This article is cited in 1 paper

Cotorsion Dimensions and Hopf Algebra Actions

Xiuli Chen, Haiyan Zhu, Fang Li

Zhejiang University

Abstract: Let $H$ be a finite-dimensional Hopf algebra over a field $k$, and let $A$ be an $H$-module algebra. In this paper, we discuss the cotorsion dimension of the smash product $A\mathbin{\#}H$. We prove that
$$ \mathrm{l.cot.D}(A\mathbin{\#}H) \leq \mathrm{l.cot.D}(A) + \mathrm{r.D}(H), $$
which generalizes the result of group rings. Moreover, we give some sufficient conditions for which
$$ \mathrm{l.cot.D}(A\mathbin{\#}H) =\mathrm{l.cot.D}(A). $$
As applications, we study the invariants of IF properties and Gorenstein global dimensions.

Keywords: Hopf algebra, cotorsion dimension, smash product, projective dimension, Gorenstein dimension.

UDC: 512.5

Received: 08.09.2010
Revised: 06.01.2011

DOI: 10.4213/mzm8985

 English version: Mathematical Notes, 2013, 93:4, 616–623

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