Dimension and depth inequalities over complete intersections

TL;DR

This paper investigates inequalities involving dimensions and depths of modules over complete intersection rings, extending Hochster's theta invariant to detect equality cases and exploring related depth and complexity theories.

Contribution

It introduces an extension of Hochster's theta invariant to identify when the dimension inequality becomes an equality, and develops parallel theories with depth and complexity.

Findings

Extension of Hochster's theta invariant developed

Nonvanishing of the invariant detects equality cases

Parallel theory with depth and complexity

Abstract

For a pair of finitely generated modules $M$ and $N$ over a codimension $c$ complete intersection ring $R$ with $\ell(M\otimes_RN)$ finite, we pay special attention to the inequality $\dim M+\dim N \leq \dim R +c$. In particular, we develop an extension of Hochster's theta invariant whose nonvanishing detects equality. In addition, we consider a parallel theory where dimension and codimension are replaced by depth and complexity, respectively.