Unstable Cohen–Macaulay Algebras

Abstract

Abstract. We characterize Cohen-Macaulay algebras in the category K fg of unstable Noetherian algebras over the Steenrod algebra via the depth of the P * -invariant ideals. This allows us to transfer 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 *

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

“…Moreover, by Lemma 6.2, the left and the right algebra have the same depth. Thus by Theorem 2.1 in [5] the results follows (cf. Corollary 2.2 loc.cit.…”

“…4 If there is no possible confusion we will omit the subscript and just write λ = λ H . 5 If λ H ∈ N 0 exists, then H is called ∆-finite. This is a weaker condition than Noetherianess.…”

Untitled

“…, i λ ∈ N 0 (see Section 1.2 in [3]). 5 In this case, any λ + 1 derivations are linearly dependent over H (Proposition 1.1.7 in [3]). The ∆-length λ H is at most the Krull dimension of H over F (cf.…”