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

Luchezar L. Avramov

Review from Zentralblatt MATH:

The paper begins with the introduction of the notion of locally complete intersection (l.c.i) for homomorphisms of noetherian rings.

If $\varphi:(R,m)\to(S,n)$ is a local homomorphism, then by a previous work of {\it L. L. Avramov}, {\it H.-B. Foxby} and {\it B. Herzog} [J. Algebra 164, No. 1, 124-145 (1994; Zbl 0798.13002)], the composition $\overline \varphi:R \to\widehat S$ of $\varphi$ with the completion map $S\to \widehat{S}$ factors as $\varphi'\circ \dot\varphi$, $\dot\varphi :R'\to\widehat S$, $\varphi': R\to R'$, where $\dot\varphi$ is flat, $\varphi'$ is surjective, $R'$ is complete and the ring $R'/mR'$ is regular. Then $\varphi$ is called a complete intersection if $\text{Ker } \varphi'$ is generated by a regular sequence. A homomorphism of noetherian rings $\varphi: R\to S$ is l.c.i. if at each prime ideal $q$ of $S$ the induced local homomorphism $\varphi_q: R_{q \cap R}\to S_q$ is a complete intersection.

With this definition, L. Avramov establishes a very general form of a conjecture of Quillen: If $S$ has a finite resolution by flat $R$-modules and the cotangent complex $L(S/R)$ is quasi-isomorphic to a bounded complex of flat $S$-modules, then $\varphi$ is l.c.i. The proof uses an interplay of commutative algebra and differential homological algebra. The family of l.c.i. homomorphisms is shown to be stable under composition, decomposition, flat base extension, localization and completion.

Reviewed by Y.Felix

Keywords: André-Quillen homology; locally complete intersection; homomorphisms; cotangent complex; localization; completion

Classification (MSC2000): 13D03 13H10 14M10 14B25 13E05 13B10

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