing but the Koszul complex 1.0.2.

Lemma. We have adjoint functors

[figure: {dg A-modules}; ? ⊗_{τ} C = R; {cocomplete dg C-comodules}; L =? ⊗_{τ} A]

4. (Co-)derived categories

Let DA be the derived category of A, i.e. the localization of the category of dg A-modules at the class of all quasi-isomorphisms.

To define the coderived category of C, we will need to replace the quasi-isomor-phisms by a different class of morphisms. To define these, we need the cobar construction ΩC: this is the graded tensor algebra on the shift \overline{C}[−1] endowed with the unique differential such that for each homogeneous element c ∈ \overline{C}[−1], we have

d(c) = −d_{C}(c) + ∑(−1)^{|c_{(1)}|}c_{(1)} ⊗ c_{(2)} ,

where we have used Sweedler’s notation and |c_{(1)}| denotes the degree of c_{(1)}. The cobar construction is endowed with the canonical twisting cochain τ_{0} : C → ΩC given by the evident map. We define a morphism f : M → N of dg C-comodules to be a weak equivalence if its image under the functor ? ⊗_{τ_{0}} ΩC defined in the preceding section is a quasi-isomorphism. In other words, the morphism

f ⊗_{τ_{0}} ΩC : M ⊗_{τ_{0}} ΩC → N ⊗_{τ_{0}} ΩC

should be a quasi-isomorphism. We define the coderived category DC of C to be the localization of the category of cocomplete dg C-comodules at the class of all weak equivalences.

Proposition. We have a pair of induced adjoint functors

[figure: DA; ? ⊗_{τ} C = R; DC; L =? ⊗_{τ} A]

Theorem. [5, Ch. 2] The following are equivalent (i) The functors L and R are equivalences. (ii) The canonical morphism

A ⊗_{τ} C ⊗_{τ} A → A

is a quasi-isomorphism. (iii) The map τ induces a quasi-isomorphism ΩC → A.

Moreover, in this case, A is determined by C up to quasi-isomorphism; C is determined by A up to weak equivalence^{1}; we have

H_{∗}C = Tor_{∗}^{A}(k, k) and H^{∗}A = Ext_{C}^{∗}(k, k).

We define (A, C, τ) to be a Koszul-Moore triple if the conditions (i)-(iii) hold. The theorem shows that for each given dg coalgebra C, there is at least one Koszul-Moore triple (ΩC, C, τ_{0}). Dually, one can show that the bar construction [4] yields a Koszul-Moore triple for each given algebra A.

Let us consider the example 2.4, where A = SV, C = ΛV and τ is the natural morphism. Then (A, C, τ) is indeed a Koszul-Moore triple by condition (ii): here, the canonical morphism is the bimodule Koszul resolution of A = SV. Thus we do have

D(A) →^{∼} D(C).

Suppose that dim V < ∞. Then dim C < ∞ and each C-comodule is cocomplete. Moreover, we have an isomorphism of categories

{dg C-comodules} →^{∼} {left DC-modules}

given by sending a dg C-comodule N to the dg left DC-module with the same underlying space and whose multiplication is given by the natural composition

DC ⊗ N → DC ⊗ N ⊗ C → DC ⊗ C ⊗ N → N.

So we obtain an equivalence

D(SV) →^{∼} {left dg Λ(DV)-modules}[W^{−1}]

where W denotes the class of morphisms which corresponds to the weak equivalences. This equivalence sends a dg SV -module M to the left Λ(DV)-module N ⊗ ΛV endowed with the Koszul differential.

To see that the weak equivalences form a strictly smaller class than the quasiisomorphisms, let us further specialize to the case where V is of dimension 1, i.e. V = kx and SV = k[x]. We obtain an equivalence

D(k[x]) →^{∼} {dg left k[ξ]-modules}[W^{−1}]

where ξ is of degree 1 and dξ = 0. The equivalence sends the module

k_{λ} = k[x]/(x − λ) , λ ∈ k ,

to the dg module

… → 0 → k →^{d} k → 0 → …

concentrated in degrees −1 and 0, where d is the multiplication by λ and ξ acts by the graded endomorphism of degree 1 which is given by the identity of k. Note that for each λ ≠ 0, the image of k_{λ} is quasi-isomorphic to 0, which corresponds to the fact that

Tor_{∗}^{k[x]}(k, k_{λ}) = 0.

However, the image of k_{λ} is never weakly equivalent to 0, since k_{λ} ≠ 0.

1A morphism of dg augmented coalgebras f : C → C′ is a weak equivalence if Ωf is a quasiisomorphism

References

1. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.

2. Gunnar Floystad, Koszul duality and equivalences of categories, arXiv:math.RA/0012264.

3. Vladimir Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), no. 2-3, 209–250.

4. Dale Husemoller, John C. Moore, and James Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185.

5. Kenji Lefèvre-Hasegawa, Sur les A_{∞}-catégories, Thèse de doctorat, Université Denis Diderot – Paris 7, November 2003, arXive:math.CT/0310337.

6. John C. Moore, Differential homological algebra, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 335–339.

UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

   E-mail address: keller@math.jussieu.fr www.math.jussieu.fr/  keller