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Serre depth and local cohomology

Antonino Ficarra

Abstract

We introduce a fundamental homological invariant, called \emph{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.

Serre depth and local cohomology

Abstract

We introduce a fundamental homological invariant, called \emph{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 -th Serre depth of a finitely generated module measures the deviation of from satisfying Serre's condition . 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 is a homomorphic image of a Gorenstein ring, we show that a finitely generated -module is equidimensional and satisfies if and only if its -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.
Paper Structure (5 sections, 35 theorems, 153 equations)

This paper contains 5 sections, 35 theorems, 153 equations.

Table of Contents

  1. Basic properties of the Serre depth
  2. Invariance under completion
  3. Serre's conditions
  4. Gröbner degenerations
  5. Eventual behaviour of the Serre depth

Key Result

Theorem A

Let $(R,\mathfrak{m})$ be a Noetherian local ring or a standard graded $K$-algebra. Let $M$ be a finitely generated $R$-module, which we assume is homogeneous if $R$ is a $K$-algebra. Consider the following statements. Then (b)$\Rightarrow$(a), and the converse implication holds if $R$ is a homomorphic image of a Gorenstein ring. Furthermore, in (a) the condition that $M$ is equidimensional can b

Theorems & Definitions (67)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Conjecture G
  • Definition 1.1
  • Theorem 1.3
  • Proposition 1.4
  • ...and 57 more