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Nilpotent Elements in Hochschild Cohomology

  • Karin ErdmannEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)

Abstract

We study the algebra \(A=K\langle x, y\rangle /(x^2, y^2, (xy)^k+q(yx)^k)\) over the field K where \(k\ge 1\) and where \(0\ne q \in K\). We determine a minimal projective bimodule resolution of A. In the case when q is not a root of unity, we compute its Hochschild cohomology. In particular, we show that for \(n\ge 3\), the nth part \(HH^n(A)\) has dimension \(k-1\) if char(K) does not divide k. We also show that every element in \(HH^n(A)\) for \(n\ge 1\) is nilpotent. This is motivated by the problem of understanding why the finite generation condition (Fg) fails, which is needed to ensure the existence of support varieties.

Keywords

Algebra of dihedral type Socle deformation Bimodule resolution Hochschild cohomology Nilpotent elements in Hochschild cohomology 

MR Subject Classification

16E40 16S80 20C20 16G20 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Mathematical Institute, University of OxfordOxfordUK

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