Galois bimodules and integrality of PI comodule algebras over invariants
Galois bimodules and integrality of PI comodule algebras over invariants
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Galois 双模和 PI 余模代数在不变量上的完整性
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发表时间：
2013
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通讯作者：
P. Etingof
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作者：
P. Etingof
设A是代数闭域k上有限维Hopf代数K的余代数，A^K是其不变量子代数.设Z是A中的中心子代数，A是具有商域Q的整环。设Qotimes_Z A是Q上的中心单代数，A是一个无挠生成Z-模且Z在Q中整闭，或者A是一个有限投射Z-模.然后证明了A和Z是中心不变量Zcap A^K的子环上的积分环。更一般地，我们证明了在相同的假设下，如果Z是一个具有商环Q的约化代数，并且Qotimes_Z A是一个具有中心Q的半单代数，则这一陈述是有效的。特别地，该陈述对于K在素PI代数A上的余作用成立，该素PI代数A的中心Z是k上的整闭的π生成域。这推广了S.当A是可交换的时，Skryabin。为了证明，我们发展了中心上有限的半单代数上的伽罗瓦双模的理论。
Let A be a comodule algebra for a finite dimensional Hopf algebra K over an algebraically closed field k, and let A^K be the subalgebra of invariants. Let Z be a central subalgebra in A, which is a domain with quotient field Q. Assume that Qotimes_Z A is a central simple algebra over Q, and either A is a finitely generated torsion-free Z-module and Z is integrally closed in Q, or A is a finite projective Z-module. Then we show that A and Z are integral over the subring of central invariants Zcap A^K. More generally, we show that this statement is valid under the same assumptions if Z is a reduced algebra with quotient ring Q, and Qotimes_Z A is a semisimple algebra with center Q. In particular, the statement holds for a coaction of K on a prime PI algebra A whose center Z is an integrally closed finitely generated domain over k. This generalizes the results of S. Skryabin in the case when A is commutative. For the proof, we develop a theory of Galois bimodules over semisimple algebras finite over the center.