Geodesic
Cotorsion Dimensions and Hopf Algebra Actions
Matematičeskie zametki, Tome 93 (2013) no. 4, pp. 614-623

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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, smash product, projective dimension, Gorenstein dimension.
Mots-clés : cotorsion dimension 
@article{MZM_2013_93_4_a11,
     author = {Xiuli Chen and Haiyan Zhu and Fang Li},
     title = {Cotorsion {Dimensions} and {Hopf} {Algebra} {Actions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {614--623},
     publisher = {mathdoc},
     volume = {93},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_4_a11/}
}
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Xiuli Chen; Haiyan Zhu; Fang Li. Cotorsion Dimensions and Hopf Algebra Actions. Matematičeskie zametki, Tome 93 (2013) no. 4, pp. 614-623. http://geodesic.mathdoc.fr/item/MZM_2013_93_4_a11/