\(p\)-adic Hodge theory for rigid-analytic varieties (Q2849818)

From the introduction:NEWLINENEWLINE``This paper starts to investigate to what extent \(p\)-adic comparison theorems stay true for rigid-analytic varieties. Up to now, such comparison isomorphisms were mostly studied for schemes over \(p\)-adic fields, but we intend to show here that the whole theory extends naturally to rigid analytic varieties over \(p\)-adic fields. This is of course in analogy with classical Hodge theory which most naturally is formulated in terms of complex-analytic spaces.''NEWLINENEWLINETheorem 1.1. Let \(K\) be a complete algebraically closed extension of \({\mathbb Q}_p\), let \(X/K\) be a proper smooth rigid analytic variety, and let \({\mathbb L}\) be an \({\mathbb F}_p\)-local system on \(X_{\text{ét}}\). Then \(H^i(X_{\text{ét}},{\mathbb L})\) is a finite dimensional \({\mathbb F}_p\)-vector space for all \(i>0\), which vanishes for \(i>\dim(X)\).NEWLINENEWLINE``We build on Faltings' theory of almost étale extensions, amplified by the theory of perfectoid spaces. [...] We introduce the pro-étale site \(X_{\text{proét}}\) whose open subsets are roughly of the form \(V\to U\to X\), where \(U\to X\) is some étale morphism, and \(V\to U\) is an inverse limit of finite étale maps. Then the local structure of \(X\) in the pro-étale topology is simpler, namely it is locally perfectoid. This amounts to extracting lots of \(p\)-power roots of units in the tower \(V\to U\).''NEWLINENEWLINETheorem 1.2. Let \(X\) be a connected affinoid rigid-analytic variety over \(K\). Then \(X\) is a \(K(\pi,1)\) for \(p\)-torsion coefficients, i.e. for all \(p\)-torsion local systems \({\mathbb L}\) on \(X\), the natural mapNEWLINE\[NEWLINEH_{\text{cont}}^i(\pi_1(X,x),{\mathbb L}_x)\longrightarrow H^i(X_{\text{ét}},{\mathbb L})NEWLINE\]NEWLINEis an isomorphism. Here, \(x\in X(K)\) is a base point, and \(\pi_1(X,x)\) denotes the profinite étale fundamental group.NEWLINENEWLINE``This theorem implies that \(X\) is `locally contractible' in the pro-étale site, at least for \(p\)-torsion coefficients.''NEWLINENEWLINETheorem 1.3. In the situation of Theorem 1.1, there is an almost isomorphism of \({\mathcal O}_K\)-modules for all \(i\geq0\), NEWLINE\[NEWLINEH^i(X_{\text{ét}},{\mathbb L})\otimes {\mathcal O}_K/p\longrightarrow H^i(X_{\text{ét}},{\mathbb L}\otimes {\mathcal O}_X^+/p).NEWLINE\]NEWLINE More generally, assume that \(f:X\to Y\) is a proper smooth morphism of rigid analytic varieties over \(K\), and \({\mathbb L}\) is an \({\mathbb F}_p\)-local system on \(X_{\text{ét}}\). Then there is an almost isomorphism for all \(i\geq0\), NEWLINE\[NEWLINE(R^if_{\text{ét*}}{\mathbb L})\otimes{\mathcal O}_Y^+/p\longrightarrow R^if_{\text{ét*}}({\mathbb L}\otimes{\mathcal O}_X^+/p).NEWLINE\]NEWLINENEWLINENEWLINE``[...] we introduce sheaves on \(X_{\text{proét}}\), which we call period sheaves, as their values on pro-étale covers of \(X\) give period rings. Among them is the sheaf \({\mathbb B}_{dR}^+\), which is the relative version of Fontaine's ring \(B_{dR}^+\). Let \({\mathbb L}\) be [a] lisse \({\mathbb Z}_p\)-sheaf on \(X\). In our setup, we can define it as a locally free \(\hat{\mathbb Z}_p\)-module on \(X_{\text{proét}}\), where \(\hat{\mathbb Z}_p=\text{lim}_{\leftarrow}{\mathbb Z}/p^n {\mathbb Z}\) as sheaves on \(X_{\text{proét}}\). Then \({\mathbb L}\) gives rise to a \(B_{dR}^+\)-local system \({\mathbb M}={\mathbb L}\otimes_{\hat{\mathbb Z}_p}{\mathbb B}_{dR}^+\) on \(X_{\text{proét}}\), and it is a formal consequence of Theorem 1.3 that NEWLINE\[NEWLINEH^i(X_{\text{ét}},{\mathbb L})\otimes B_{dR}^+\cong H^i(X_{\text{proét}},{\mathbb M}).NEWLINE\]NEWLINENEWLINENEWLINETheorem 1.5. Let \(X\) be a smooth rigid analytic variety over \(k\), where \(k\) is a complete discretely valued nonarchimedean extension of \({\mathbb Q}_p\) with perfect residue field. Then there is a fully faithful functor from the category of filtered \({\mathcal O}_X\)-modules with an integrable connection satisfying Griffiths transverslity, to the category of \({\mathbb B}_{dR}^+\)-local systems.''NEWLINENEWLINETheorem 1.6. Let \(k\) is a complete discretely valued nonarchimedean extension of \({\mathbb Q}_p\) with perfect residue field \(\kappa\), and algebraic closure \(\bar{k}\), and let \(X\) be a proper smooth rigid-analytic variety over \(k\). For any lisse \({\mathbb Z}_p\)-sheaf \({\mathbb L}\) on \(X\) with associated \({\mathbb B}_{dR}^+\)-local system \({\mathbb M}\), we have a \(\text{Gal}(\overline{k}/k)\)-equivariant isomorphism NEWLINE\[NEWLINEH^i(X_{\overline{k}},{\mathbb L})\otimes_{{\mathbb Z}_p}B_{dR}^+\cong H^i(X_{\overline{k}},{\mathbb M}).NEWLINE\]NEWLINEIf \({\mathbb L}\) is de Rham, with associated filtered module with integrable connection \(({\mathcal E},\nabla,\text{Fil}^{\bullet})\), then the Hodge-de Rham spectral sequenceNEWLINE\[NEWLINEH_{\text{Hodge}}^{i-j,j}(X,{\mathcal E})\Rightarrow H_{dR}^i(X,{\mathcal E})NEWLINE\]NEWLINE degenerates. Moreover, \(H^i(X_{\overline{k}},{\mathbb L})\) is a de Rham representation of \(\text{Gal}(\overline{k}/k)\) with associated filtered \(k\)-vector space \(H_{dR}^i(X,{\mathcal E})\). In particular, there is a \(\text{Gal}(\overline{k}/k)\)-equivariannt isomorphismNEWLINE\[NEWLINEH^i(X_{\overline{k}},{\mathbb L})\otimes_{{\mathbb Z}_p}\hat{\overline{k}}\cong\bigoplus_j H_{\text{Hodge}}^{i-j,j}(X,{\mathcal E})\otimes_k \hat{\overline{k}}(j).NEWLINE\]NEWLINENEWLINENEWLINE``Interestingly, no `Kähler' assumption is necessary for this result in the \(p\)-adic case.''NEWLINENEWLINE[..]NEWLINENEWLINE``We note that this proof [of theorem 1.6] is direct: All desired isomorphisms are proved by a direct argument, and not by producing a map between two cohomology theories and then proving that it has to be an isomorphism by abstract arguments.''NEWLINENEWLINE[..]NEWLINENEWLINE``It also turns out that our methods are flexible enough to handle the relative case.''