Support Varieties for Finite Dimensional Algebras
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We give a presentation of the theory of support varieties for finite dimensionalalgebras A using the Hochschild cohomology ring. Our presentation is especiallyfocused on the finite generation hypotheses an algebra must satisfy to have an adequatetheory of support varieties, as well as the consequences of these hypothesesfor the complexity of the modules of such an algebra. To demonstrate that certainaspects of the corresponding theory for group algebras can be recovered, we showthat by assuming the finite generation hypotheses we can prove that every closedhomogeneous variety is the variety of some module. Following this, we investigatewhether a result of Purin in  concerning the complexity of trivial extensions ofhereditary algebras can be generalized: Firstly, using a result of Benson and somewell-known results concerning radical square zero algebras, we give an examplethat shows that an algebra A can be of finite representation type while its trivialextension T(A) has infinite complexity, hence showing that a straightforward generalizationof Purin's result is not available. After this, we derive a weak bound onthe length of the terms of the minimal T(A)-projective resolution of an A-moduleconsidered as a T(A)-module. Following this, we utilize the proof of a result ofGuo et al. in  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 anda result by Dichi and Sangare in , we are able to show that if a selfinjectivealgebra A satisfies the finite generation hypotheses (Fg) then the complexity ofT(A) is exactly one greater than that of A.