广义位形空间的同调稳定性和稠密性
Homological stability and densities of generalized configuration spaces
摘要：
证明了截断多项式代数中系数格式的分解同调计算其广义位形空间的上同调。利用交换代数与李代数之间的Koszul对偶，得到了后者上同调的新表达式.因此，我们得到了一个统一的和概念性的方法来处理同调稳定性，同调密度和算术密度的广义配置空间。我们的研究结果分类，概括，并在事实上提供了一个概念上的理解的巧合出现在工作的Farb-沃尔夫森-伍德。我们的计算稳定的同调密度也产生合理的同伦类型，回答了一个问题所提出的Vakil-木材。我们的方法取决于同调Chevalley复合物的同调稳定性的研究，这是独立的兴趣。
英文摘要：
We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.
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