Nonvanishing cohomology and classes of Gorenstein rings

By David A. Jorgensen, Liana Sega. 2003

67 citations recorded by the source corpus.

Abstract We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ , there exists a positive integer n M such that for all finitely generated Λ -modules N , if Ext Λ i ( M , N )=0 for all i ≫0, then Ext Λ i ( M , N )=0 for all i ⩾ n M . Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2433b9dabdcd02cdb09e630d61ce9f2ed8ed3f3f

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In Algebra and Number Theory

In Geometry and Topology

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Citations in the corpus

Independence of the Total Reflexivity Conditions for Modules (2004, methodology): “Related rings were used by Gasharov and Peeva in [6] and then by the authors in [7] to disprove various conjectures.”
Algebras that satisfy Auslander’s condition on vanishing of cohomology (2008, background): “In [29] Jorgensen and Şega showed that the conjecture fails, even for commutative local finite dimensional k-algebras: one counterexample (R,m) is Gorenstein with m4 = 0, another is not Gorenstein and has m3 = 0 and length R = 8.”
Finiteness in derived categories of local rings (2004, result): “This aspect of our work is in the same spirit as that of Jorgensen and Şega [21], who have discovered various classes of local rings strictly contained between complete intersections and Gorenstein rings, which nevertheless retain some traces of the homological properties of the former.”
The radius of a subcategory of modules (2011, background): “For a local ring R, Jorgensen and Şega [17] introduced the uniform Auslander condition: (UAC) : There exists an integer n such that for all R-modules M,N with ExtiR(M,N) = 0 for all i≫ 0 one has Ext i R(M,N) = 0 for all i ≥ n.”