Suppose $F$ is a finite extension of ${\mathbb{Q}}_p$, $G$ is the group of $F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori subgroup of $G$. We construct two spectral sequences relating derived functors on mod-$p$ representations of $G$ to the analogous functors on Hecke modules coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of Ollivier–Vignéras, we provide a link between the right adjoint of parabolic induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a ‘Poincaré duality spectral sequence’ relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules $H^i(I_1,\pi )$.

中文翻译：

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pro-p-Iwahori 上同调的函子性质

认为 $F$ 是一个有限的扩展 ${\mathbb{问}}_{磷}$, $G$ 是一组 $F$- 连接还原点 $F$-组，和 ${\mathrm{一世}}_1$ 是一个亲$磷$-岩堀亚群 $G$. 我们构建了两个谱序列，这些谱序列与 mod 上的派生函子相关$磷$ 的表示 $G$ 到来自 pro- 的 Hecke 模块上的类似函子$磷$-岩堀上同调。更具体地说：(1) 使用 Ollivier-Vignéras 的结果，我们提供了对 pro-$磷$-Iwahori 上同调和 Emerton 派生的普通部分函子；(2) 我们建立了一个“庞加莱对偶谱序列”，在亲$磷$-Iwahori 上同调到 Kohlhaase 的更高平滑对偶的函子。作为应用程序，我们计算 Hecke 模块的各种示例$H^{\mathrm{一世}}({\mathrm{一世}}_1,\pi )$.