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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 2

### Support varieties and representation type of self-injective algebras

Pages: 197 – 215

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n2.a13

#### Authors

#### Abstract

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in “Support varieties and representation type of small quantum groups,” *Internat. Math. Res. Notices* 2010, no. 7, 1346–1362. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.

#### Keywords

support variety; Hochschild cohomology; complexity; representation type; wild; tame; block; self-injective algebra; Hecke algebra; reduced universal enveloping algebra; small half-quantum group; Nichols algebra; quantum symmetric algebra; Hopf algebra

#### 2010 Mathematics Subject Classification

16D50, 16E40, 16G10, 16G60, 16L60, 16T05, 17B35, 17B37, 20C08

Published 25 January 2012