Crystalline cohomology

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In mathematics, crystalline cohomology is a Weil cohomology theory for schemes discovered by Grothendieck (in his letter to Tate (Grothendieck 1966) and his lecture (Grothendieck 1968)) and developed by Pierre Berthelot (1974). Its values are modules over rings of Witt vectors over the base field.

Crystalline cohomology is closely related to the (algebraic) de Rham cohomology introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p.

The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal extensions of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic p to characteristic 0 and employing an appropriate version of algebraic de Rham cohomology.

Contents

[edit] Applications

For schemes in characteristic p, crystalline cohomology theory can handle questions about p-torsion in cohomology groups better than l-adic cohomology. This makes it a natural backdrop for much of the work on p-adic L-functions.

Crystalline cohomology, from the point of view of number theory, fills a gap in the l-adic cohomology information, which occurs exactly where there are 'bad primes', traditionally the preserve of ramification theory and an important handle on arithmetic problems. Conjectures with wide scope on making this into formal statements were enunciated by Jean-Marc Fontaine. Progress on these has led to the development of p-adic Hodge theory.

[edit] de Rham cohomology

De Rham cohomology solves the problem of finding an algebraic definition of the cohomology groups (singular cohomology)

Hi(X,C)

for X a smooth complex variety. These group are the cohomology of the complex of smooth differential forms on X (with complex number coefficients) as these form a resolution of the constant sheaf C.

The algebraic de Rham cohomology is defined to be the hypercohomology of the complex of algebraic forms (Kähler differentials) on X. The smooth i-forms form an acyclic sheaf, so the hypercohomology of the complex of smooth forms is the same as its cohomology, and the same is true for algebraic sheaves of i-forms over affine varieties, but algebraic sheaves of i-forms over non-affine varieties can have non-vanishing higher cohomology groups, so the hypercohomology can differ from the cohomology of the complex.

For smooth complex varieties Grothendieck (1963) showed that the algebraic de Rham cohomology is isomorphic to the usual smooth de Rham cohomology and therefore (by de Rham's theorem) to the cohomology with complex coefficients. This definition of algebraic de Rham cohomology is available for algebraic varieties over any field k.

[edit] Coefficients

If X is a variety over an algebraically closed field of characteristic p > 0, then the l-adic cohomology groups for

l≠p

any prime number (other than p) give satisfactory cohomology groups of X, with coefficients in the ring Zl of l-adic integers. It is not possible in general to find similar cohomology groups with coefficients in the p-adic numbers (or the rationals, or the integers).

The classic reason is that if X is a supersingular elliptic curve, then its ring of endomorphisms generates a quaternion algebra over Q that is non-split, at p and infinity. If X has a cohomology group over the p-adic integers with the expected dimension 2, the ring of endomorphisms would have a 2-dimensional representation; and this is not possible as it is non-split at p. A quite subtle point is that if X is a supersingular elliptic curve over the prime field, with p elements, then its crystalline cohomology is a free rank 2 module over the p-adic integers. The argument given does not apply in this case, because some of the endomorphisms of supersingular elliptic curves are only defined over a quadratic extension of the field of order p.

Grothendieck's crystalline cohomology theory gets round this obstruction because it takes values in the ring of Witt vectors over the ground field. So if the ground field is the algebraic closure of the field of order p, its values are modules over the maximal unramified extension of the p-adic integers, a larger ring containing k-th roots of unity for all k not divisible by p, rather than over the p-adic integers.

[edit] Motivation

One idea for defining a Weil cohomology theory of a variety X over a field k of characteristic p is to 'lift' it to a variety X* over the ring of Witt vectors of k (which gives back X on reduction mod p), then take the de Rham cohomology of this lift. The problem is that it is not at all obvious that this cohomology is independent of the choice of lifting.

The idea of crystalline cohomology in characteristic 0 is to find a direct definition of a cohomology theory as the cohomology of constant sheaves on a suitable site

Inf(X)

over X, called the infinitesimal site and then show it is the same as the de Rham cohomology of any lift.

The site Inf(X) is a category whose objects can be thought of as some sort of generalization of the conventional open sets of X. In characteristic 0 its objects are infinitesimal thickenings U→T of Zariski open subsets U of X. This means that U is the closed subscheme of of a scheme T defined by a nilpotent sheaf of ideals on T; for example, Spec(Z/pZ)→ Spec(Z/pnZ).

Grothendieck showed that for smooth schemes X over C, the cohomology of the sheaf OX is the same as the usual (smooth or algebraic) de Rham cohomology.

[edit] Crystalline cohomology

In characteristic p the most obvious analogue of the crystalline site defined above in characteristic 0 does not work. The reason is roughly that in order to prove exactness of the de Rham complex, one needs some sort of Poincaré lemma; whose proof in turn uses integration, and integration requires various divided powers, which exist in characteristic 0 but not always in characteristic p. Grothendieck solved this problem by defining objects of the crystalline site of X to be roughly infinitesimal thickenings of Zariski open subsets of X, together with a divided power structure structure giving the needed divided powers.

We will work over the ring Wn = W/pnW of Witt vectors of length n over a perfect field k of characteristic of characteristic p>0. For example, k could be the finite field of order p, and Wn is then the ring Z/pnZ. (More generally one can work over a base scheme S which has a fixed sheaf of ideals I with a divided power structure.) If X is a scheme over Wn, then the crystalline site of X relative to Wn, denoted Cris(X/Wn), has as its objects pairs U→T consisting of a closed immersion of a Zariski open subset U of X into some scheme T defined by a sheaf of ideals J, together with a divided power structure on J compatible with the one on Wn.

Crystalline cohomology of a scheme X over k is defined to be the inverse limit

H^i(X/W)=\lim_{\leftarrow}H^i(X/W_n)

where

Hi(X / Wn) = Hi(Cris(X / Wn),O)

is the cohomology of the crystalline site of X/Wn with values in the sheaf of rings O = OX/Wn.

A key point of the theory is that the crystalline cohomology of a smooth scheme X over k can often be calculated in terms of the algebraic de Rham cohomology of a proper and smooth lifting of X to a scheme Z over W. There is a canonical isomorphism

H^i(X/W) = H^i_{DR}(Z/W) \quad(= H^i(Z,\Omega_{Z/W}^*)= \lim_{\leftarrow}H^i(Z,\Omega_{Z/W_n}^*))

of the crystalline cohomology of X with the de Rham cohomology of Z over the formal scheme of W (an inverse limit of the hypercohomology of the complexes of differential forms). Conversely the de Rham cohomology of X can be recovered as the reduction mod p of its crystalline cohomology.

[edit] Crystals

If X is a scheme over S then the sheaf OX/S is defined by OX/S(T) = coordinate ring of T, where we write T as an abbreviation for an object U→T of Cris(X/S).

A crystal on the site Cris(X/S) is a sheaf F of OX/S modules that is rigid in the following sense:

for any map f between objects T, T′ of Cris(X/S), the natural map from f*F(T) to F(T′) is an isomorphism.

This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology.

An example of a crystal is the sheaf OX/S.

The term crystal attached to the theory was a metaphor inspired by certain properties of algebraic differential equations. These had played a role in p-adic cohomology theories (precursors of the crystalline theory, introduced in various forms by Bernard Dwork, Monsky, Washnitzer, Lubkin and Nick Katz) particularly in Dwork's work. Such differential equations can be formulated easily enough by means of the algebraic Koszul connections, but in the p-adic theory the analogue of analytic continuation is more mysterious (since p-adic discs fall apart rather than overlap). By decree, a crystal would have the 'rigidity' and the 'propagation' notable in the case of the analytic continuation of complex analytic functions. (Cf. also the rigid analytic spaces introduced by John Tate, in the 1960s, when these matters were actively being debated.)

[edit] See also

[edit] References

  • Berthelot, Pierre Cohomologie cristalline des schémas de caractéristique p>0. Lecture Notes in Mathematics, Vol. 407. Springer-Verlag, Berlin-New York, 1974. 604 pp.
  • Berthelot, Pierre; Ogus, Arthur Notes on crystalline cohomology. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. vi+243 pp. ISBN 0-691-08218-9
  • Grothendieck, A. Letter to Atiyah, Oct 14 1963, published as On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 29 1966 95--103.
  • Grothendieck, A. Letter to J. Tate (May, 1966).
  • A. Grothendieck, Crystals and the de Rham cohomology of schemes, p. 306-358 in Dix exposés sur la cohomologie des schémas, by J. Giraud, A. Grothendieck, S.L. Kleiman, M. Raynaud, Amsterdam, North Holland, 1968. Advanced studies in pure mathematics volume 3.
  • Illusie, Luc Report on crystalline cohomology. Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 459--478. Amer. Math. Soc., Providence, R.I., 1975.
  • Illusie, Luc Cohomologie cristalline (d'après P. Berthelot). Séminaire Bourbaki (1974/1975: Exposés Nos. 453-470), Exp. No. 456, pp. 53--60. Lecture Notes in Math., Vol. 514, Springer, Berlin, 1976.
  • Illusie, Luc Crystalline cohomology. Motives (Seattle, WA, 1991), 43-70, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. MR95a:14021
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