Commutative algebra inspired by modularity lifting

TL;DR

The paper surveys how commutative algebra concepts arising from modularity lifting—notably congruence modules and the Wiles defect—inform the passage from minimal to non-minimal deformation problems. It develops a unifying framework around $R_\Sigma$ and $\mathbb{T}_\Sigma$, their patching, and derived-action extensions, to produce numerical criteria that certify freeness and complete intersection properties in patched rings. A key result is the positivity and rigidity of the Wiles defect, $\delta_{\lambda}(M)$, which vanishes precisely in the complete intersection/free-summand situation, and a deformation/ Koszul-homology approach to define a defect module for finitely and infinitely generated modules. These ideas extend to higher codimensions and to derived actions on complexes, enabling broader modularity-lifting applications and suggesting new algebraic invariants with potential number-theoretic impact.

Abstract

This article gives an overview of some recent results in commutative algebra that are inspired by the work of Wiles, Taylor and Wiles, Diamond, Lenstra and others on the modularity of elliptic curves.

Theorems & Definitions (12)

• Proposition 2.1
• Proof 1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.6
• Theorem 2.7
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4