Vanishing of the André-Quillen homology Module<math>H₂(A,B,G (I))</math>

Abstract

Let $I$ be an ideal of a commutative Noetherian ring $A$, $A\supset \Q$, $B=A/I$ and ${\bf G}(I)$ the associated graded ring to $I$. It is known that $H_{2}(A,B,B)=0$ is equivalent to $I$ being syzygetic. We prove that the vanishing of $\hd $ is equivalent to $I$ being of linear type and $\sigma _{3,q}:\extp{3}{B}{I/I^{2}}\otimes _{B}I^{q}/I^{q+1}\rightarrow \tor{3}{A}{B}{A/I^{q+1}}$, the $(3,q)$-antisymmetrization morphism, being surjective for all $q\geq 0$. Using this and a theorem of Ulrich on a conjecture of Herzog, we deduce that, in a regular local ring $A$, a Gorenstein, licci ideal $I$ verifies $\hd =0$ if and only if $I$ is a complete intersection. Thus, we characterize perfect (respectively, Gorenstein) ideals of grade two (respectively, three) with $\hd =0$ as those ideals which are of linear type (respectively, complete intersection). With any grade, but small deviation, we show that a licci ideal, generically a complete intersection and of deviation one, verifies $\hd =0$. This is not true for licci ideals of linear type and of deviation two.