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$F$-Invariants of Stanley-Reisner Rings

Wágner Badilla-Céspedes

Abstract

In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.

$F$-Invariants of Stanley-Reisner Rings

Abstract

In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases.

Paper Structure

This paper contains 15 sections, 37 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Stanley-Reisner Rings
  3. $F$-Thresholds
  4. Definition and First Properties
  5. $F$-Thresholds in Stanley-Reisner Rings
  6. The Ideal $J_e$
  7. Cartier Contraction.
  8. The Ideal $\mathfrak{q}_e$ in Stanley-Reisner Rings
  9. Cartier Threshold of $\mathfrak{a}$ with Respect to $J$
  10. Definition and First Properties
  11. Relation Between $c^J(\mathfrak{a})$ and ${\operatorname{ct}} _J(\mathfrak{a})$
  12. Cartier Thresholds in Stanley-Reisner Rings
  13. $a$-Invariants and Regularity
  14. Definitions and Property
  15. Regularity in Stanley-Reisner Rings

Key Result

Theorem A

Let $\mathfrak{a}, \;J$ be two ideals in a Stanley-Reisner ring $R$, such that $\mathfrak{a} \subseteq J$, and $J$ is a radical ideal. Then, the Cartier threshold of $\mathfrak{a}$ with respect to $J$ is a rational number.

Theorems & Definitions (86)

  • Theorem A: see Theorem \ref{['pro1series']} and Corollary \ref{['mainresult']}
  • Theorem B: see Theorem \ref{['pro5']}
  • Theorem C: see Theorem \ref{['thmregularity']}
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: DSNBP
  • ...and 76 more