Vanishing of cohomology over Cohen--Macaulay rings

2010/06/04 by Lars Winther Christensen, Christensen, Lars Winther, Henrik Holm +1
Mathematics · #13D07 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13D07 #msc:13H10

paper · pdf · doi:10.48550/arxiv.1006.1006

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

arxiv created 2012/02/23 · arxiv updated 2012/02/24

Abstract

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.

Related

• On the vanishing of local cohomology of the absolute integral closure in positive characteristic
• Vanishing of (co)homology, freeness criteria, and the Auslander-Reiten conjecture for Cohen-Macaulay Burch rings
• Algebras that satisfy Auslander's condition on vanishing of cohomology
• On the vanishing of self extensions over Cohen-Macaulay local rings
• Local cohomology annihilators and Macaulayfication