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On the global dimension of fixed rings


Author: Martin Lorenz
Journal: Proc. Amer. Math. Soc. 106 (1989), 923-932
MSC: Primary 16A72; Secondary 16A60
DOI: https://doi.org/10.1090/S0002-9939-1989-0972235-6
MathSciNet review: 972235
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Abstract: Let $ G$ be a finite group acting on a $ k$-algebra $ R$, and let $ S = {R^G}$ denote the fixed subring of $ R$. Our main interest is in the case where $ \left\vert G \right\vert$ is not invertible in $ R$. Instead, we assume that $ R$ is flat over $ S$ and that the trivial $ kG$-module $ k$ has a periodic projective resolution. (For a field $ k$ of characteristic $ p$, the latter condition holds precisely if the Sylow $ p$-subgroups of $ G$ are cyclic or generalized quaternion.) We use a periodicity result for Extgroups, established here in a more general setting that is independent of group actions, to estimate the global dimension of $ S$ in this case.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0972235-6
Article copyright: © Copyright 1989 American Mathematical Society

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