通过同调支撑的局部Bousfield类
Local Bousfield classes via homological support
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中文总结 AI 辅助
本文在张量三角化范畴中研究$A$-局部范畴的Bousfield类，揭示同调支撑/余支撑的分类作用，将结果应用于色度同伦理论，回答了相关问题并得到新的上同调Bousfield类例子。
中文摘要 AI 辅助
给定一个大张量三角化范畴中的对象$A$，我们研究相关局部化（即$A$-局部对象构成的张量三角化范畴）的同调与上同调Bousfield类。我们证明：当满足$A$-相对形式的同调检测性质时，同调支撑能精确对$A$-局部范畴的同调Bousfield类进行分类；进一步证明该性质成立当且仅当$A$与同调剩余域的余积是Bousfield等价的。类似地，用同调余支撑对上同调Bousfield类的分类严格更强，它等价于$A$-相对形式的同调分层——这种分层与上同调Bousfield类分类的等价性，即使在绝对情形下也是全新的结果。另一意外结论是：同调分层还等价于同调Bousfield类的分类，且所有上同调Bousfield类均为同调类。将这些结果应用于色度同伦理论，可对Morava K-理论余积的任意谱局部化的同调Bousfield类进行分类，覆盖了诸多受关注的局部化情形。我们还完全刻画了这类色度局部化何时为相对同调分层的，得到了非同调类的上同调Bousfield类的新例子，特别回答了Wolcott关于调和谱范畴的问题，这些例子通过展示具有空同调余支撑的局部谱构造得到。
英文摘要
Given an object $A$ in a big tensor-triangulated category, we study the homological and cohomological Bousfield classes of the associated localization: the tensor-triangulated category of $A$-local objects. We show that the homological support classifies the homological Bousfield classes of the $A$-local category precisely when an $A$-relative form of the homological detection property holds. Moreover, we prove that this holds if and only if $A$ is Bousfield equivalent to a coproduct of homological residue fields. The analogous classification of cohomological Bousfield classes by homological cosupport is strictly stronger: it is equivalent to an $A$-relative form of homological stratification. This equivalence between stratification and the classification of cohomological Bousfield classes is new even in the absolute case. A further surprise is that stratification is also equivalent to the classification of homological Bousfield classes together with the statement that every cohomological Bousfield class is homological. Applied to chromatic homotopy theory, these results classify the homological Bousfield classes of any localization of spectra with respect to a coproduct of Morava $K$-theories. This covers many localizations of interest. We also completely characterize when such chromatic localizations are relatively homologically stratified. This yields new examples of cohomological Bousfield classes that are not homological. In particular, it answers a question of Wolcott concerning the category of harmonic spectra. Our examples are produced by exhibiting local spectra with empty homological cosupport.
