A generalized Koszul theory and its application

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Kerry Anne Holmes
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1 University of California, Riverside June 26, 2013

2 Motivation There are many structures with natural gradings, where the degree 0 components are not semisimple. For example:

3 Motivation There are many structures with natural gradings, where the degree 0 components are not semisimple. For example: Tensor algebras A = A 0 A 1 (A 1 A0 A 1 )... with non-semisimple A 0.

4 Motivation There are many structures with natural gradings, where the degree 0 components are not semisimple. For example: Tensor algebras A = A 0 A 1 (A 1 A0 A 1 )... with non-semisimple A 0. Γ = Ext A (M, M) of a module M.

5 Motivation There are many structures with natural gradings, where the degree 0 components are not semisimple. For example: Tensor algebras A = A 0 A 1 (A 1 A0 A 1 )... with non-semisimple A 0. Γ = Ext A (M, M) of a module M. A polynomial ring A = R[Y ] with R being a finite-dimensional algebra.

6 Motivation There are many structures with natural gradings, where the degree 0 components are not semisimple. For example: Tensor algebras A = A 0 A 1 (A 1 A0 A 1 )... with non-semisimple A 0. Γ = Ext A (M, M) of a module M. A polynomial ring A = R[Y ] with R being a finite-dimensional algebra. Some locally finite k-linear categories whose objects have non-simisimple endomorphism algebras.

7 Motivation The classical Koszul theory does NOT apply to the above examples. Thus we develop a generalized Koszul theory, which: preserves many classical results such as the Koszul duality;

8 Motivation The classical Koszul theory does NOT apply to the above examples. Thus we develop a generalized Koszul theory, which: preserves many classical results such as the Koszul duality; can apply to the above examples;

9 Motivation The classical Koszul theory does NOT apply to the above examples. Thus we develop a generalized Koszul theory, which: preserves many classical results such as the Koszul duality; can apply to the above examples; and has a close relation to the classical theory.

10 Motivation The classical Koszul theory does NOT apply to the above examples. Thus we develop a generalized Koszul theory, which: preserves many classical results such as the Koszul duality; can apply to the above examples; and has a close relation to the classical theory. There are already some generalized theories by Woodcock, Madsen, Green, Reiten and Solberg.

11 Motivation In the classical Koszul theory, the semisimple condition guarantees the splitting of short exact sequences 0 P Q R 0 where P and Q are projective A 0 -modules. This condition is unnecessarily strong. A suitable condition is that A 0 has finitistic dimension 0. Remark: when A 0 has finitistic dimension 0, the T -Koszul theory coincides with the generalized theory we will describe later.

12 Setup Let A = i 0 A i be a graded k-algebra such that: A is generated in degrees 0 and 1, i.e., A i A j = A i+j

13 Setup Let A = i 0 A i be a graded k-algebra such that: A is generated in degrees 0 and 1, i.e., A i A j = A i+j A is locally finite, i.e, dim k A i <, i, j 0;

14 Setup Let A = i 0 A i be a graded k-algebra such that: A is generated in degrees 0 and 1, i.e., A i A j = A i+j A is locally finite, i.e, dim k A i <, i, j 0; Replace the condition gldim A 0 = 0 by fidim A 0 = 0 and fidim A o 0 = 0.

15 Setup Let A = i 0 A i be a graded k-algebra such that: A is generated in degrees 0 and 1, i.e., A i A j = A i+j A is locally finite, i.e, dim k A i <, i, j 0; Replace the condition gldim A 0 = 0 by fidim A 0 = 0 and fidim A o 0 = 0. Remark: Semisimple, self-injective, or finite dimensional local algebras satisfy the weaker condition.

16 Generalized Koszul modules All graded A-modules M will be locally finite and bounded below, i.e., dim k M i < for i Z and M i = 0 when i << 0. We say M is generated in degree s if M = A M s.

17 Generalized Koszul modules All graded A-modules M will be locally finite and bounded below, i.e., dim k M i < for i Z and M i = 0 when i << 0. We say M is generated in degree s if M = A M s. A graded A-module M is a generalized Koszul module if M has a graded projective resolution... P i... P 1 P 0 M 0 such that P i is generated in degree i.

18 Generalized Koszul modules All graded A-modules M will be locally finite and bounded below, i.e., dim k M i < for i Z and M i = 0 when i << 0. We say M is generated in degree s if M = A M s. A graded A-module M is a generalized Koszul module if M has a graded projective resolution... P i... P 1 P 0 M 0 such that P i is generated in degree i. A is a generalized Koszul algebra if A 0 = A/ i 1 A i is a Koszul module.

19 Generalized Koszul modules All graded A-modules M will be locally finite and bounded below, i.e., dim k M i < for i Z and M i = 0 when i << 0. We say M is generated in degree s if M = A M s. A graded A-module M is a generalized Koszul module if M has a graded projective resolution... P i... P 1 P 0 M 0 such that P i is generated in degree i. A is a generalized Koszul algebra if A 0 = A/ i 1 A i is a Koszul module. M is quasi-koszul if Ext A (M, A 0) is generated in degree 0 as an Ext A (A 0, A 0 )-module. A is a quasi-koszul algebra if A 0 is a quasi-koszul module.

20 An example Let A be the following algebra with relations αδ = ρα = α, δ 2 = ρ 2 = 0. Put all endomorphisms in degree 0 and all non-endomorphisms in degree 1. δ x α y ρ α Then: P x = x 0 x 0 y 1 P y = y 0. y y 0 1 We check that it is a generalized Koszul algebra.

21 Generalized results Theorem: A is a generalized Koszul algebra if and only if it is a projective A 0 -module and quasi-koszul.

22 Generalized results Theorem: A is a generalized Koszul algebra if and only if it is a projective A 0 -module and quasi-koszul. Theorem: A is quasi-koszul if and only if so is the opposite algebra A o.

23 Generalized results Theorem: A is a generalized Koszul algebra if and only if it is a projective A 0 -module and quasi-koszul. Theorem: A is quasi-koszul if and only if so is the opposite algebra A o. Theorem: If A is a generalized Koszul algebra, then it is a quadratic algebra.

24 Generalized results Theorem: A is a generalized Koszul algebra if and only if it is a projective A 0 -module and quasi-koszul. Theorem: A is quasi-koszul if and only if so is the opposite algebra A o. Theorem: If A is a generalized Koszul algebra, then it is a quadratic algebra. Theorem: If A is a generalized Koszul algebra, then E A = Ext A (, A 0) gives a duality between the category of generalized Koszul A-modules and the category of generalized Koszul Γ-modules. That is, if M is a generalized Koszul A-module, then E A (M) is a generalized Koszul Γ-module, and E Γ E A M = Ext Γ (E AM, Γ 0 ) = M as graded A-modules. In particular, Γ is a generalized Koszul algebra.

25 A relation to the classical theory Let r = rad A 0 and R = ArA be the two-sided ideal generated by r.

26 A relation to the classical theory Let r = rad A 0 and R = ArA be the two-sided ideal generated by r. Define Ā = A/R. For a graded A-module M, define M = M/RM.

27 A relation to the classical theory Let r = rad A 0 and R = ArA be the two-sided ideal generated by r. Define Ā = A/R. For a graded A-module M, define M = M/RM. Theorem: A is a generalized Koszul algebra if f it is a projective A 0 -module and Ā is a classical Koszul algebra. A graded A-module M is generalized Koszul if f it is a projective A 0 -module and M is classical Koszul.

28 Skew group algebras Let A = i 0 A i be as before, and let G be a finite group whose elements act on A as grade-preserving algebra automorphisms. The skew group algebra AG = A k kg as vector spaces, and its multiplication is determined by (ag) (bh) = ag(b)gh for a, b A and g, h G, where g(b) is the image of b under the action of g.

29 Skew group algebras Let A = i 0 A i be as before, and let G be a finite group whose elements act on A as grade-preserving algebra automorphisms. The skew group algebra AG = A k kg as vector spaces, and its multiplication is determined by (ag) (bh) = ag(b)gh for a, b A and g, h G, where g(b) is the image of b under the action of g. Many people have studied representations of skew group algebras. Most of them assumed that kg is semisimple.

30 Koszul properties of skew group algebras Martinez has shown that if kg is semisimple and A is a classical Koszul algebra, then AG is Koszul as well. We drop the semisimple condition and obtain:

31 Koszul properties of skew group algebras Martinez has shown that if kg is semisimple and A is a classical Koszul algebra, then AG is Koszul as well. We drop the semisimple condition and obtain: Theorem: Let M be a graded AG-module and let Λ = Ext A (M, M). Then:

32 Koszul properties of skew group algebras Martinez has shown that if kg is semisimple and A is a classical Koszul algebra, then AG is Koszul as well. We drop the semisimple condition and obtain: Theorem: Let M be a graded AG-module and let Λ = Ext A (M, M). Then: The AG-module M k kg is a generalized Koszul AG-module if and only if as an A-module M is generalized Koszul. In particular, AG is a generalized Koszul algebra over (AG) 0 = A 0 k kg if and only if A is a generalized Koszul algebra.

33 Koszul properties of skew group algebras Martinez has shown that if kg is semisimple and A is a classical Koszul algebra, then AG is Koszul as well. We drop the semisimple condition and obtain: Theorem: Let M be a graded AG-module and let Λ = Ext A (M, M). Then: The AG-module M k kg is a generalized Koszul AG-module if and only if as an A-module M is generalized Koszul. In particular, AG is a generalized Koszul algebra over (AG) 0 = A 0 k kg if and only if A is a generalized Koszul algebra. If as an A-module M is generalized Koszul, then Ext AG (M k kg, M k kg) is isomorphic to the skew group algebra ΛG as graded algebras.

34 Extension algebras of standard modules Quasi-hereditary algebras are introduced by CPS. Standardly stratified algebras generalize this notion without assuming that the endomorphism algebra of each standard module is k.

35 Extension algebras of standard modules Quasi-hereditary algebras are introduced by CPS. Standardly stratified algebras generalize this notion without assuming that the endomorphism algebra of each standard module is k. Let A be standardly stratified and be the direct sum of all standard modules. It is interesting to study Γ = Ext A (, ). Note that A need not be graded, but Γ is naturally graded.

36 Extension algebras of standard modules Quasi-hereditary algebras are introduced by CPS. Standardly stratified algebras generalize this notion without assuming that the endomorphism algebra of each standard module is k. Let A be standardly stratified and be the direct sum of all standard modules. It is interesting to study Γ = Ext A (, ). Note that A need not be graded, but Γ is naturally graded. As an analogue of Koszul modules defined by linearly graded resolutions, we define linearly filtered modules which have linearly filtered resolutions.

37 Extension algebras of standard modules Quasi-hereditary algebras are introduced by CPS. Standardly stratified algebras generalize this notion without assuming that the endomorphism algebra of each standard module is k. Let A be standardly stratified and be the direct sum of all standard modules. It is interesting to study Γ = Ext A (, ). Note that A need not be graded, but Γ is naturally graded. As an analogue of Koszul modules defined by linearly graded resolutions, we define linearly filtered modules which have linearly filtered resolutions. Theorem: Suppose that all summands of are linearly filtered and End A ( ) = as A-modules. If M is linearly filtered, then Ext A (M, ) is a Koszul Γ-module. In particular, Γ is a generalized Koszul algebra.

38 Triangulate equivalences BGS has shown D b g (A) is triangulated equivalent to D b g (Γ) under some conditions.

39 Triangulate equivalences BGS has shown D b g (A) is triangulated equivalent to D b g (Γ) under some conditions. Martinez, Marzochuk and etc give further investigations using categorification and Rickard-Morita theory.

40 Triangulate equivalences BGS has shown D b g (A) is triangulated equivalent to D b g (Γ) under some conditions. Martinez, Marzochuk and etc give further investigations using categorification and Rickard-Morita theory. Can one apply their methods and results to this generalized framework and obtain triangulated equivalences between some full subcategories of D b g (A) and D b g (Γ)? A possible candidate is the homotopy category of graded A-modules which are all projective A 0 -modules.

A generalized Koszul theory and its applications in representation theory

A generalized Koszul theory and its applications in representation theory A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Liping Li IN PARTIAL FULFILLMENT