When is Ext*(M,N) finitely generated as a Ext*(M,M) module?

Question

Let A be a finite dimensional algebra over a field k and M,N a finitely generated A-module. Im searching for examples where the module $ Ext^{o} (M,N) $ is a finitely generated $ Ext^{o}(M,M) $ -module(via yonedaproduct) for every finitely generated A-modules M,N.

This is the case for example when A is a cocommutative Hopfalgebra by a result of Friedlander and Suslin and its a conjecture that this also holds for a general Hopfalgebra. see for example: http://www.math.uiuc.edu/K-theory/0085/paper.pdf

Answer 1

This is true whenever condition (Fg) from the theory of support varieties holds.

A finite dimensional algebra $A$ with Jacobson radical $J$ is said to satisfy condition (Fg) if the Hochschild cohomology ring ${\rm{HH}}^{\ast}(A)$ of $A$ is Noetherian and ${\rm{Ext}}^{\ast}_A(A/J,A/J)$ is finitely generated as an ${\rm{HH}}^{\ast}(A)$-module. The ring ${\rm{HH}}^{\ast}(A)$ acts on ${\rm{Ext}}^{\ast}_A(M,N)$ through the graded center of ${\rm{Ext}}^{\ast}_A(M,M)$ as explained in section 3 of [Solberg, Support varieties for modules and complexes]. Condition (Fg) implies that ${\rm{Ext}}^{\ast}_A(M,N)$ is finitely generated as an ${\rm{HH}}^{\ast}(A)$-module [Propositions 5.5 and 5.7 of that paper]. This in turn implies that ${\rm{Ext}}^{\ast}_A (M,N)$ is finitely generated as an ${\rm{Ext}}^{\ast}_A(M,M)$-module.

See the introduction to arXiv:1003.2867 for classes of algebras that satisfy condition (Fg). In particular (Fg) holds when $A$ is a representation finite self-injective algebra over an algebraically closed field.