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Annals of Mathematics, II. Series Vol. 150, No. 2, pp. 455487 (1999) 

Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homologyLuchezar L. AvramovReview 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, 124145 (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 quasiisomorphic 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: AndréQuillen homology; locally complete intersection; homomorphisms; cotangent complex; localization; completion Classification (MSC2000): 13D03 13H10 14M10 14B25 13E05 13B10 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
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