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. In algebraic geometry, a motive (or sometimes motif) refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ...

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.

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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ...

de Rham cohomology

De Rham cohomology solves the problem of finding an algebraic definition of the cohomology groups (singular cohomology) In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

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. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

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. In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ...

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. In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... This article is about algebraic varieties. ...

Coefficients

If X is a variety over an algebraically closed field of characteristic p > 0, then the l-adic cohomology groups for l any prime number other than p give satisfactory cohomology groups of X, with coefficients in the ring Z_l 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). In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... The title given to this article is incorrect due to technical limitations. ...

The classic reason (due to Serre) 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. In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebra A over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. ... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots...

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. In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

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. This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. ...

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). In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. ...

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

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. In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β unknown. ...

We will work over the ring W_n = 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 W_n 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 W_n, then the crystalline site of X relative to W_n, denoted Cris(X/W_n), 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 W_n.

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

where

Hⁱ(X / W_n) = Hⁱ(Cris(X / W_n),O)

is the cohomology of the crystalline site of X/W_n with values in the sheaf of rings O = O_X/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

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.

Crystals

If X is a scheme over S then the sheaf O_X/S is defined by O_X/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 O_X/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 O_X/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.) Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. ... Nick Katz (Nicholas M. Katz) is an American mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ...

See also

• Motivic cohomology

Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by Grothendieck during the 1960s. ...

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