论文标题
在pro- $ p $ iwahori子组的mod $ p $同时;
On the Mod $p$ Cohomology of Pro-$p$ Iwahori Subgroups of $\operatorname{SL}_{n}(\mathbb Q_{p})$
论文作者
论文摘要
本文可以看作是对作者论文一部分的更新。我们研究pro- $ p $ iwahori子组的mod $ p $共同体，$ \ operatatorName {sl} _ {n} _ {n}（\ Mathbb q_ {p}）$（和$ \ operatorNorname {gl} $ n = 3 $。在这里，我们使用频谱序列$ e_ {1}^{s，t} = h^{s，t}（\ mathfrak {g}，\ mathbb {f} _ {p}） $ i $的订购基础，这为我们提供了$ \ mathfrak {g} = \ mathbb {f} _ {p} \ otimes _ {\ mathbb {f} _ {p} _ {p} [π]} \ perperatornAme {gr} i $计算的， $ h^{s，t}（\ mathfrak {g}，\ mathbb {f} _ {p}）$。我们注意到，乘法频谱序列$ e_ {1}^{s，t} = h^{s，t}（\ mathfrak {g}，\ mathbb {f} _ {p} _ {p}）$在第一页上倒在每个页面上的所有映射，都必须在每个页面上折叠，并允许我们允许我们的组成部分，并描述我们的coh off up cop and coh and coh and coh and coh and coh and coh。最后，我们注意到与$ \ mathbb {q} _ {p} $上的中央分区代数的共同体相关，并指出了一些未来的研究方向。
This paper can be seen as an update to part of the author's dissertation. We study the mod $p$ cohomology of the pro-$p$ Iwahori subgroups $I$ of $\operatorname{SL}_{n}(\mathbb Q_{p})$ (and $\operatorname{GL}_{n}(\mathbb{Q}_{p})$) for $n=2$ and $n=3$. Here we use the spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p}) \Longrightarrow H^{s+t}(I,\mathbb{F}_{p})$ due to Sorensen, and we do explicit calculations with an ordered basis of $I$, which gives us a basis of $\mathfrak{g} = \mathbb{F}_{p} \otimes_{\mathbb{F}_{p}[π]} \operatorname{gr} I$ that we use to calculate $H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$. We note that the multiplicative spectral sequence $E_{1}^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_{p})$ collapses on the first page by noticing that all maps on each page are necessarily trivial, and this allows us to describe the above group cohomology groups and all cup products. Finally we note some connections to cohomology of central division algebras over $\mathbb{Q}_{p}$ and point out some future research directions.
