Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

Abstract

We consider a one point extension algebra B of a quiver algebra Aq over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.

Keywords

Hochschild cohomology / quantum-like relation / one point extension

References

[1] Bergh P A, Erdmann K. Homology and cohomology of quantum complete intersections. Algebra Number Theory, 2008, 2: 501–522

[2] Bergh P A, Oppermann S. Cohomology of twisted tensor products. J Algebra, 2008, 320: 3327–3338

[3] Erdmann K, Holloway M, Snashall N, Solberg Ø,Taillefer R. Support varieties for selfinjective algebras. K-Theory, 2004, 33: 67–87

[4] Green E L, Marcos E N, Snashall N. The Hochschild cohomology ring of a one point extension. Comm Algebra, 2003, 31: 2019–2037

[5] Obara D. Hochschild cohomology of quiver algebras defined by two cycles and a quantum-like relation. Comm Algebra, 2012, 40: 1724–1761

[6] Obara D. Hochschild cohomology of quiver algebras defined by two cycles and a quantum-like relation, II. Comm Algebra, 2015, 43: 3404–3446

[7] Snashall N. Support varieties and the Hochschild cohomology ring modulo nilpotence. In: Proceedings of the 41st Symposium on Ring Theory and Representation Theory, Shizuoka, Japan, Sept 5–7, 2008; Fujita H, ed. Symp Ring Theory Represent Theory Organ Comm, Tsukuba, 2009, 68–82

[8] Snashall N, Solberg Ø.Support varieties and Hochschild cohomology rings. Proc Lond Math Soc, 2004, 88: 705–732

[9] Solberg Ø.Support varieties for modules and complexes. In: Trends in Representation Theory of Algebras and Related Topics. Contemp Math, Vol 406. Providence: Amer Math Soc, 2006, 239–270

[10] Xu F. Hochschild and ordinary cohomology rings of small categories. Adv Math, 2008, 219: 1872–1893

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Higher Education Press and Springer-Verlag Berlin Heidelberg