Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space
Author:
Elmar Große-Klönne
Journal:
J. Algebraic Geom. 14 (2005), 391-437
DOI:
https://doi.org/10.1090/S1056-3911-05-00402-9
Published electronically:
March 28, 2005
MathSciNet review:
2129006
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Abstract: We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed characteristic. For this we introduce log rigid cohomology and generalize the so-called Hyodo-Kato isomorphism to versions for non-proper $Y$, for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of $Y$. We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel’d’s symmetric space $X$ and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of $X$ given by de Shalit (2005).
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berkoV. Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Invent. Math 137 (1999), no.1, 1–84.
bercoP. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, Première partie, Prépublication IRMAR 96-03, Université de Rennes (1996).
berfiP. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), 329–377.
chiweB. Chiarellotto, Weights in rigid cohomology applications to unipotent $F$-isocrystals, Ann. Scient. École Norm. Sup. (4) 31 (1998), no. 5, 683–715.
dsE. de Shalit, Residues on buildings and de Rham cohomology of $p$-adic symmetric domains, Duke Math. J. 106 (2001), no.1, 123–191.
deshE. de Shalit, The $p$-adic monodromy-weight conjecture for $p$-adically uniformized varieties, Compositio Math. 141 (2005), 101–120.
elkR. Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Scient. École Norm. Sup. 6, pp. 553–604 (1973).
genA. Genestier, Espaces symétriques de Drinfeld, Astérisque 234 (1990).
crelleE. Große-Klönne, Rigid analytic spaces with overconvergent structure sheaf, J. reine angew. Math. 519 (2000), 73–95.
findagE. Große-Klönne, Finiteness of de Rham cohomology in rigid analysis, Duke Math. J. 113 (2002), no.1, 57–91.
monoE. Große-Klönne, The Cech filtration and monodromy in log crystalline cohomology, preprint.
coloE. Große-Klönne, Compactifications of log morphisms, Tohoku Math. J. 56 (2004), 79–104.
hyoinO. Hyodo, A note on $p$-adic étale cohomology in the semi-stable reduction case, Invent. Math 91, 543–557 (1988).
hyokaO. Hyodo and K. Kato, Semi-stable Reduction and Crystalline Cohomology with Logarithmic Poles, Asterisque 223, SMF, Paris (1994), 221–268.
illastL. Illusie, Réduction semi-stable ordinaire, cohomologie étale $p$-adique et cohomologie de de Rham d’aprés Bloch-Kato et Hyodo, Astérisque 223, SMF, Paris (1994), 209–220.
illordL. Illusie, Ordinarité des intersections complètes générales, The Grothendieck Festschrift, vol. II, 375–405, Progress in Math. vol. 87, Birkhäuser (1990).
iovspiA. Iovita and M. Spiess, Logarithmic differential forms on $p$-adic symmetric spaces, Duke Math. J. 110 (2001), no.2, 253–278.
itoT. Ito, Weight-Monodromy conjecture for $p$-adically uniformized varieties, preprint 2003.
fkatoF. Kato, Log smooth Deformation Theory. Tohoku Math. J. 48 (1996), 317–354.
kaloK. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory, J. Hopkins Univ. Press (1989), 191–224.
kidrR. Kiehl, Die de Rham Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper, Publ. Math. I.H.E.S. 33 (1967), 5–20.
merD. Meredith, Weak formal schemes, Nagoya Math. J., Vol.45, (1972), 1–38.
mokrA. Mokrane, La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. 72 (1993), 301–337.
musG. A. Mustafin, Non-Archimedean uniformization, Math. USSR Sbornik 34, 187–214 (1987).
oglogA. Ogus, $F$-crystals on schemes with constant log structure, Compositio Math. 97 (1995), 187–225.
periB. Perrin-Riou, Représentations $p$-adiques ordinaires, Astérisque 223, SMF, Paris (1994), 185–220.
rzM. Rapoport and T. Zink, Period spaces for $p$-divisible groups, Annals of Math. Studies 141, Princeton Univ. Press (1996).
schnP. Schneider, The cohomology of local systems on $p$-adically uniformized varieties, Math. Ann. 293, 623–650 (1992).
ssP. Schneider and U. Stuhler, The cohomology of $p$-adic symmetric spaces, Invent. Math. 105 (1991), no.1, 47–122.
shihoA. Shiho, Crystalline Fundamental groups II — Log Convergent Cohomology and Rigid Cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163.
steenJ. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1976), 229-257.
tateJ. Tate, $p$-divisible groups, Proc. Conf. Local Fields, Driebergen, Springer-Verlag, (1967), 158–183.
Additional Information
Elmar Große-Klönne
Affiliation:
Mathematisches Institut der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email:
klonne@math.uni-muenster.de
Received by editor(s):
April 18, 2003
Received by editor(s) in revised form:
October 15, 2004
Published electronically:
March 28, 2005
Additional Notes:
Partly supported by Deutsche Forschungs Gemeinschaft
