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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.

MSC:

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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References:

[1] M. ARTIN - A. GROTHENDIECK - J. L. VERDIER, Théorie des topos et cohomologie étale des schémas II, SGA4, Lecture Notes in Math., 270 (1972), Springer-Verlag. MR354653
[2] P. BERTHELOT, Cohomologie cristalline des schémas de caractéristique pD0, Lecture Notes in Math., 407 (1974), Springer-Verlag. Zbl0298.14012 MR384804 · Zbl 0298.14012
[3] P. BERTHELOT, Géométrie rigide et cohomologie des variété algébriques de caractéristique p, Bull. Soc. Math. France, Mémoire, 23 (1986), pp. 7-32. Zbl0606.14017 MR865810 · Zbl 0606.14017
[4] P. BERTHELOT, Finitude et pureté cohomologique en cohomologie rigide (avec un appendice par Aise Johan de Jong), Invent. Math., 128 (1997), pp. 329-377. Zbl0908.14005 MR1440308 · Zbl 0908.14005
[5] P. BERTHELOT, Dualité de Poincaré et formule de Künneth en cohomologie rigide, C. R. Acad. Sci. Paris, 325, Série I (1997), pp. 493-498. Zbl0908.14006 MR1692313 · Zbl 0908.14006
[6] P. BERTHELOT, A note on < Cohomologie rigide> (1989).
[7] P. BERTHELOT, Cohomologie rigide et cohomologie rigide à supports propres Première Partie, preprint 96-03, Université de Rennes (1996). MR865810
[8] S. BOSCH - U. GÜNTZER - R. REMMERT, Non-Archimedean Analysis, Grundlehren der math. Wissenschaften 261, Springer-Verlag 1984. Zbl0539.14017 MR746961 · Zbl 0539.14017
[9] A. GROTHENDIECK, Sur quelques points d’algèbre homologique, Tôhoku Math. J., 9 (1957), pp. 119-221. Zbl0118.26104 MR102537 · Zbl 0118.26104
[10] A. GROTHENDIECK, Revêtements Etales et Groupe Fondamental, SGA1, Lecture Notes in Math., 224 (1971), Springer-Verlag. MR354651
[11] A. GROTHENDIECK - J. DIEUDONNÉ, Eléments de Géométrie Algébrique I, Grundlehren der math. Wissenschaften 166, Springer-Verlag 1971. Zbl0203.23301 · Zbl 0203.23301
[12] L. GRUSON - M. RAYNAUD, Critéres de platitude et de projectivité, Invent. Math., 13 (1971), pp. 1-89. Zbl0227.14010 MR308104 · Zbl 0227.14010
[13] P. GABRIEL - M. ZISMAN, Calculus of Fractions and Homotopy theory, Ergebnisse der Math. und Ihrer Grenzgebiete, band 35 (1967). Zbl0186.56802 MR210125 · Zbl 0186.56802
[14] R. HARTSHORNE, Algebraic Geometry, Graduate texts in Math. 52, SpringerVerlag (1977). Zbl0367.14001 MR463157 · Zbl 0367.14001
[15] L. ILLUSIE, Complexe cotangent et déformations II, Lecture Notes in Math., 283 (1972), Springer-Verlag. Zbl0238.13017 MR491681 · Zbl 0238.13017
[16] R. KIEHL, Theorem A und Theorem B in der nichtarchimedischen Funktionentheories, Invent. Math., 2 (1967), pp. 256-273. Zbl0202.20201 MR210949 · Zbl 0202.20201
[17] Z. MEBKHOUT, Sur le théorème de finitude de la cohomologie p-adique d’une variété affine non singulière, Amer. J. Math., 119 (1997), pp. 1027-1081. Zbl0926.14007 MR1473068 · Zbl 0926.14007
[18] J. S. MILNE, Étale cohomology, Princeton University Press, 1980. Zbl0433.14012 MR559531 · Zbl 0433.14012
[19] M. NAGATA, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ., 3 (1963), pp. 89-102. Zbl0223.14011 MR158892 · Zbl 0223.14011
[20] M. RAYNAUD, Géométrie analytique rigide, Bull. Soc. Math. France, Mémoire, 39-40 (1974), pp. 319-327. Zbl0299.14003 MR470254 · Zbl 0299.14003
[21] J. TATE, Rigid analytic spaces, Invent. Math., 12 (1971), pp. 257-289. Zbl0212.25601 MR306196 · Zbl 0212.25601
[22] N. TSUZUKI, Cohomological descent in rigid cohomology, to appear in the volume for the Dwork Trimester in Italy. Zbl1073.14026 MR2099093 · Zbl 1073.14026
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