Vanishing of cotangent functors
type: Item
instance of: scholarly article
title: Vanishing of cotangent functors
zbMATH Open document ID: 0657.13023
DOI: 10.1007/BF01160888
author: Bernd Ulrich
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173895
review text: The main results of the paper are concerned around \textit{J. Herzog}'s question [see ``Homological properties of the modules of differentials'', Colecao Atas Soc. Brasileira Mat. 14 (1981)]: Let P be a regular local ring containing the rationals, let I be an ideal, and set \(R=P/I\). Then does the vanishing of the cotangent modules \(T_ i(R/P,R)\) for all \(i\geq 2\) imply that I is a complete intersection? Let I be in the linkage class of a complete intersection. Let \(\omega_ R\) denote the canonical module of R. Then it is shown: \((1)\quad I\quad is\) a Gorenstein ideal, provided \(T_ 4(R/P,R)=0\) or \(T_ 2(R/P,\omega_ R)=0\). \((2)\quad I\quad is\) a complete intersection, if in addition \(T_ 3(R/P,R)=0.\) There are similar results for upper cotangent modules, e.g., the equivalence of the following conditions is shown: \((a)\quad T^ 6(R/P,R)=T^{10}(R/P,R)=0.\quad (b)\quad R\quad is\) Gorenstein and \(T^{10}(R/P,R)=0\). \((c)\quad I\quad is\) a complete intersection. In order to prove most of his results the author uses a structure theorem for ideals in the linkage class of a complete intersection [see \textit{C. Huneke} and the author, Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)].
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 13D10
zbMATH DE Number: 4073141
zbMATH Keywords: vanishing of the cotangent modules
zbMATH Keywords: linkage class of a complete intersection
reviewed by: Peter Schenzel
MaRDI profile type: Publication
cites work: Q3663419
cites work: The Koszul algebra of a codimension 2 embedding
cites work: Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3
cites work: Q3207091
cites work: Deformation von Cohen-Macaulay Algebren.
cites work: Q3964688
cites work: Cotangent functors of curve singularities
cites work: On the symmetric and Rees algebra of an ideal generated by a d-sequence
cites work: Linkage and the Koszul Homology of Ideals
cites work: Numerical invariants of liaison classes
cites work: Divisor Class Groups and Deformations
cites work: The structure of linkage
cites work: Almost complete intersections are not Gorenstein rings
cites work: Liaison des variétés algébriques. I
cites work: Liaison and deformation
cites work: Q3781839
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description: scientific article; zbMATH DE number 4073141