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On a relative dependency formula

Shashi Ranjan Sinha

Abstract

Celikbas, Liang and Sadeghi established a one-sided inequality for the relative version of Jorgensen's dependency formula and questioned whether it would be an equality. In this paper, we show that the inequality can be indeed strict, and prove a relative dependency formula. Along the way, we obtain some bounds on s(M,N), a notion related to the vanishing of relative homology of finitely generated modules M and N over a local ring R, under specific assumptions.

On a relative dependency formula

Abstract

Celikbas, Liang and Sadeghi established a one-sided inequality for the relative version of Jorgensen's dependency formula and questioned whether it would be an equality. In this paper, we show that the inequality can be indeed strict, and prove a relative dependency formula. Along the way, we obtain some bounds on s(M,N), a notion related to the vanishing of relative homology of finitely generated modules M and N over a local ring R, under specific assumptions.

Paper Structure

This paper contains 4 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tate and Relative homology
  4. Proof of the main results

Key Result

Proposition 1

CLS Let $M$ and $N$ be $R\hbox{-}$modules such that $M$ has finite Gorenstein dimension. Then $s(M,N) \leq t(M,N)$.

Theorems & Definitions (19)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 1
  • Proposition 9
  • ...and 9 more