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Bounds on Gorenstein Dimensions and Exceptional Complete Intersection Maps

Hossein Faridian

Abstract

We prove that if $f:R \rightarrow S$ is a local homomorphism of noetherian local rings of finite flat dimension and $M$ is a non-zero finitely generated $S$-module whose Gorenstein flat dimension over $R$ is bounded by the difference of the embedding dimensions of $R$ and $S$, then $M$ is a totally reflexive $S$-module and $f$ is an exceptional complete intersection map. This is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat dimension. We also prove two analogues involving Gorenstein injective dimension.

Bounds on Gorenstein Dimensions and Exceptional Complete Intersection Maps

Abstract

We prove that if is a local homomorphism of noetherian local rings of finite flat dimension and is a non-zero finitely generated -module whose Gorenstein flat dimension over is bounded by the difference of the embedding dimensions of and , then is a totally reflexive -module and is an exceptional complete intersection map. This is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat dimension. We also prove two analogues involving Gorenstein injective dimension.
Paper Structure (3 sections, 14 theorems, 39 equations)

This paper contains 3 sections, 14 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions and Observations
  3. Main Results

Key Result

Theorem A

Let $f:(R,\mathfrak{m}) \rightarrow (S,\mathfrak{n})$ be a local homomorphism of noetherian local rings with $\operatorname{fd}_{R}(S)< \infty$, and $M$ a non-zero finitely generated $S$-module with $\operatorname{Gfd}_{R}(M)\leq \operatorname{edim}(R)- \operatorname{edim}(S)$. Then the following as

Theorems & Definitions (31)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 21 more