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 $\varphi\colon R\to S$ of commutative noetherian rings. It is defined in terms of the structure of $\varphi$ in a formal neighborhood of each point of $\mathrm{Spec} S$. We characterize the l.c.i. property by different conditions on the vanishing of the André-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 $R$-mdoules and the cotangent complex $\mathrm{L}(S|R)$ is quasi-isomorphic to a bounded complex of flat $S$-modules, then $\varphi$ 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 rproper 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.


Luchezar L. Avramov