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, explained in Grothendieck's letter to Tate (1966), 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 Dwork, Monsky, Washnitzer, Lubin and 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 tend to be disjoint 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 Tate, in the 1960s, when these matters were actively being debated.)
Read more about this topic: Crystalline Cohomology
Famous quotes containing the word crystals:
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