Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups II

Abstract

We continue our study of the homological properties of the purely inseparable extensions $ \mathrm{H} \hookrightarrow\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$ of integrally closed unstable Noetherian integral domains over the Steenrod algebra. It turns out that the projective dimension of $\mathrm {H}$ is a lower bound for the projective dimension of $\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$. Furthermore, $\operatorname{depth} (\mathrm{H}) \geq\operatorname{depth}(\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}})$, where $\operatorname{depth}$ denotes the depth. Moreover, both algebras have the same global dimension. We apply these results to extension $\mathbb F[V_{\bullet}]^G \hookrightarrow\mathbb F[V]^G$ of rings of invariants.