Bousfield Localization Functors and Hopkins' Chromatic Splitting Conjecture
Bousfield Localization Functors and Hopkins' Chromatic Splitting Conjecture
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布斯菲尔德定位函子和霍普金斯色分裂猜想
DOI：
10.1090/conm/181/02036
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发表时间：
1993
影响因子：
2
通讯作者：
Mark Hovey
中科院分区：
文献类型：
--
作者：
Mark Hovey
本文试图理解稳定同伦理论中的Bousfield局部化函子。在本文中，对于素数p，所有的谱都是p-局部的。回想一下，如果E是谱，则如果E∧X为空，则谱X是E-非循环的。如果一个E-非循环谱到它的每个映射都是空的，则该谱是E-局部的。映射X→Y是E等价的，如果它在E∗上诱导同构，或者等价地，如果纤维是E无圈的。在[Bou79]中，Bousfield证明了存在一个称为E-局部化的函子，它将一个谱X取为E-局部谱Lex，以及一个自然变换X-→lex，它是一个E-同构。研究Lex就是研究E所看到的那部分同伦理论。这些局部化函子在同伦理论中一直是非常重要的。Ravenel[Rav84]证明了有限谱关于所有Morava K-理论∨n<；∞K(N)的楔形是局部的。这从概念上证明了从EilenbergMacLane谱HFP到有限谱X不存在非平凡映射。Hopkins和Ravenel后来将其推广到色收敛定理[Rav92]。如果我们像往常一样用Ln表示关于第一个n+1 Morava K-理论K(0)∨···∨K(N)的局部化，色收敛定理说对有限X，塔.。。πiLnX→πILN−1X.。。→πiL0X同构于常数塔{πIX}。特别地，X是LNX的逆极限。
This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is E-acyclic if E ∧X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X → Y is an E-equivalence if it induces an isomorphism on E∗, or equivalently, if the fibre is E-acyclic. In [Bou79], Bousfield shows that there is a functor called E-localization, which takes a spectrum X to an E-local spectrum LEX, and a natural transformation X → LEX which is an E-isomorphism. Studying LEX is studying that part of homotopy theory which E sees. These localization functors have been very important in homotopy theory. Ravenel [Rav84] showed, among other things, that finite spectra are local with respect to the wedge of all the Morava K-theories ∨ n<∞K(n). This gave a conceptual proof of the fact that there are no non-trivial maps from the EilenbergMacLane spectrum HFp to a finite spectrum X. Hopkins and Ravenel later extended this to the chromatic convergence theorem [Rav92]. If we denote, as usual, the localization with respect to the first n + 1 Morava K-theories K(0) ∨ · · · ∨ K(n) by Ln, the chromatic convergence theorem says that for finite X, the tower . . . πiLnX → πiLn−1X . . . → πiL0X is pro-isomorphic to the constant tower {πiX}. In particular, X is the inverse limit of the LnX.