| dc.description.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. | |
