Last time we defined crystalline cohomology by defining the crystalline site. Our setup was to fix a PD-scheme and look at an -scheme to which extends. Remember that the objects of the site were triples that we short-handed with . Today we’ll start by re-examining these objects. We can give a little better description which will help us figure out what a sheaf on this site is. Recall that the full subcategory of the category of presheaves (of sets) on the site is a topos which we’ll denote .
Suppose is an object of , i.e. a sheaf. This just means it is a functor that also satisfies for any covering the following sequence being exact .
Let’s fix some object and some sheaf . If is a Zariski open set then define and . There is an inclusion , which by construction is a map in the category , so by virtue of being a contravariant functor we get . The contravariant functor is a sheaf on the Zariski site of which we denote .
We have a proposition that the data of a sheaf on is equivalent to the data that for ever -PD thickening we assign a Zariski sheaf on denoted conspicuously by and for every morphism we assign a map satisfying two compatibility conditions.
1) If is another map then we have a commutative diagram
2) If is an open immersion, the map is an isomorphism.
Using this new characterization it is much easier to describe what the structure sheaf in this site should be, since to give a sheaf I need to give on every thickening a Zariski sheaf and we have a natural Zariski structure sheaf . This assignment can be seen to satisfy the conditions and hence we have a crystalline sheaf. Note that this was not the only choice that works, though. We’ve already pointed out that , so we could also assign which is another crystalline sheaf that we’ll denote .
For our last example, let’s just create a new sheaf from our two above by taking . This is actually a sheaf of P.D. ideals in and we’ll call it . By construction it sits in an exact sequence .
The last thing to point out for today is that this alternate way of thinking about crystalline sheaves is that it really gives us a way of thinking in Zariski terms which we’re already familiar with. In particular, if is a map of crystalline sheaves we can check whether it is an isomorphism (or surjective or injective) on stalks by checking for each and each -P.D. thickening of a neighborhood if the map is an isomorphism.