Dimension and depth inequalities over complete intersections

TL;DR

This work extends Serre/Hochster-type dimension inequalities to complete intersections of arbitrary codimension by defining a higher-codimension theta invariant $ heta_r^R(M,N)$ via backward differences of Hochster's theta. A central device is the principal lifting theorem, which reduces complexity from $r$ to $r-1$ while preserving $ heta_r^R(M,N)$, enabling an inductive bridge from hypersurfaces to arbitrary codimension. In parallel, a depth-based theory replaces dimension with depth and codimension with complexity, yielding the bound ${ m depth }M+{ m depth }N\u2264 ext{ dim }R+r$ and linking invariants to depth-inequality equalities and Serre-type conjectures in higher codimension. The paper also develops the notion of intersection liftability, obtaining nonnegativity and sharpness statements for $ heta_r^R(M,N)$ and connecting these ideas to Dao’s invariants and related Tor-length phenomena. Thorough Appendix material supplies the machinery for principal liftings, Eisenbud operators, and superficial elements that underlie the main results.

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.

Theorems & Definitions (61)

• Theorem 1.2
• proof
• Theorem 1.3
• Theorem 1.4
• proof
• Definition 1.8
• Remark 1.10
• Proposition 2.1
• proof
• Definition 2.2