Serre depth and local cohomology

TL;DR

This paper introduces Serre depth, a new homological invariant that generalizes depth and stratifies Serre's conditions, providing tools to analyze modules over local rings and graded algebras with applications to initial ideals and powers.

Contribution

The paper defines Serre depth, explores its properties, invariance, and relations to Serre's conditions, extending known results from polynomial rings to more general rings and modules.

Findings

Serre depth is invariant under completion.

Modules over Gorenstein images satisfy (S_r) iff Serre depth equals dimension.

Serre depths of monomial ideals stabilize for large powers.

Abstract

We introduce a fundamental homological invariant, called Serre depth, which stratifies Serre's conditions in the same way that depth stratifies the Cohen-Macaulay property. We study the Serre depths of modules over arbitrary Noetherian local rings and over standard graded algebras over a field, extending the polynomial ring case due to Muta and Terai. Under mild hypotheses, we show that the $r$-th Serre depth of a finitely generated module $M$ measures the deviation of $M$ from satisfying Serre's condition $(S_r)$. The main results of the paper can be summarized as follows: (i) We establish the basic properties of Serre depth and prove that it is invariant under completion. (ii) If the base ring $R$ is a homomorphic image of a Gorenstein ring, we show that a finitely generated $R$-module $M$ is equidimensional and satisfies $(S_r)$ if and only if its $r$-th Serre depth equals its Krull dimension. Analogous statements are obtained for schemes. (iii) For a homogeneous ideal in a standard graded polynomial ring over a field, we compare its Serre depths with those of its initial ideal. (iv) We characterize the Serre depths of a monomial ideal in terms of its skeletons and prove that the Serre depths of sufficiently large powers of a monomial ideal stabilize; the proof uses Presburger arithmetic.