Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
Abstract
Classical definitions of locally complete intersection (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 phi : R \longrightarrow S of commutative noetherian rings. It is defined in terms of the structure of phi 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 AndréQuillen homology of the Ralgebra 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 Rmodules and the cotangent complex \cot SR is quasiisomorphic to a bounded complex of flat Smodules, then phi 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, 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 classical assumptions. For instance, the stability of l.c.i. homomorphisms under decomposition settles an open case in Fulton's treatment of orientations of morphisms of schemes.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 1999
 arXiv:
 arXiv:math/9909192
 Bibcode:
 1999math......9192A
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Rings and Algebras
 EPrint:
 33 pages, published version