VANISHING OF THE ANDRE-QUILLEN HOMOLOGY MODULE H-2(A,B,G(I))

Let I be an ideal of a commutative Noetherian ring A, A superset of Q, B = A/I and 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 v anishing of H-2(A, B, G(I)) is equivalent to I being of linear type an d sigma(3,q) : A(3)(B)(I/I-2) x (B) I-q/I-q+1 --> Tor(3)(A)(B, A/I-q+1 ), the (3, q)-antisymmetrization morphism, being surjective for all q greater than or equal to 0. Using this and a theorem of Ulrich on a co njecture of Herzog, we deduce that, in a regular local ring A, a Goren stein, licci ideal I verifies H-2(A, B, G(I)) = 0 if and only if I is a complete intersection. Thus, we characterize perfect (respectively, Gorenstein) ideals of grade two (respectively, three) with H-2(A, B, G (I)) = 0 as those ideals which are of linear type (respectively, compl ete intersection). With any grade, but small deviation, we show that a licci ideal, generically a complete intersection and of deviation one , verifies H-2(A, B, G(I)) = 0. This is not true for licci ideals of l inear type and of deviation two.