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Towards non–abelian $p$–adic Hodge theory in the good reduction case
About this Title
Martin C. Olsson
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 210, Number 990
ISBNs: 978-0-8218-5240-8 (print); 978-1-4704-0607-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00625-2
Published electronically: September 15, 2010
MSC: Primary 14-XX; Secondary 11-XX
Table of Contents
Chapters
- 1. Introduction
- 2. Review of some homotopical algebra
- 3. Review of the convergent topos
- 4. Simplicial presheaves associated to isocrystals
- 5. Simplicial presheaves associated to smooth sheaves
- 6. The comparison theorem
- 7. Proofs of –
- 8. A base point free version
- 9. Tangential base points
- 10. A generalization
- A. Exactification
- B. Remarks on localization in model categories
- C. The coherator for algebraic stacks
- D. $\widetilde B_{\mathrm {cris}}(V)$-admissible implies crystalline.
Abstract
We develop a non–abelian version of $p$–adic Hodge Theory for varieties (possibly open with “nice compactification”) with good reduction. This theory yields in particular a comparison between smooth $p$–adic sheaves and $F$–isocrystals on the level of certain Tannakian categories, $p$–adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups.- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI 10.1007/BF01390174
- Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354653
- Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 0491705
- David Blanc, New model categories from old, J. Pure Appl. Algebra 109 (1996), no. 1, 37–60. MR 1386812, DOI 10.1016/0022-4049(95)00078-X
- Benjamin A. Blander, Local projective model structures on simplicial presheaves, $K$-Theory 24 (2001), no. 3, 283–301. MR 1876801, DOI 10.1023/A:1013302313123
- Spencer Bloch and Igor Kříž, Mixed Tate motives, Ann. of Math. (2) 140 (1994), no. 3, 557–605. MR 1307897, DOI 10.2307/2118618
- A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
- P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168, DOI 10.1007/978-1-4613-9649-9_{3}
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- Daniel Dugger, Sharon Hollander, and Daniel C. Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51. MR 2034012, DOI 10.1017/S0305004103007175
- Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Gerd Faltings, Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80. MR 1463696
- Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705, DOI 10.1090/S0894-0347-1988-0924705-1
- Gerd Faltings, Almost étale extensions, Astérisque 279 (2002), 185–270. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922831
- Jean-Marc Fontaine, Représentations $p$-adiques semi-stables, Astérisque 223 (1994), 113–184 (French). With an appendix by Pierre Colmez; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293972
- Jean-Marc Fontaine, Sur certains types de représentations $p$-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), no. 3, 529–577 (French). MR 657238, DOI 10.2307/2007012
- Jean-Marc Fontaine, Cohomologie de de Rham, cohomologie cristalline et représentations $p$-adiques, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 86–108 (French). MR 726422, DOI 10.1007/BFb0099959
- Jean-Marc Fontaine, Le corps des périodes $p$-adiques, Astérisque 223 (1994), 59–111 (French). With an appendix by Pierre Colmez; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293971
- Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612, DOI 10.1007/978-3-0348-8707-6
- Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
- Richard M. Hain, Torelli groups and geometry of moduli spaces of curves, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93) Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 97–143. MR 1397061
- Richard Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. MR 1431828, DOI 10.1090/S0894-0347-97-00235-X
- Richard M. Hain, Completions of mapping class groups and the cycle $C-C^-$, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 75–105. MR 1234261, DOI 10.1090/conm/150/01287
- Richard Hain and Eduard Looijenga, Mapping class groups and moduli spaces of curves, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 97–142. MR 1492535, DOI 10.1090/pspum/062.2/1492535
- Richard Hain and Makoto Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of $\Bbb P^1-\{0,1,\infty \}$, Compositio Math. 139 (2003), no. 2, 119–167. MR 2025807, DOI 10.1023/B:COMP.0000005077.42732.93
- V. A. Hinich and V. V. Schechtman, On homotopy limit of homotopy algebras, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 240–264. MR 923138, DOI 10.1007/BFb0078370
- Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- Luc Illusie, Autour du théorème de monodromie locale, Astérisque 223 (1994), 9–57 (French). Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293970
- Luc Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271–322. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922832
- J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87. MR 906403, DOI 10.1016/0022-4049(87)90100-9
- Roy Joshua, Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks, Adv. Math. 209 (2007), no. 1, 1–68. MR 2294217, DOI 10.1016/j.aim.2006.04.005
- Franz W. Kamber and Philippe Tondeur, Invariant differential operators and cohomology of Lie algebra sheaves, Differentialgeometrie im Großen (Tagung, Math. Forschungsinst., Oberwolfach, 1969) Ber. Math. Forschungsinst. Oberwolfach, Heft 4, Bibliographisches Inst., Mannheim, 1971, pp. 177–230. MR 0370616
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Kazuya Kato and Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\textbf {C}$, Kodai Math. J. 22 (1999), no. 2, 161–186. MR 1700591, DOI 10.2996/kmj/1138044041
- L. Katzarkov, T. Pantev, and B. Toën, Schematic homotopy types and non-abelian Hodge theory, Compos. Math. 144 (2008), no. 3, 582–632. MR 2422341, DOI 10.1112/S0010437X07003351
- L. Katzarkov, T. Pantev, and B. Toën, Algebraic and topological aspects of the schematization functor, Compos. Math. 145 (2009), no. 3, 633–686. MR 2507744, DOI 10.1112/S0010437X09004096
- Kiran S. Kedlaya, Fourier transforms and $p$-adic ‘Weil II’, Compos. Math. 142 (2006), no. 6, 1426–1450. MR 2278753, DOI 10.1112/S0010437X06002338
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
- Jean-Yves Étesse and Bernard Le Stum, Fonctions $L$ associées aux $F$-isocristaux surconvergents. I. Interprétation cohomologique, Math. Ann. 296 (1993), no. 3, 557–576 (French). MR 1225991, DOI 10.1007/BF01445120
- Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Hiroaki Nakamura, Galois rigidity of profinite fundamental groups [translation of Sūgaku 47 (1995), no. 1, 1–17; MR1362515 (98d:14027)], Sugaku Expositions 10 (1997), no. 2, 195–215. Sugaku Expositions. MR 1600655
- Arthur Ogus, $F$-crystals, Griffiths transversality, and the Hodge decomposition, Astérisque 221 (1994), ii+183. MR 1280543
- Arthur Ogus, The convergent topos in characteristic $p$, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 133–162. MR 1106913, DOI 10.1007/978-0-8176-4576-2_{5}
- Martin C. Olsson, $F$-isocrystals and homotopy types, J. Pure Appl. Algebra 210 (2007), no. 3, 591–638. MR 2324594, DOI 10.1016/j.jpaa.2006.10.006
- Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747–791 (English, with English and French summaries). MR 2032986, DOI 10.1016/j.ansens.2002.11.001
- Martin C. Olsson, On Faltings’ method of almost étale extensions, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 811–936. MR 2483956, DOI 10.1090/pspum/080.2/2483956
- J. P. Pridham, Galois actions on homotopy groups, preprint (2008).
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- Neantro Saavedra Rivano, Catégories tannakiennes, Bull. Soc. Math. France 100 (1972), 417–430 (French). MR 320015
- Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
- Atsushi Shiho, Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), no. 4, 509–656. MR 1800845
- Atsushi Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163. MR 1889223
- Atsushi Shiho, Crystalline fundamental groups and $p$-adic Hodge theory, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 381–398. MR 1738868, DOI 10.1090/crmp/024/19
- R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
- Bertrand Toën, Champs affines, Selecta Math. (N.S.) 12 (2006), no. 1, 39–135 (French, with English summary). MR 2244263, DOI 10.1007/s00029-006-0019-z
- —, Dualité de Tannaka supérieure, preprint available at http://math.unice.fr/˜toen.
- Takeshi Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233–411. MR 1705837, DOI 10.1007/s002220050330
- —, Crystalline sheaves, syntomic cohomology, and $p$–adic polylogarithms, notes from a seminar at Cal Tech on Feb. 20, 2001.
- Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104. MR 2223406
- Vadim Vologodsky, Hodge structure on the fundamental group and its application to $p$-adic integration, Mosc. Math. J. 3 (2003), no. 1, 205–247, 260 (English, with English and Russian summaries). MR 1996809, DOI 10.17323/1609-4514-2003-3-1-205-247