On the global dimension of fixed rings

Lorenz, Martin

doi:10.1090/S0002-9939-1989-0972235-6

On the global dimension of fixed rings
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Proc. Amer. Math. Soc. 106 (1989), 923-932 Request permission

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 $\left | G \right |$ 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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