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Support Varieties for Finite Dimensional Algebras

Sandøy, Mads Hustad
Master thesis
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http://hdl.handle.net/11250/2433755
Utgivelsesdato
2016
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  • Institutt for matematiske fag [1390]
Sammendrag
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.
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