Support Varieties for Finite Dimensional Algebras

Abstract

We give a presentation of the theory of support varieties for finite dimensional algebras A using the Hochschild cohomology ring. Our presentation is especially focused on the finite generation hypotheses an algebra must satisfy to have an adequate theory of support varieties, as well as the consequences of these hypotheses for the complexity of the modules of such an algebra. To demonstrate that certain aspects of the corresponding theory for group algebras can be recovered, we show that by assuming the finite generation hypotheses we can prove that every closed homogeneous variety is the variety of some module. Following this, we investigate whether a result of Purin in [29] concerning the complexity of trivial extensions of hereditary algebras can be generalized: Firstly, using a result of Benson and some well-known results concerning radical square zero algebras, we give an example that shows that an algebra A can be of finite representation type while its trivial extension T(A) has infinite complexity, hence showing that a straightforward generalization of Purin's result is not available. After this, we derive a weak bound on the length of the terms of the minimal T(A)-projective resolution of an A-module considered as a T(A)-module. Following this, we utilize the proof of a result of Guo et al. in [22] in giving a description of the syzygies and the minimal T(A)- projective resolution of an A-module considered as a T(A)-module. Using this and a result by Dichi and Sangare in [16], we are able to show that if a selfinjective algebra A satisfies the finite generation hypotheses (Fg) then the complexity of T(A) is exactly one greater than that of A.