No date set.

## On the mod $p$ cohomology of unipotent groups

### Daniel Kongsgaard

#### University of California, San Diego, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8192f6b35d5d677e3929bd9b3e9be64f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8192f6b35d5d677e3929bd9b3e9be64f = 'dk&#111;ngsg&#97;' + '&#64;'; addy8192f6b35d5d677e3929bd9b3e9be64f = addy8192f6b35d5d677e3929bd9b3e9be64f + '&#117;csd' + '&#46;' + '&#101;d&#117;'; var addy_text8192f6b35d5d677e3929bd9b3e9be64f = 'dk&#111;ngsg&#97;' + '&#64;' + '&#117;csd' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak8192f6b35d5d677e3929bd9b3e9be64f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8192f6b35d5d677e3929bd9b3e9be64f + '\'>'+addy_text8192f6b35d5d677e3929bd9b3e9be64f+'<\/a>';

Let $\mathcal{G}$ be a split and connected reductive $\mathbb{Z}_p$-group and let $\mathcal{N}$ be the unipotent radical of a Borel subgroup. In this poster we study the cohomology with trivial $\mathbb{F}_p$-coefficients of the nilpotent pro-$p$-group $N = \mathcal{N}(\mathbb{Z}_p)$ and the Lie algebra $\mathfrak{n} = \operatorname{Lie}(\mathcal{N}_{\mathbb{F}_p})$. We proceed by arguing that $N$ is a $p$-valued group using ideas of Schneider (unpublished) and Zábrádi, which by a result of Sørensen gives us a spectral sequence $E_1^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_p) \Longrightarrow H^{s+t}(N,\mathbb{F}_p)$, where $\mathfrak{g} = \mathbb{F}_p \otimes_{\mathbb{F}_p[\pi]} \operatorname{gr} N$ is the graded Lie $\mathbb{F}_p$-algebra attached to $N$ as in Lazards work. We then argue that $\mathfrak{g} \cong \mathfrak{n}$ by looking at the Chevalley constants, and, using results of Polo and Tilouine and ideas from Große-Klönne, we show that the dimensions of the $\mathbb{F}_p$-cohomology of $\mathfrak{n}$ and $N$ agree, which allows us to conclude that the spectral sequence collapses on the first page.