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

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 Andr\'e-Quillen 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 \cot SR is quasi-isomorphic to a bounded complex of flat S-modules, 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.Comment: 33 pages, published versio

Quasi-complete intersection homomorphisms

Extending a notion defined for surjective maps by Blanco, Majadas, and Rodicio, we introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally complete intersection homomorphisms, while sharing many of its remarkable properties.Comment: Final version, to appear in the special issue of Pure and Applied Mathematics Quarterly dedicated to Andrey Todorov. The material in the first four sections has been reorganized and slightly expande

Homology over local homomorphisms

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a `contracting' endomorphism -- such as the Frobenius endomorphism -- to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.Comment: To appear in the American Journal of Mathematics; new version has minor changes in the presentation; table of content removed; 52 page