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Local cohomology tables of sequentially almost Cohen-Macaulay modules

Cheng Meng

Abstract

Let $R$ be a polynomial ring over a field. We introduce the concept of sequentially almost Cohen-Macaulay modules and describe the extremal rays of the cone of local cohomology tables of finitely generated graded $R$-modules which are sequentially almost Cohen-Macaulay, and describe some cases when the local cohomology table of a module of dimension 3 has a nontrivial decomposition.

Local cohomology tables of sequentially almost Cohen-Macaulay modules

Abstract

Let be a polynomial ring over a field. We introduce the concept of sequentially almost Cohen-Macaulay modules and describe the extremal rays of the cone of local cohomology tables of finitely generated graded -modules which are sequentially almost Cohen-Macaulay, and describe some cases when the local cohomology table of a module of dimension 3 has a nontrivial decomposition.
Paper Structure (9 sections, 54 theorems, 35 equations, 1 table)

This paper contains 9 sections, 54 theorems, 35 equations, 1 table.

Table of Contents

  1. introduction
  2. notations
  3. Notations on the tables
  4. Convex cones in a $\mathbb{Q}$-vector space
  5. Some Lemmas on tables in series form
  6. The dimension filtration
  7. Projective dimension 1 case
  8. The $\Gamma$ functor
  9. Decomposition in dimension 3

Key Result

Theorem 1.1

Let $R = k[x_1,\ldots,x_n]$ be a standard graded polynomial ring. The extremal rays of the cone generated by Betti tables of finitely generated graded $R$-modules are given by the modules with a pure resolution, and every Betti table in the cone decomposes in the following way. For every finitely ge

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: de2021decomposition, Theorem 4.6
  • Theorem A: Theorem \ref{['4.19']}
  • Theorem B: Theorem \ref{['theorem-all local cohomology tables']}
  • Theorem C: Theorem \ref{['6.8']}
  • Theorem D: Theorem \ref{['6.14']}
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 92 more