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Homological properties of the relative Frobenius morphism

Peter M. McDonald

Abstract

This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map $\varphi$ of finite flat dimension, the results relate homological properties of the relative Frobenius of $\varphi$ to those of the fibers of $\varphi$. The focus is on the complete intersection property and the Gorenstein property.

Homological properties of the relative Frobenius morphism

Abstract

This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map of finite flat dimension, the results relate homological properties of the relative Frobenius of to those of the fibers of . The focus is on the complete intersection property and the Gorenstein property.
Paper Structure (3 sections, 14 theorems, 50 equations)

This paper contains 3 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Simplicial rings
  3. Homological dimension

Key Result

Theorem 1.1

Let $\varphi\colon R\to S$ be a map of $F$-finite local rings of positive characteristic and let $k_R$ be the residue field of $R$. If $\varphi$ has finite flat dimension, then where $\bar{S'}:=S\otimes_R^\mathbb{L} k'$ for $k'$ any finite, purely inseparable extension of $k_R$.

Theorems & Definitions (37)

  • Theorem 1.1: \ref{['main']}
  • Corollary 1.2: \ref{['flatfrob']}, \ref{['CI']}
  • Remark 1
  • Remark 2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 27 more