Vanishing of cohomology over Cohen--Macaulay rings

Details Vanishing of cohomology over Cohen--Macaulay rings Vanishing of cohomology over Cohen--Macaulay rings

2010-06-04

arxiv.org/abs/1006.1006v2

Mathematics

Commutative Algebra

Updated references. Final version to appear in Manuscripta Math.; 9 pp

Scientific paper

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings - colloquially called AC rings - that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen--Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen-Macaulay AC rings.

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