Stably reflexive modules and a lemma of Knudsen

Details Stably reflexive modules and a lemma of Knudsen Stably reflexive modules and a lemma of Knudsen

2011-10-18

arxiv.org/abs/1110.3909v2

Mathematics

Commutative Algebra

22 pages. The statement in Thm. 6.1 (iv) has been corrected (earlier Prop. 6 (iv)). The proof of 6.1 has been expanded and so

Scientific paper

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.

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