Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
复制标题
局部完全交同态和Quillen关于余切同调消失的猜想
DOI：
10.2307/121087
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发表时间：
1999
影响因子：
4.9
通讯作者：
L. Avramov
中科院分区：
文献类型：
--
作者：
L. Avramov
交换环的局部完全交（l.c.i）同调性的经典缩差仅限于本质上为flnite型或∞的映射。本文所引入的概念对同态异构体具有重要意义。交换诺瑟环的S。在spec的每个点的正式邻域中，它的结构被简化了。我们在r -代数s的Andre-Quillen同态消失的不同条件下刻画了l.c.i.性质，其中一个描述建立了Quillen猜想的一个非常一般的形式，这个猜想甚至对flnite型同态开放：如果S在r -模处具有无穷大分辨率，且余切复形L（SjR）拟同构于S-模处的无穷大有界复形，则S是l.c.i。该证明使用了交换代数、微分梯度同调代数和同伦理论的混合方法。在多种操作下，包括合成、分解、基变化时的∞、局部化和补全，l.c.i.的性质是稳定的。目前的框架允许以适当的概括性说明结果；其中许多是新的，即使是经典的假设。例如，l.c.i.同态在分解下的稳定性解决了Fulton处理方案的态射取向的一个开放情况。
Classical deflnitions of locally complete intersection (l.c.i.) homomor- phisms of commutative rings are limited to maps that are essentially of flnite type, or ∞at. The concept introduced in this paper is meaningful for homo- morphisms ': Ri! S of commutative noetherian rings. It is deflned in terms of the structure of ' in a formal neighborhood of each point of SpecS .W e characterize the l.c.i. property by difierent conditions on the vanishing of the Andre-Quillen homology of the R-algebra S. One of these descriptions estab- lishes a very general form of a conjecture of Quillen that was open even for homomorphisms of flnite type: If S has a flnite resolution by ∞at R-modules and the cotangent complex L(SjR) is quasi-isomorphic to a bounded complex of ∞at S-modules, then ' is l.c.i. The proof uses a mixture of methods from commutative algebra, difierential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a variety of operations, including composition, decomposition, ∞at base change, localization, and com- pletion. The present framework allows for the results to be stated in proper generality; many of them are new even with classical assumptions. For in- stance, the stability of l.c.i. homomorphisms under decomposition settles an open case in Fulton's treatment of orientations of morphisms of schemes.