同调代数中的另一点：不连续群作用的对偶性
Another point in homological algebra: Duality for discontinuous group actions
摘要：
本文考虑可压缩$n$维流形$X$上群$G$的不连续运算.令$E$是特征为0的域$k$上的$G$的有限维表示。令$\mathcal{E}$是与$E$相关联的商空间$Y=G \setminus X$上的层。设$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$是紧支集上同调在$H^{\bullet}(Y;\mathcal{E})$中的像.在$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$和$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E}^*)$（$\mathcal{E}^*$是与$E$对偶的表示相关联的层）都是有限维的情况下，我们建立了$H^{m}_{\textbf{!}}(Y;\mathcal{E})$和$H^{n-m}_{\textbf{!}}(Y;\mathcal{E}^{\ast})$之间的非退化对偶。我们还表明，这种对偶是兼容的Hecke运营商。
英文摘要：
We consider discontinuous operations of a group $G$ on a contractible $n$-dimensional manifold $X$. Let $E$ be a finite dimensional representation of $G$ over a field $k$ of characteristics 0. Let $\mathcal{E}$ be the sheaf on the quotient space $Y=G \setminus X$ associated to $E$. Let $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ be the image in $H^{\bullet}(Y;\mathcal{E})$ of the cohomology with compact support. In the cases where both $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ and $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E}^*)$ ($\mathcal{E}^*$ being the the sheaf associated to the representation dual to $E$) are finite dimensional, we establish a non-degenerate duality between $H^{m}_{\textbf{!}}(Y;\mathcal{E})$ and $H^{n-m}_{\textbf{!}}(Y;\mathcal{E}^{\ast})$. We also show that this duality is compatible with Hecke operators.
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