Stably reflexive modules and a lemma of Knudsen

In his fundamental work on the stack M¯g,n 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 his appendix to [21]. Then we give a ‘coordinate free’ generalisation of his lemma, generalise a construction used in Knudsenʼs proof concerning versal families of pointed algebras, and show that Knudsenʼs 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.