The cotangent complex of a morphism

Abstract

Introduction.Let A -> B -y C be a sequence of ring homomorphisms.(Unless otherwise noted all rings will be commutative with unit, and all modules and homomorphisms will be unitary.)Let QBM, QCM, Q.c/B denote the respective modules of Kahler differentials ( §1.1).Then one obtains easily the following exact sequence of C-modules:FlBIA <g>B C -y QCM -> Qc/S -> 0.Also if C=B\I, where Tis an ¡deal in B, then Qc/B=0 and one obtains the exact sequencewhere d is induced from the universal derivation d of B into 0BM ( §1.1).In §2 we show that (0.1) and (0.2) are parts of a nine term exact sequence.To be precise, let M be a C-module.Then we can form two exact sequences (0.3) QBM ®BM-» nclA ®cM-y Qc/B <g>c M -y 0, (0.4) 0 -> DerB(C, M) -> Der^C, M) -> Der"(5, M) (see §1.1 for Der).We define functors TIB\A, M) and T\B\A, M) i=0, 1, 2 (for any Ä-module M) such that T0(B¡A, M) = ClBIA <g)B M and T°(BjA, M) = F>erA(B, M), and the T¡ (resp.T') fit into nine term exact sequences extending (0.3) (resp.(0.4)).The groups F¡ and Ti are formed by taking homology and cohomology of a three term complex, the Cotangent Complex of B over A. Also, under suitable finiteness conditions, the vanishing of the functor T1(B/A, ■) (resp.TX(BIA, ■)) is equivalent to B being "smooth" (formerly "simple") over A, and the vanishing of T2(B\A, ■) (resp.T2(B\A, ■) is equivalent to B being a "locally complete intersection over A." These and other vanishing criteria are discussed in §3.The lacobian criterion for nonsingularity of a variety is obtained as a natural consequence of these criteria.In §4 we apply the functors Ti to the study of infinitesimal deformations, and obstructions thereto.In §5 we explain the role of the F¡ in a reformulation of the Grothendieck-Riemann-Roch Theorem.Many other authors have studied, in many different guises, the homology and cohomology theories of commutative algebras, and have obtained some of the results which we prove in this paper.However, since our definitions are not the same (although in some cases equivalent to) those of previous authors, we have