Locally complete intersection homomorphisms and a conjecture of Quillen onthe vanishing of cotangent homology

Classical definitions of locally complete intersection (l.c.i,) homomorphis ms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homom orphisms cp : R --> S of commutative noetherian rings. It is defined in ter ms of the structure of cp in a formal neighborhood of each point of SpecS. We characterize the l.c.i. property by different conditions on the vanishin g of the Andre-Quillen homology of the R-algebra S. One of these descriptio ns establishes a very general form of a conjecture of Quillen that was open even for homomorphisms of finite type: If S has a finite resolution by fla t R-modules and the cotangent complex L(S\R) is quasi-isomorphic to a bound ed complex of flat S-modules, then cp is l.c.i. The proof uses a mixture of methods from commutative algebra, differential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a var iety of operations, including composition, decomposition, flat base change, localization, and completion. The present framework allows for the results to be stated in proper generality; many of them are new even with classica l assumptions. For instance, the stability of l.c.i. homomorphisms under de composition settles an open case in Fulton's treatment of orientations of m orphisms of schemes.