The Uniqueness Of Minimal Acyclic Complexes

In this paper, we discuss conditions for uniqueness among minimal acyclic complexes of finitely generated free modules over a commutative local ring which share a common syzygy module. Although such uniqueness occurs over Gorenstein rings, the question has been asked whether two minimal acyclic complexes in general can be isomorphic to the left and non-isomorphic to the right. We answer the question in the negative for certain cases, including periodic complexes, sesqui-acyclic complexes, and certain rings with radical cube zero. In particular, we investigate the question for graded algebras with Hilbert series $HR(t)=1+et+(e-1)t2$, and such monomial algebras possessing a special generator.