Stably reflexive modules and a lemma of Knudsen

TL;DR

This paper generalizes and refines the concept of stably reflexive modules introduced by Knudsen, providing new theoretical tools for algebraic geometry, especially in the study of stable curves and singularities.

Contribution

It offers an alternative definition of stably reflexive modules, generalizes Knudsen's lemma and stabilisation construction, and extends Cohen-Macaulay approximation theorems in a broader context.

Findings

Generalized Knudsen's lemma in a coordinate-free manner

Extended stabilisation construction to plane curve singularities

Proved new approximation theorems for flat families

Abstract

In his fundamental work on the stack of stable n-pointed genus g curves, Finn F. Knudsen introduced the concept of a stably reflexive module in order to prove a key technical lemma. We propose an alternative definition and generalise the results in the appendix to his article. Then we give a `coordinate free' generalisation of his lemma, generalise a construction used in Knudsen's proof concerning versal families of pointed algebras, and show that Knudsen's stabilisation construction works for plane curve singularities. In addition we prove approximation theorems generalising Cohen-Macaulay approximation with stably reflexive modules in flat families. The generalisation is not covered (even in the closed fibres) by the Auslander-Buchweitz axioms.