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Crystalline condition for Ainf-cohomology and ramification bounds

Algebra and Number Theory Seminar

Crystalline condition for Ainf-cohomology and ramification bounds
Series: Algebra and Number Theory Seminar
Location: Zoom Meeting
Presenter: Pavel Coupek, Purdue University

Let $p>2$ be a prime and let $X$ be a proper smooth formal scheme over $\mathcal{O}_K$ where $K/\mathbb{Q}_p$ is a local number field. In this talk, we describe a series of conditions $(\mathrm{Cr}_s)$ that provide control on the Galois action on the Breuil--Kisin cohomology $\mathrm{R}\Gamma_{\Delta}(X/\mathfrak{S})$ inside the $A_{\inf}$--cohomology $\mathrm{R}\Gamma_{\Delta}(X_{\mathbb{C}_K}/A_{\inf})$. When $s=0$, the resulting condition is essentially the crystallinity criterion of Gee and Liu for Breuil--Kisin--Fargues $G_K$--modules, and it leads to an alternative proof of crystallinity of the $p$--adic \'{e}tale cohomology $H^i_{\mathrm{et}}(X_{\mathbb{C}_K}, \mathbb{Q}_p)$. Adapting a strategy of Caruso and Liu, the conditions $(\mathrm{Cr}_s)$ for higher $s$ then lead to upper bounds on ramification of the mod $p$ \'{e}tale cohomology $H^i_{\mathrm{et}}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$, expressed in terms of $i, p$ and $e=e(K/\mathbb{Q}_p)$ that work without any restrictions on the size of $i$ and $e$.