Overholonomicity of overconvergent $$F$$-isocrystals over smooth varieties.(English)Zbl 1276.14031

From the introduction:
“Let [...] $$k$$ be a perfect residue field of characteristic $$p>0$$ [...] In order to define a good category of $$p$$-adic coefficients over $$k$$-varieties stable under cohomological operations, Berthelot introduced the notion of arithmetic $${\mathcal D}$$-modules and their cohomological operations. [...] Also, he defined holonomic $$F$$-complexes of arithmetic $${\mathcal D}$$-modules. Virrion checked the stability of holonomicity under the dual functor. Berthelot conjectured its stability under the other Grothendieck operations. [...]
In order to avoid these conjectures and to get a category of $$F$$-complexes of arithmetic $${\mathcal D}$$-modules that satisfies these stability conditions, the first step was to introduce the notion of overcoherence [...] To improve the stability properties, we defined the category of overholonomic $$F$$-complexes over $$k$$-varieties [...] We got the stability of overholonomicity by direct images, extraordinary direct images, extraordinary inverse images, and inverse images. Moreover, it is already known that this category of $$p$$-adic coefficients is not zero since it contains unit-root overconvergent $$F$$-isocrystals [...] Now it remains to check the stability of overholonomicity by (internal and external) tensor products. The second step [...] established that $$F$$-complexes dévissable in overconvergent $$F$$-isocrystals are stable under tensor products. The third step is to prove that the notions (still with Frobenius structure) of overcoherence, overholonomicity and d{e}vissability in overconvergent $$F$$-isocrystals are identical. With what we have proved in the first and second steps, the equality between the overholonomicity and d{e}vissability in overconvergent $$F$$-isocrystals implies that the overholonomicity is stable under Grothendieck’s aforesaid six cohomological operations.”
The purpose of the present paper is to perform this indicated third step. For this, the decisive geometric input needed is Kedlaya’s semistable reduction theorem [K. Kedlaya, Semistable reduction for overconvergent $$F$$-isocrystals, I, II, III, IV].
From the introduction:
”The technical key point is a comparison theorem between the relative logarithmic rigid cohomology and rigid cohomology [...] This fundamental key point was checked by the second author, and the fact that this implies the overholonomicity of log-extendable overconvergent $$F$$-isocrystals was checked by the first one.”
It should be mentioned that, besides arithmetic $${\mathcal D}$$-modules themselves, also various other basic constructions originally introduced by Berthelot (e.g. quasi coherence on formal schemes, modified here as quasi coherence on formal log-schemes) play an important role here.

MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology
Full Text:

References:

 [1] Y. André and F. Baldassarri, De Rham Cohomology of Differential Modules on Algebraic Varieties, Basel: Birkhäuser, 2001, vol. 189. · Zbl 0995.14003 [2] F. Baldassarri and P. Berthelot, ”On Dwork cohomology for singular hypersurfaces,” in Geometric Aspects of Dwork Theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 177-244. · Zbl 1117.14022 [3] F. Baldassarri and B. Chiarellotto, ”On Christol’s theorem. A generalization to systems of PDEs with logarithmic singularities depending upon parameters,” in $$p$$-Adic Methods in Number Theory and Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1992, vol. 133, pp. 1-24. · Zbl 0768.12006 [4] F. Baldassarri and B. Chiarellotto, ”Algebraic versus rigid cohomology with logarithmic coefficients,” in Barsotti Symposium in Algebraic Geometry, San Diego, CA: Academic Press, 1994, vol. 15, pp. 11-50. · Zbl 0833.14010 [5] P. Berthelot, ”Cohomologie rigide et théorie des $$\mathcalD$$-modules,” in $$p$$-Adic Analysis, New York: Springer-Verlag, 1990, vol. 1454, pp. 80-124. · Zbl 0722.14008 [6] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres. Première Partie, 1996. [7] P. Berthelot, ”$$\mathcalD$$-modules arithmétiques. I. Opérateurs différentiels de niveau fini,” Ann. Sci. École Norm. Sup., vol. 29, iss. 2, pp. 185-272, 1996. · Zbl 0886.14004 [8] P. Berthelot, $$\mathcalD$$-Modules Arithmétiques. II. Descente par Frobenius, , 2000, vol. 81. · Zbl 0948.14017 [9] P. Berthelot, ”Introduction à la théorie arithmétique des $$\mathcalD$$-modules,” in Cohomologies $$p$$-Adiques et Applications Arithmétiques, II, , 2002, vol. 279, pp. 1-80. · Zbl 1098.14010 [10] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, New York: Springer-Verlag, 1984, vol. 261. · Zbl 0539.14017 [11] D. Caro, ”$$\mathcalD$$ modules arithmétiques surcohérents. Application aux fonctions $$L$$,” Ann. Inst. Fourier $$($$Grenoble$$)$$, vol. 54, iss. 6, pp. 1943-1996, 2004. · Zbl 1129.14030 [12] D. Caro, ”Dévissages des $$F$$-complexes de $$\mathcalD$$-modules arithmétiques en $$F$$-isocristaux surconvergents,” Invent. Math., vol. 166, iss. 2, pp. 397-456, 2006. · Zbl 1114.14011 [13] D. Caro, ”Fonctions $$L$$ associées aux $$\mathcalD$$-modules arithmétiques. Cas des courbes,” Compos. Math., vol. 142, iss. 1, pp. 169-206, 2006. · Zbl 1167.14012 [14] D. Caro, ”$$F$$-isocristaux surconvergents et surcohérence différentielle,” Invent. Math., vol. 170, iss. 3, pp. 507-539, 2007. · Zbl 1203.14025 [15] D. Caro, Sur la stabilité par produits tensoriels des $$F$$-complexes de $$\mathcalD$$-modules arithmétiques, 2007. [16] D. Caro, ”$$\mathcalD$$-modules arithmétiques surholonomes,” Ann. Sci. École Norm. Supér., vol. 42, iss. 1, pp. 141-192, 2009. · Zbl 1168.14013 [17] D. Caro, ”$$\mathcalD$$-modules arithmétiques associés aux isocristaux surconvergents. Cas lisse,” Bull. Soc. Math. France, vol. 137, iss. 4, pp. 453-543, 2009. · Zbl 1300.14021 [18] D. Caro, ”Log-isocristaux surconvergents et holonomie,” Compos. Math., vol. 145, iss. 6, pp. 1465-1503, 2009. · Zbl 1308.14018 [19] D. Caro, ”Une caractérisation de la surcohérence,” J. Math. Sci. Univ. Tokyo, vol. 16, iss. 1, pp. 1-21, 2009. · Zbl 1213.14041 [20] B. Chiarellotto and N. Tsuzuki, ”Cohomological descent of rigid cohomology for étale coverings,” Rend. Sem. Mat. Univ. Padova, vol. 109, pp. 63-215, 2003. · Zbl 1167.14306 [21] G. Christol, ”Un théorème de transfert pour les disques singuliers réguliers,” in $$p$$-Adic Cohomology, , 1984, vol. 119-120, p. 5, 151-168. · Zbl 0553.12014 [22] R. Crew, ”Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve,” Ann. Sci. École Norm. Sup., vol. 31, iss. 6, pp. 717-763, 1998. · Zbl 0943.14008 [23] R. Crew, ”Arithmetic $$\mathcalD$$-modules on a formal curve,” Math. Ann., vol. 336, iss. 2, pp. 439-448, 2006. · Zbl 1131.14018 [24] A. J. de Jong, ”Smoothness, semi-stability and alterations,” Inst. Hautes Études Sci. Publ. Math., vol. 83, pp. 51-93, 1996. · Zbl 0916.14005 [25] B. Dwork, G. Gerotto, and F. J. Sullivan, An introduction to $$G$$-functions, Princeton, NJ: Princeton Univ. Press, 1994. · Zbl 0830.12004 [26] K. S. Kedlaya, ”Semistable reduction for overconvergent $$F$$-isocrystals on a curve,” Math. Res. Lett., vol. 10, iss. 2-3, pp. 151-159, 2003. · Zbl 1057.14024 [27] K. S. Kedlaya, ”Full faithfulness for overconvergent $$F$$-isocrystals,” in Geometric Aspects of Dwork Theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 819-835. · Zbl 1087.14018 [28] K. S. Kedlaya, ”More étale covers of affine spaces in positive characteristic,” J. Algebraic Geom., vol. 14, iss. 1, pp. 187-192, 2005. · Zbl 1065.14020 [29] K. S. Kedlaya, ”Semistable reduction for overconvergent $$F$$-isocrystals. I. Unipotence and logarithmic extensions,” Compos. Math., vol. 143, iss. 5, pp. 1164-1212, 2007. · Zbl 1144.14012 [30] K. S. Kedlaya, ”Semistable reduction for overconvergent $$F$$-isocrystals. II. A valuation-theoretic approach,” Compos. Math., vol. 144, iss. 3, pp. 657-672, 2008. · Zbl 1153.14015 [31] K. S. Kedlaya, ”Semistable reduction for overconvergent $$F$$-isocrystals. III. Local semistable reduction at monomial valuations,” Compos. Math., vol. 145, iss. 1, pp. 143-172, 2009. · Zbl 1184.14031 [32] K. S. Kedlaya, ”Semistable reduction for overconvergent $$F$$-isocrystals, IV: local semistable reduction at nonmonomial valuations,” Compos. Math., vol. 147, iss. 2, pp. 467-523, 2011. · Zbl 1230.14023 [33] R. Kiehl, ”Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie,” Invent. Math., vol. 2, pp. 256-273, 1967. · Zbl 0202.20201 [34] B. Le Stum, Rigid cohomology, Cambridge: Cambridge Univ. Press, 2007. · Zbl 1131.14001 [35] M. Nagata, Local Rings, New York: Interscience Publishers a division of John Wiley & Sons, 1962, vol. 13. · Zbl 0123.03402 [36] C. Noot-Huyghe and F. Trihan, ”Sur l’holonomie de $$\mathcalD$$-modules arithmétiques associés à des $$F$$-isocristaux surconvergents sur des courbes lisses,” Ann. Fac. Sci. Toulouse Math., vol. 16, iss. 3, pp. 611-634, 2007. · Zbl 1213.14043 [37] A. Shiho, ”Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site,” J. Math. Sci. Univ. Tokyo, vol. 7, iss. 4, pp. 509-656, 2000. · Zbl 0984.14009 [38] A. Shiho, ”Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology,” J. Math. Sci. Univ. Tokyo, vol. 9, iss. 1, pp. 1-163, 2002. · Zbl 1057.14025 [39] N. Tsuzuki, ”On the Gysin isomorphism of rigid cohomology,” Hiroshima Math. J., vol. 29, iss. 3, pp. 479-527, 1999. · Zbl 1019.14007 [40] N. Tsuzuki, ”Morphisms of $$F$$-isocrystals and the finite monodromy theorem for unit-root $$F$$-isocrystals,” Duke Math. J., vol. 111, iss. 3, pp. 385-418, 2002. · Zbl 1055.14022 [41] A. Virrion, ”Dualité locale et holonomie pour les $$\mathcalD$$-modules arithmétiques,” Bull. Soc. Math. France, vol. 128, iss. 1, pp. 1-68, 2000. · Zbl 0955.14015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.