Cohomological descent of rigid cohomology for étale coverings. (English) Zbl 1167.14306

Let \(k\) be a field of characteristic \(p>0\). Let \(V\) be a complete discrete valuation ring with fraction field \(K\) of characteristic \(0\) and with \(k\) as its residue field. For a separated \(k\)-scheme \(X\) of finite type, P. Berthelot defined the category of overconvergent isocrystals on \(X\) with values in \(K\) as those \(p\)-adic coefficient categories which should be viewed as analogues of lisse \( l\)-adic sheaves (\( l\neq p\)). For such an overconvergent isocrystal \(E\) on \(X\) he defined the rigid cohomology groups \(H_{\text{rig}}^*(X,E)\), which are \(K\)-vector spaces. They are conjectured to be finite-dimensional: P. Berthelot proved this in the case where \(E\) is the constant overconvergent isocrystal and \(X\) is smooth [Invent. Math. 128, 329–377 (1997; Zbl 0908.14005)]; the reviewer then proved it for general \(X\) (still with \(E\) constant) [Duke Math. J. 113, No. 1, 57–91 (2002; Zbl 1057.14023)]; for general \(E\) but smooth \(X\) the finiteness theorem is due to Kedlaya; for general \(E\) and \(X\) it seems to be open at present.
In the paper under review the authors develop the theory of cohomological descent as a means to compute \(H_{\text{rig}}^*(X,E)\) in terms of the rigid cohomology of \(E\) on étale hypercoverings of \(X\). This is an important contribution to the foundations of rigid cohomology. In fact they establish such an étale descent theory for the computation of the cohomology of coherent module sheaves over algebras of overconvergent functions. As rigid cohomology is the cohomology of de Rham complexes with coefficients in locally free module sheaves over algebras of overconvergent functions, this implies the said étale descent for rigid cohomology.
Let us indicate slightly more technically what they do, without giving full definitions here. A triple \(\mathfrak X=(X,\overline{X},{\mathcal X})\) is a sequence of \(V\)-morphisms \(X\overset{j}{}{\overline X}\overset{i}{}{\mathcal X}\), where \({\mathcal X}\) is a formal \(V\)-scheme of topologically finite type, \({\overline{X}}\) and \(X\) are \(k\)-schemes of finite type, \(j\) is an open immersion and \(i\) is a closed immersion. Such objects are at the core of the definition of rigid cohomology: Associated to \({\mathcal X}\) is a \(K\)-rigid space \({\mathcal X}_K\) together with a specialization map sp\(: {\mathcal X}_K\to {\mathcal X}_k\). Set \(]X[_{\mathcal X}=\text{sp}^{-1}(X)\) and \(]{\overline{X}}[_{\mathcal X}=\text{sp}^{-1}(\overline{X})\). Let \(j^{\dagger}{\mathcal O}_{{\mathcal X}_K}\) be the sheaf on \(]\overline{X}[_{\mathcal X}\) consisting of holomorphic functions on \(]X[_{\mathcal X}\) which extend to holomorphic functions on some (not prescribed) strict neighbourhood of \(]X[_{\mathcal X}\) in \(]\overline{X}[_{\mathcal X}\). If \({\mathcal X}/V\) is smooth, then the cohomology of the de Rham complex on \(]\overline{X}[_{\mathcal X}\) with coefficients in \(j^{\dagger}\to{\mathcal O}_{{\mathcal X}_K}\) is the rigid cohomology \(H_{\text{rig}}^*(X/K)\) of \(X\).
The main result of the paper now states that if \(w_{\bullet}:\mathfrak Y_{\bullet}\to\mathfrak X\) is the hypercovering – with simplicial triple \(\mathfrak Y_{\bullet}=(Y_{\bullet},\overline{Y}_{\bullet}, {\mathcal Y}_{\bullet})\) – which is the Čech diagram obtained from a morphism of triples \(w: \mathfrak Y\to\mathfrak X\), then \(w_{\bullet}\) is universally cohomological descendable (“universal” means that base changes are allowed) if the following holds: \(w\) is smooth around \(Y\), \(\overline{Y}\to \overline{X}\) is étale surjective and \(Y=X\times_{\overline{X}}\overline{Y}\). Similarly if \(w\) is smooth around \(Y\), \({Y}\to {X}\) is étale surjective and \(\overline{Y}\to \overline{X}\) is proper.
All the details are worked out in full and the machinery of cohomological descent, adapted to the present situation, is thoroughly presented. As one application a definition of rigid cohomology in terms of hypercoverings is given, equivalent to Berthelot’s one. It is shown that the covering spectral sequence associated with a hypercovering behaves well with respect to Frobenius structures.
The results of the paper are important ingredients in Tsuzuki’s proof that rigid cohomology satisfies cohomological descent for proper hypercoverings [N. Tsuzuki, Invent. Math. 151, No. 1, 101–133 (2003; Zbl 1085.14019)]. This allowed him to re-prove the above-mentioned finiteness of rigid cohomology with constant coefficients in the non-smooth case.
A descent theory for overconvergent isocrystals has been established in [J.-Y. Etesse, Ann. Sci. Ec. Norm. Sup., IV. Sér. 35, No. 4, 575–603 (2002; Zbl 1060.14028)]. There, different methods are used.


14F30 \(p\)-adic cohomology, crystalline cohomology
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14G22 Rigid analytic geometry
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