Journal of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6834(e) ISSN 0894-0347(p)

-adic periods and derived de Rham cohomology

Author: A. Beilinson
Journal: J. Amer. Math. Soc. 25 (2012), 715-738
MSC (2010): Primary 14F30, 14F40; Secondary 14F20
Posted: January 27, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that derived de Rham cohomology of Illusie satisfies the -adic Poincaré lemma in h-topology. This yields a new construction of the -adic period map and a simple proof of Fontaine's C $_{\text {dR}}$ conjecture.

References

[B] A. Beilinson, On the crystalline period map, math. AG 1111.3316 (2011).
[Ber] Laurent Berger, Représentations 𝑝-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219–284 (French, with English summary). MR 1906150 (2004a:14022), http://dx.doi.org/10.1007/s002220100202
[Bh1] Bhargav Bhatt, Derived direct summands, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Princeton University. MR 2753219
[Bh2] B. Bhatt, -adic derived de Rham cohomology, 2012.
[Col] P. Colmez, Les nombres algébriques sont denses dans B $^{+}_{\text {dR}}$ , Périodes -adiques, Astérisque 223, Soc. Math. France, 1994, pp. 103-111.
[Con] B. Conrad, Cohomological descent, http://math.stanford.edu/˜conrad/.
[D] Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 0498552 (58 #16653b)
[dJ1] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020 (98e:14011)
[dJ2] A. Johan de Jong, Families of curves and alterations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 599–621. MR 1450427 (98f:14019)
[Fa1] Gerd Faltings, 𝑝-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705 (89g:14008), http://dx.doi.org/10.1090/S0894-0347-1988-0924705-1
[Fa2] Gerd Faltings, Almost étale extensions, Astérisque 279 (2002), 185–270. Cohomologies 𝑝-adiques et applications arithmétiques, II. MR 1922831 (2003m:14031)
[FC] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353 (92d:14036)
[Far] L. Fargues, Letter to L. Illusie, 2010.
[F1] 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 (84d:14010), http://dx.doi.org/10.2307/2007012
[F2] Jean-Marc Fontaine, Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux, Invent. Math. 65 (1981/82), no. 3, 379–409 (French). MR 643559 (83k:14023), http://dx.doi.org/10.1007/BF01396625
[F3] 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 (95k:11086)
[F4] 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 (95g:14024)
[Gr] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103. MR 0199194 (33 #7343)
[GV] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin, 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 (50 #7131)
[HS] 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 (89d:55052), http://dx.doi.org/10.1007/BFb0078370
[Ill1] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin, 1971 (French). MR 0491680 (58 #10886a)
[Ill2] Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin, 1972 (French). MR 0491681 (58 #10886b)
[J] Uwe Jannsen, On the 𝑙-adic cohomology of varieties over number fields and its Galois cohomology, Galois groups over 𝑄 (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 315–360. MR 1012170 (90i:11064)
[K1] 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 (99b:14020)
[K2] Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725 (95g:14056), http://dx.doi.org/10.2307/2374941
[N1] Wiesława Nizioł, Semistable conjecture via 𝐾-theory, Duke Math. J. 141 (2008), no. 1, 151–178. MR 2372150 (2008m:14035), http://dx.doi.org/10.1215/S0012-7094-08-14114-6
[N2] Wiesława Nizioł, 𝑝-adic motivic cohomology in arithmetic, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 459–472. MR 2275605 (2007k:14029)
[N3] Wiesława Nizioł, On uniqueness of 𝑝-adic period morphisms, Pure Appl. Math. Q. 5 (2009), no. 1, 163–212. MR 2520458 (2011c:14058)
[Ol] Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. MR 2195148 (2006j:14017), http://dx.doi.org/10.1007/s00208-005-0707-6
[R] M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 27–76 (French). MR 0282993 (44 #227)
[RG] Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 0308104 (46 #7219)
[SD] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin, 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 (50 #7131)
[SV] Andrei Suslin and Vladimir Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), no. 1, 61–94. MR 1376246 (97e:14030), http://dx.doi.org/10.1007/BF01232367
[T] Michael Temkin, Stable modification of relative curves, J. Algebraic Geom. 19 (2010), no. 4, 603–677. MR 2669727 (2011j:14064), http://dx.doi.org/10.1090/S1056-3911-2010-00560-7
[Ts1] 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 (2000m:14024), http://dx.doi.org/10.1007/s002220050330
[Ts2] Takeshi Tsuji, Semi-stable conjecture of Fontaine-Jannsen: a survey, Astérisque 279 (2002), 323–370. Cohomologies 𝑝-adiques et applications arithmétiques, II. MR 1922833 (2003g:14028)
[V1] J.-L. Verdier, Topologies et faisceaux, Théorie des topos et cohomologie étale de schémas (SGA 4), Tome 1, Lect. Notes in Math. 269, Springer-Verlag, 1972, pp. 219-264.
[V2] J.-L. Verdier, Fonctorialité de catégories de faisceaux, Théorie des topos et cohomologie étale de schémas (SGA 4), Tome 1, Lect. Notes in Math. 269, Springer-Verlag, 1972, pp. 265-298.
[V3] J.-L. Verdier, Cohomologie dans les topos, Théorie des topos et cohomologie étale de schémas (SGA 4), Tome 2, Lect. Notes in Math. 270, Springer-Verlag, 1972, pp. 1-82.
[Y] G. Yamashita, Théorie de Hodge -adique pour les variétés ouvertes, C. R. A. S. 349 (21-22) (2011), 1127-1130.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|