The cotangent complex of a morphism

Abstract

(see ?1.1 for Der). We define functors Ti(B/A, M) and Ti(B/A, M) i=0, 1, 2 (for any B-module M) such that TO(B/A, M) = QB/A 0B M and TO(B/A, M) = DerA(B, M), and the Ti (resp. Ti) fit into nine term exact sequences extending (0.3) (resp. (0.4)). The groups Ti and Ti are formed by taking homology and cohomology of a three term complex, the Cotangent Complex of B over Also, under suitable finiteness conditions, the vanishing of the functor T1(B/A, -) (resp. T1(B/A, .)) 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 Jacobian 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 Ti 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