Wiles defect for modules and criteria for freeness

TL;DR

This paper refines criteria for module freeness over local rings by introducing the Wiles defect, providing a formula linking the defect of modules to that of the rings, enhancing understanding of module structure.

Contribution

It introduces the Wiles defect as a new tool to refine existing criteria for module freeness over complete intersection rings, extending previous results.

Findings

Formulated a formula expressing the Wiles defect of a module in terms of the ring's Wiles defect.

Refined Diamond's numerical criterion for module freeness using the Wiles defect.

Provided insights into the relationship between module properties and ring invariants.

Abstract

F. Diamond proved a numerical criterion for modules over local rings to be free modules over complete intersection rings. We formulate a refinement of these results using the notion of Wiles defect. A key step in the proof is a formula that expresses the Wiles defect of a module in terms of the Wiles defect of the underlying ring.