BIREP Summer School on Koszul Duality – Talks by Professor V. Mazorchuk

No. Title Description
1
2
3
 The category of linear complexes
Quadratic duality
Koszul duality		 The aim of this series of lectures is to describe the approach to Koszul duality based on the study of the category of linear complexes of projective modules. The first lecture will provide necessary preliminaries about the category of linear complexes of projective modules. The second lecture will describe the general quadratic duality theorem for any positively graded algebra. The last lecture will then restrict to the special case of Koszul algebras for which quadratic duality becomes Koszul duality, that is an equivalence of certain triangulated categories.

BIREP Summer School on Koszul Duality – Talks by the participants

No. Title Description
1 Koszul rings and their duals The aim of this talk is to introduce some basic definitions, properties and results.
Quadratic and Koszul rings should be defined, as well as the quadratic dual $A^!$ and the Koszul dual $E(A)$. Various facts should be included for instance that Koszul implies quadratic, that $E(A)$ is again Koszul, and that $E(E(A)) = A$ (Theorem 1.2.5., [BGS96]).
The reference is [BGS96], section 2 up to 2.11.
2 The Koszul duality functor This talk should explain Theorem 1.2.6, [BGS96] (Koszul duality as derived equivalence) together with its proof which can be found in loc. cit. section 2.12-2.14.
3 Necessary and sufficient conditions Survey properties which are necessary and/or sufficient for an algebra to be Koszul.
For instance the numerical criterion for Koszulness should be proven ([BGS96] Thm 2.11.1). In particular it should be shown that the product of the Hilbert series of $A$ and the Poincare series of $A^!$ is $1$ (see [Rei03]).
Further possibilities are:
  1. Characterizations of Koszulness, [Kra], Thm 2, Prop. 10.
  2. A Koszul algebra has finite global dimension if and only if its Koszul dual $A^!$ is a finite dimensional $k$-vector space [Kra] Cor 6, Thm 1.
  3. The PBW-property implies Koszulness, cp. [PP05].
4 Examples from combinatorics Combinatorial problems provide a rich source of Koszul algebras. The aim of this talk is to provide further examples of Koszul algebras.
Several nice examples are outlined in [Rei03] and it is suggested that the speaker explains some subset of the following examples (although of course if the speaker has their own examples in mind the suggestions are not binding):
  1. A semigroup ring $K[\Lambda ]$ is Koszul if and only if $\Lambda$ is Cohen-Macaulay poset. Examples include Veronese subalgebras, Segre subalgebras, and Hibi rings. Further details can be found in [PRS98].
  2. Algebras from walks
  3. Stanley-Reisner rings
  4. Orlik-Solomon algebras
5 BGG correspondence This talk should cover the original inspiration for Koszul duality, namely the relationship between the symmetric and exterior algebras.
It would be good to cover the relation provided by Koszul duality in both the setting of the module categories and the traditional form which relates the bounded derived category of coherent sheaves on projective space to the stable category of the exterior algebra.
The original reference is [BGG78] and a more modern write up can be found in [EFS03].
6 The preprojective algebra is Koszul (except for Dynkin quivers) There are several sources, the speaker should choose the approach which they prefer:
  1. [BBK02] Prop. 4.2, Cor 4.3 (here: assumption that the quiver has no oriented cycles)
  2. [MOV06] Thm 1.3a, Thm 2.3a, the article is interesting for someone interested in geometric constructions (it uses a quantum version of the MacKay correspondence)
  3. [EE05] is very short
7 Perverse sheaves on the flag variety and category $\mathcal{O}$
Part 1		 The aim of these two talks is to understand the main results of [BGS96], section 3 and the concept of mixedness in section 4 until 4.4. The first talk covers section 3.1-3.4.
Give quick definitions of the bounded derived category with constructible cohomology and the perverse t-structure and the category of perverse sheaves with respect to a stratification. Also the simple objects should be described and that the category of perverse sheaves is an abelian length category (e.g. [HTT08], chapter 8, [Ara01]).
Then explain the Schubert cell decomposition of $G/P$. Prove that the category of perverse sheaves with respect to this stratification has enough projectives and calculate some extension groups, [BGS96], Thm 3.3.1, Thm 3.4.1, Prop 3.4.2.
8 Perverse sheaves on the flag variety and category $\mathcal{O}$
Part 2		 Give a short introduction to category $\mathcal{O}$. Explain [BGS96], section 1.3.
The main result is [BGS96] Thm 3.5.3 which extends Prop 3.5.2. Koszulness of the rings $A_Q, A^Q$. Explain the notion of mixedness and what it means in the considered example (see section 4.1-4.4). A generalization can be found in [Bac99].
9 Koszulness from Frobenius splitting The aim of this talk is to prove Theorem 2.2 of [Bez95]. The paper is mostly self contained, but it would be useful to provide some background and further details on some of the proofs involving homological algebra in categories of coherent sheaves.
Another source is [IM94].
10 dg-categories and derived categories This is an introductory talk intended to provide the background for the talks which follow.
To begin with, dg-categories, dg-functors, and natural transformations of dg-functors should be introduced. The dg-category of dg-modules over a dg-category can then be explained as well as the notions of quasi-isomorphism, acyclicity, and K-projectivity of dg-modules. Finally, the derived category of a dg-category should be introduced and some of its elementary properties should be explained e.g., that the compact objects are given by the perfect complexes. Enhancements of triangulated categories by dg-categories should be briefly mentioned.
Suggested references for this talk are [Kel94], and Section 3 of [KL12] which provides a helpful, if brief, overview.
11 Generators for triangulated categories The first part of the talk should cover classical generators for triangulated categories, following for example [Rou08], as well as giving examples (for instance that for a ring $R$ the category of perfect complexes $\mathsf{D}^\mathrm{perf}(R)$ is clasically generated by $R$ and that the perfect complexes over a dg-category with finitely many objects is clasically generated by the sum of the representable dg-modules).
The second part of the talk should give an introduction to Keller's theorem [Kel94] Theorem 4.3 that an algebraic triangulated category $\mathsf{T}$ with a generator $g$ is equivalent to the derived category of the dg-endomorphism algebra of $g$ and the corresponding statement for essentially small algebraic triangulated categories (a complete proof is not expected, a sketch of the ideas would be more than sufficient). In particular, the special case where the dg-endomorphism algebra of $g$ is formal and so one just gets a tilting object should be covered.
12 Examples of Koszul dualities via dg categories The aim of this talk is to give some examples of Koszul duality from the point of view of dg-categories using the machinery developed in the preceding two talks. The first, and canonical example (which occurs earlier), is the BGG correspondence relating exterior and symmetric algebras: it should be shown that this example can be understood by analysing the endomorphism dga of the trivial module in the stable category of the exterior algebra.
More generally, one can use these techniques to understand Theorem 36 [MVM10], which should be explained.
13 Koszulness from exceptional collections The aim of this talk is to present a proof of Corollary 7.3 of [Bon89], relating the Koszul property of the endomorphism algebra of a strong exceptional collection to the left dual of this collection being strong.
Basically the talk should consist of introducing the necessary concepts (such as strong exceptional collections and left duals) and proving the corollary, as well as providing examples. Useful references are [Bon89], [Bon90], for the method of helices, see also [BP93].
14 Koszul duality of strict polynomial functors This talk should give some details on another instance of Koszul duality, namely its avatar for strict polynomial functors.
A rough outline of Ringel/Koszul duality in this context should be given, with a focus on the relationship between divided, symmetric, and exterior powers which gives rise to it.
The original source is [Tou13] but an excellent overview is given in [Kra12].

References:

  • [Ara01] A. Arabia: Faisceaux pervers sur les variétés algébriques complexes. Correspondance de Springer (d'après Borho-MacPherson)
  • [Bac99] E. Backelin: Koszul duality for parabolic and singular category $\mathcal{O}$, Representation Theory 3 139-152
  • [BBK02] S. Brenner, M.C.R. Butler, A. King: Periodic algebras which are almost Koszul, Algebras and Representation Theory 5 (4) 331-367
  • [Bez95] R. Bezrukavnikov: Koszul Property and Frobenius splitting of Schubert varieties
  • [BGG78] J. Bernstein, I. Gelfand, S. Gelfand: Algebraic vector bundles on Pn and problems of linear algebra, Functional Analysis and Its Applications 12 (3), 66-67
  • [BGS96] A. Beilinson, V. Ginzburg, W. Soergel: Koszul Duality Patterns in Representation Theory, Journal of the American Mathematical Society 9, 473-527
  • [Bon89] A.I. Bondal: Representations of associative algebras and coherent sheaves, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 53 (1) 25-44
  • [Bon90] A.I. Bondal: Helices, representations of quivers and {K}oszul algebras, London Math. Soc. Lecture Note Ser. 148 75-95
  • [BP93] A.I. Bondal, A.E. Polishchuk: Homological properties of associative algebras: the method of helices, Rossi\u\i skaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya 57 (2) 3-50
  • [EE05] P. Etingof, C.H. Eu: Koszulity and the Hilbert series of prepojective algebras
  • [EFS03] D. Eisenbud, G. Fløystad, F.O. Schreyer: Sheaf cohomology and free resolutions over exterior algebras, Transactions of the American Mathematical Society 355 (11) 4397-4426
  • [HTT08] R. Hotta, K. Takeuchi, T. Tanisaki: $D$-modules, perverse sheaves, and representation theory}, Progress in Mathematics 236
  • [IM94] S.P. Inamdar, V.B. Mehta: Frobenius splitting of {S}chubert varieties and linear syzygies, American Journal of Mathematics 116 (6) 1569-1586
  • [Kel94] B. Keller: Deriving DG categories, Annales scientifiques de l'École Normale Supérieure 27 (1), 63-102
  • [KL12] A. Kuznetsov, V. Lunts: Categorical resolutions of irrational singularities
  • [Kra] U. Kraehmer: Notes on Koszul Algebras
  • [Kra12] H. Krause: Koszul, Ringel, and Serre duality for strict polynomial functors, Compositio Mathematica 149 (6), 996- 1018
  • [MOV06] A. Malkin, V. Ostrik, M. Vybornov: Quiver varieties and Lusztig's algebra, Advances in Mathematics 203 (2) 514-536
  • [MVM10] R. Martinez Villa, A. Martsinkovsky: Stable projective homotopy theory of modules, tails, and Koszul duality, Communications in Algebra 3941-3973
  • [PP05] A.E. Polishchuk, L. Positselski: Quadratic Algebras, University Lecture Series 37, American Mathematical Society (2005)
  • [PRS98] I. Peeva, V. Reiner, B. Sturmfels : How To Shell A Monoid, Mathematische Annalen 310 (2) 379-393
  • [Rei03] V. Reiner: Koszul algebras in combinatorics
  • [Rou08] R. Rouquier: Dimensions of triangulated categories, Journal of K-Theory. K-Theory and its Applications in Algebra, Geometry, Analysis & Topology 1 (2) 193-256
  • [Tou13] A. Touzé: Ringel duality and derivatives of non-additive functors, Journal of Pure and Applied Algebra 217 (9) 1642-1673

Please contact the organisers for questions concerning the talks or in case you have further suggestions for topics that might be missing.