Unstable Cohen–Macaulay Algebras

Abstract

We characterize Cohen-Macaulay algebrasin the category Kfg of unstable Noetherian algebras over the Steenrod algebra via the depth of the P ∗ - invariant ideals . Thisallowsusto trans fer the Cohen-Macaulay property to suitable subalgebras. We apply this to rings of invariants of finite groups and to the P ∗ -inseparable closure. For over 50 years the Steenrod algebra P ∗ and algebras over it have proven to be decisive tools in algebraic topology. Applications occur in cohomology theory, invariant theory of finite groups, and most recently in algebraic geometry. This makes it desirable to have a completely algebraic (and not topological) approach to the subject. In this paper we follow the path begun in (12), (13), (14) and (16) of building a complete P ∗ -invariant commutative algebra. We find a P ∗ -invariant version of the classical characterization of Cohen-Macaulay algebras. This allows us to show that Cohen-Macaulayness is inherited by subalgebras provided that the P ∗ -invariant prime ideal spectra are in bijective correspondence and the extension is integral. This applies in particular to an unstable algebra and its P ∗ -inseparable closure. It also applies to rings of polynomial invariants of finite groups.