Holmstrom Crystalline cohomology

Crystalline cohomology

The group $H^q_{crys}(X / W(k))$ is isomorphic to the cohomology $H^q(X, W_{\bullet} \Omega_X^*)$ of the de Rham-Witt complex. See Bloch: Algebraic K-theory and crystalline cohomology, and Illusie: Complexe de de Rham-Witt et cohomologie cristalline.

MR0978241 (90c:11042) Étesse, Jean-Yves(F-RENNB-isomorphism) Rationalité et valeurs de fonctions $L$ en cohomologie cristalline. (French. English summary) [Rationality and values of -functions in crystalline cohomology] Ann. Inst. Fourier (Grenoble) 38 (1988), no. 4, 33–92. 11G25 (14F30 14G10) PDF Doc Del Clipboard Journal Article Make Link

This paper studies the $L$-series of $F$-crystals on schemes over finite fields. The author first derives various compatibilities for cycle classes in the crystalline cohomology of a locally free $\scr O$-module, defined using either Poincaré duality or the Gysin map, and applies the compatibilities to prove a Lefschetz trace formula, thereby generalizing results of \n P. Berthelot\en [Cohomologie cristalline des schemas de caracteristique , Lecture Notes in Math., 407, Springer, Berlin, 1974; MR0384804 (52 #5676)]. This then enables him to prove, in the usual way, that the $L$-series of a locally free $F$-crystal $E$ on a scheme $X$ proper and smooth over a finite field $k$ is a rational function with coefficients in the Witt vectors of $k$. In particular, the function is meromorphic, which proves a conjecture of \n N. M. Katz\en [in Seminaire Bourbaki, 24`eme annee (1971/1972), Exp. No. 409, 167200, Lecture Notes in Math., 317, Springer, Berlin, 1973; MR0498577 (58 #16672)] in this context.

When the $F$-crystal $E$ is a unit-root crystal, the author gives an interpretation of the zeros and poles of the $L$-function of the form $q^ru$, $u$ a $p$-adic unit, in terms of the $p$-adic étale sheaves $\nu(E,r)$. For $r=0$, this again proves a conjecture of Katz in this context.

The last part of the paper is concerned with extending the reviewer’s results [Amer. J. Math. 108 (1986), no. 2, 297360; MR0833360 (87g:14019)] on the behaviour of the zeta function of $X$ near integers to the $L$-series of a unit-root $F$-crystal $E$ on $X$. Under certain standard assumptions, it is shown that L(X,E,t)\sim\break c_0·\chi(X,\bold Z_p(E,r))·q^{\chi(X,E,r)}·(1-q^rt)^{-\rho_r} as $t\to q^{-r}$, where $c_0$ is a $p$-adic unit, $\chi(X,\bold Z_p(E,r))$ is a certain Euler-Poincaré characteristic involving a regulator term, and $\chi(X,E,r)$ is defined in terms of the sheaves $E'\otimes\Omega^i_X$, where $E'$ is the $\scr O_X$-module associated with $E$.

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Crystalline cohomology

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Crystalline cohomology

AG (Algebraic geometry)

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Crystalline cohomology

Charp, Pure

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Crystalline cohomology

See also Sheaf cohomology, Weil cohomology

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Crystalline cohomology

Illusie: Complexe de de Rham-Witt et cohomologie cristalline

Berthelot: LNM407

Survey by Illusie in Motives volumes.

Gillet and Messing: Cycle classes and Riemann-Roch for crystalline cohomology

Crystalline cohomology of algebraic stacks and Hyodo-Kato cohomology . Asterisque 316.

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Crystalline cohomology

Nicholas Ring on cycle classes.

Gillet and Messing: Cycle classes and RR for crystalline cohomology. “Cycle map, factors through cycles mod algebraic equivalence” (?)

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Crystalline cohomology

http://mathoverflow.net/questions/68500/poincare-duality-for-crystalline-cohomology

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Crystalline cohomology

Chern classes in crystalline cohom, see Illusie in C R Acad approx 1969.

http://mathoverflow.net/questions/46469/can-one-compare-integral-structures-on-de-rham-and-crystalline-cohomology

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Crystalline cohomology

http://math.columbia.edu/~dejong/wordpress/?p=1908 de Jong on crystalline cohomology

http://mathoverflow.net/questions/11648/current-status-of-crystalline-cohomology

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Crystalline cohomology

http://mathoverflow.net/questions/20381/crystalline-cohomology-of-abelian-varieties

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Crystalline cohomology

Note: When starting looking at p-adic cohomology, there are various surveys/introductions by Illusie and Kedlaya to look at.

Wikipedia

A proof of the Crystalline conjecture: Niziol

A survey by Niziol on “p-adic motivic cohomology”.

Bloch: Crystals and de Rham-Witt connections: http://www.ams.org/mathscinet-getitem?mr=2074428

http://www.ams.org/mathscinet-getitem?mr=0565469 review for cryst cohom intro.

http://mathoverflow.net/questions/56753/learning-crystalline-cohomology

http://mathoverflow.net/questions/11648/current-status-of-crystalline-cohomology

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Crystalline cohomology

http://math.columbia.edu/~dejong/wordpress/?p=2227 Stacks project on finiteness and comparison with de Rham cohomology.

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Crystalline cohomology

Bhatt and de Jong article on comparison with de Rham cohomology: http://front.math.ucdavis.edu/1110.5001

Bloch: Algebraic K-theory and crystalline cohomology

Feigin, B.L., Tsygan, B.L.: Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen. 19, 52–62 (1985)

Atsushi Shiho, Crystalline fundamental groups and $p$-adic Hodge theory (381–398);

Mokrane: Cohomologie cristalline des varietes ouvertes (1993)

Yamasita: p-adic étale cohomology and crystalline cohomology of open varieties

arXiv:1205.1597 Torsion in the crystalline cohomology of singular varieties from arXiv Front: math.AG by Bhargav Bhatt This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.

nLab page on Crystalline cohomology