序数的上同调Ⅰ：基本理论与相容性结果
The Cohomology of the Ordinals I: Basic Theory and Consistency Results
摘要：
在本文中，在一个预计的两部分系列的第一，我们描述了一个组织框架的研究无穷组合。这个框架是Čech上同调。我们特别表明，Čech上同调群的序数阐明高维推广Todorcevic的散步和相干序列技术，并开始考虑这些技术的"不合理的有效性" $ω_1$。这个讨论占据了我们论文的前半部分，并且是针对一般数学受众而写的。我们在论文的后半部分转向更恰当的集合论考虑。我们描述了一些一致性结果的上同调群的序数证明其地位作为一个分次家庭的不紧性原则。我们特别表明，非平凡的上同调群的序数是在一些紧张局势与大基数，并最大限度地存在于哥德尔的模型$\mathrm{L}$。我们描述强迫添加，然后trivialize，非平凡的$n$ -cocycles，并得出结论，这些原则与那些基准的不紧性现象，存在的平方序列和故障的固定反射的一些比较。
英文摘要：
In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. We show in particular that the Čech cohomology groups of the ordinals articulate higher-dimensional generalizations of Todorcevic's walks and coherent sequences techniques, and begin to account for those techniques' `unreasonable effectiveness' on $ω_1$. This discussion occupies the first half of our paper and is written with a general mathematical audience in mind. We turn in the paper's second half to more properly set-theoretic considerations. We describe a number of consistency results on the cohomology groups of the ordinals which certify their status as a graded family of incompactness principles. We show in particular that nontrivial cohomology groups on the ordinals are in some tension with large cardinals, and are maximally extant in Gödel's model $\mathrm{L}$. We describe forcings to add, then trivialize, nontrivial $n$-cocycles, and conclude with some comparison of these principles with those benchmark incompactness phenomena, the existence of square sequences and failures of stationary reflection.
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