Integral \(p\)-adic étale cohomology of Drinfeld symmetric spaces (Q2037840)

Let \(K\) be a finite extension of \({\mathbb Q}_p\), let \(C\) be the \(p\)-adic completion of an algebraic closure \(\overline{K}\) of \(K\). Write \({\mathcal G}_K=\mathrm{Gal}(\overline{K}/K)\). Drinfeld's symmetric space of dimension \(d\) over \(K\) is the rigid analytic variety \({\mathbb H}_K^d:={\mathbb P}_K^d-\cup_{H\in{\mathcal H}}H\), where \({\mathcal H}\) is the space of \(K\)-rational hyperplanes in \(K^{d+1}\). It is equipped with an action of \(G=\mathrm{GL}_{d+1}(K)\). This article describes the integral \(p\)-adic cohomology groups \(H_{et}^i({\mathbb H}_K^d,{\mathbb Z}_p(i))\) for \(i\ge0\). Let \(\mathrm{Sp}_i({\mathbb Z}_p)\) denote the \(i\)-th generalized Steinberg representation of \(G\) with coefficients in \({\mathbb Z}_p\). Make the similar definitions with \({\mathbb F}_p\) instead of \({\mathbb Z}_p\). {Theorem 1.1} There are compatible topological isomorphisms of \(G\times {\mathcal G}_K\)-modules \[ H_{et}^i({\mathbb H}_K^d,{\mathbb Z}_p(i))\cong \mathrm{Sp}_i({\mathbb Z}_p)^*,\quad H_{et}^i({\mathbb H}_K^d,{\mathbb F}_p(i))\cong\mathrm{Sp}_i({\mathbb F}_p)^*. \] For \(i>d\), these cohomology groups are trivial. The key input for the proof of Theorem 1.1 is the pro-ordinarity of the standard semistable model \({\mathfrak X}_{{\mathcal O}_K}\) of \({\mathbb H}_K^d\) over \({\mathcal O}_K\), a result due to the reviewer; more precisely, he proved that \[ H^i({\mathfrak X}_{{\mathcal O}_K},\Omega^j_{{\mathfrak X}_{{\mathcal O}_K}})=0\text{ for }i\ge0, j\ge0, \] where \(\Omega^{\bullet}_{{\mathfrak X}_{{\mathcal O}_K}}\) is the logarithmic de Rham complex of \({\mathfrak X}_{{\mathcal O}_K}\) over \({\mathcal O}_K\). Further inputs are results of Bhatt, Morrow and Scholze, adapted by Cesnavicius and Koshiwaka, on the \(p\)-adic étale cohomology of proper rigid analytic spaces with semistable reduction, the rigidity (due to the reviewer) of \(G\)-invariant lattices in \(\mathrm{Sp}_i({\mathbb Q}_p)\) and previous work of the authors. However, to have all these inputs acting as desired, a new key actor is needed, \(A_{\mathrm{inf}}\)-cohomology. Here \(A_{\mathrm{inf}}=W({\mathcal O}_C^{\flat})\) is Fontaine's ring associated to \(C\); its construction can be generalized to produce a sheaf \(A\Omega_{\mathfrak X}\) of \(A_{\mathrm{inf}}\)-modules on \(\mathfrak X=\mathfrak X_{\mathcal O_K}\widehat{\otimes}_{\mathcal O_K}\mathcal O_C\). These objects come with a Frobenius endomorphism \(\varphi\), and the relevant technical result then reads: {Theorem 1.4} There is a topological \({\varphi}^{-1}\)-equivariant isomorphism of \(G\times {\mathcal G}_K\)-modules \[ H_{et}^i({\mathfrak X},A\Omega_{\mathfrak X}\{i\})\cong A_{\mathrm{inf}}\widehat{\otimes}_{\mathbb Z_p}\mathrm{Sp}_i({\mathbb Z}_p)^*. \]