Etale广群胚的同调，分次方法
Homology of étale groupoids, a graded approach
摘要：
本文引入分次Etale广群胚的分次同调论。对于$\mathbb Z$ -分次广胚，我们建立了分次零同调与非分次同调的正合序列。针对任意图群胚，证明了常系数分次零同调群$\mathbb Z$同构于相应Leavitt路代数的分次Grothendieck群.为此，我们考虑了原图的覆盖图的Leavitt路代数的对角代数，并直接构造了群同构。考虑到图的平凡分次，我们的结果将Matui关于无汇（有限型的移位）的有限图的零同调推广到所有的任意图。利用这些结果，我们证明了分次零同调群是有限型移位的最终共轭的完全不变量，并且可以成为解析图代数和代数图代数的统一不变量.
英文摘要：
We introduce a graded homology theory for graded étale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients $\mathbb Z$ is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
