Finite Localizations
Finite Localizations
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有限本地化
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发表时间：
1997
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通讯作者：
H. Miller
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作者：
H. Miller
这篇短文是对Doug Ravenel和Mark Mahowald和Hal Sadofsky的文章的回应。我给清洁和更一般的建设“伸缩”或“有限的本地化”，他们写Ln。我更喜欢称之为有限E（n）-局部化，并将其写成L f n，因为正如我将展示的，它可以用与Bousfield局部化完全相同的术语来表征，但增加了有限性假设。如果B是一个有限的E（n− 1）-无圈谱，它有一个vn-自映射φ：B−→ n − q B [2]，那么Ln B是B的映射望远镜;所以L f n是这个构造的推广，因为它可以应用于任何谱X。用同样的方法，有限局部化LfA可以被定义为有限谱的同伦类型的任意集合A。特别令人感兴趣的是A是某个谱E的有限E-非循环谱的集合的情况，在这种情况下，我们将相应的有限局部化记为LfE。这种局部化的构造比布兹菲尔德同调局部化的构造简单--人们可以完全在同伦范畴中工作，并且一个可数望远镜就足以构造这种局部化。事实证明，很容易证明LfA总是“粉碎”（即，自然映射X −→X（LfAS是等价的），并且与关于谱LfAS的Bousfield局部化相一致。对于任何谱E，存在一个正则映射LfEX −→LEX。关于E的“望远镜猜想”（Ravenel在[4]中为E = E（n）做了广告）是断言这个映射是等价的。它等价于要求任何E-非循环谱都有一个穷举滤子，其相关滤子是有限E-非循环谱的楔形。这个结构特征正是Bousfield检验K理论的地方，并且通过Ravenel的计算[5]我们现在知道它在大素数的情况下对E（2）是失败的。如果不变量在有限E（n）-无环上为零，并且与楔和上纤维化相容，但在所有E（n）-无环上不为零，这将是非常有趣的。
This short note is a response to the articles [7] of Doug Ravenel and [3] of Mark Mahowald and Hal Sadofsky. I give cleaner and more general constructions of the “telescopic” or “finite localization,” which they write Ln. I prefer to call this the finite E(n)-localization and write L f n for it, because, as I shall show, it can be characterized in exactly the same terms as the Bousfield localization, but with the addition of a finiteness assumption. If B is a finite E(n− 1)-acyclic spectrum with a vn-self-map φ : B−→Σ−qB [2], then LnB is the mapping telescope of B; so L f n is a generalization of this construction in that it can be applied to any spectrum X. By the same method, a finite localization LfA can be defined for any set A of homotopy types of finite spectra. Of particular interest is the case in which A is the set of finite E-acyclic spectra for some spectrum E, and in this case we will write LfE for the corresponding finite localization. The construction of this localization is simpler than that of the Bousfield homology localization— one can work entirely in the homotopy category, and a countable telescope suffices for the construction. It turns out to be easy to show that LfA is always “smashing” (i.e., the natural map X −→X ∧ LfAS is an equivalence) and coincides with Bousfield localization with respect to the spectrum LfAS . For any spectrum E, there is a canonical map LfEX −→LEX. The “telescope conjecture” for E (advertised for E = E(n) by Ravenel in [4]) is the assertion that this map is an equivalence. It is equivalent to require that any E-acyclic spectrum has an exhaustive filtration whose associated quotients are wedges of finite E-acyclic spectra. This structural feature is exactly what Bousfield checks for K-theory, and by virtue of Ravenel’s computation [5] we now know that it fails for E(2) at large primes. It would be be very interesting to have invariants vanishing on finite E(n)-acyclics and compatible with wedges and cofibrations, but not vanishing on all E(n)-acyclics.