Vanishing of cotangent functors
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)].
linkage class of a complete intersection
scientific article; zbMATH DE number 4073141
scientific article