Title: Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras Show Chinese title

Title: 上同调和下同调交叉乘积由逆独异作用和Steinberg代数

Abstract: Given a unital action $\theta $ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_\theta S$ with values in a bimodule over $A\rtimes_\theta S$. The spectral sequences involve a new kind of (co)homology of the inverse monoid $S,$ which is based on $KS$-modules. The spectral sequences take especially nice form, when $(A\rtimes_\theta S)^e $ is flat as a left (homology case) or right (cohomology case) $A^e$-module, involving also the Hochschild (co)homology of $A.$ Same nice spectral sequences are also obtained if $K$ is a commutative ring, over which $A$ is projective, and $S$ is $E$-unitary. We apply our results to the Steinberg algebra $A_K(\mathscr{G})$ over a field $K$ of an ample groupoid $\mathscr{G},$ whose unit space $\mathscr{G} ^{(0)}$ is compact. In the homology case our spectral sequence collapses on the $p$-axis, resulting in an isomorphism between the Hochschild homology of $A_K(\mathscr{G})$ with values in an $A_K(\mathscr{G})$-bimodule $M$ and the homology of the inverse semigroup of the compact open bisections of $\mathscr{G}$ with values in the invariant submodule of $M.$ Show Chinese abstract

Abstract: 给定一个逆独异点$S$在域$K$上的代数$A$上的单位作用$\theta $，我们构造（上）同调谱序列，这些序列收敛到交叉乘积$A\rtimes_\theta S$关于在$A\rtimes_\theta S$上的双模的Hochschild（上）同调。 谱序列涉及逆独异群$S,$的一种新的 (上)同调，该同调基于$KS$模块。 当$(A\rtimes_\theta S)^e $作为左（同调情况）或右（上同调情况）$A^e$-模是平坦的时候，谱序列具有特别好的形式，还涉及$A.$的 Hochschild (上)同调。如果$K$是一个交换环，$A$在其上是投射的，且$S$是$E$-单位的，也会得到同样好的谱序列。 我们将我们的结果应用于域$K$上的Steinberg代数$A_K(\mathscr{G})$，这是一个单位空间$\mathscr{G} ^{(0)}$是紧致的丰富群oid$\mathscr{G},$。 在同调情况下，我们的谱序列在$p$轴上崩溃，导致$A_K(\mathscr{G})$在取值于$A_K(\mathscr{G})$双模$M$的 Hochschild 同调与$\mathscr{G}$的紧致开双射的逆独异步的同调在$M.$的不变子模上的同调之间的同构。