Acyclic complexes of finitely generated free modules over local rings

Abstract

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.