Cohen-Macaulay invariant subalgebras of Hopf dense Galois extensions
Cohen-Macaulay invariant subalgebras of Hopf dense Galois extensions
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DOI：
10.1090/conm/728/14658
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发表时间：
2017-11
期刊：
影响因子：
--
通讯作者：
Ji-wei He;Yinhuo Zhang
中科院分区：
文献类型：
--
作者：
Ji-wei He;Yinhuo Zhang
Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left $H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from $R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module, is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and $H=\operatorname{\bf k} G$ for some finite subgroup $G\subseteq Aut(R)$, then all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of $R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.
