Abstract

. Classical definitions of l.c.i. homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms ' : R \Gamma! S of commutative noetherian rings. It is defined in terms of the structure of ' in a formal neighborhood of each point of Spec S. We characterize the l.c.i. property by different conditions on the vanishing of the Andr'eQuillen homology of the R-algebra S. One of these descriptions 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 flat R--modules and the cotangent complex L(S jR) is quasi-isomorphic to a bounded complex of flat S--modules, then ' 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 variety of operations, including composition, decomp...

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