Crystalline Site 3: De Rham Spaces


I think my posts are becoming increasingly incomprehensible, and I have a partial remedy to that. At some point I want to take a few steps backwards and examine the “calculus of divided powers”. This will help us get a feel for how we are doing something that can be thought of as positive characteristic de Rham cohomology. I also want to look at some comparison theorems, so that we can see that this actually matches up with other cohomology groups that we know how to compute and it will give us useful information. But before doing any of that, I want to show the usefulness and motivation for all of this by tying it back to the height of varieties that we defined awhile ago.

Today I’ll just do a quick post on yet another way to think about this site when we are in characteristic {0}. For any space {X} (and by space, I mean that really loosely as just an object in {\mathrm{Psh}(\mathrm{Ring}^{op})}, but you can think scheme or variety if you want) we can form the associated “de Rham space”. This is the presheaf defined by {X_{dR}(R)=X(R/\mathcal{N}(R))} where {\mathcal{N}(R)} is the nilradical.

What this is doing in effect is identifying infinitely close points. A concrete way to visualize this is to take {X=\mathbb{A}^1_k}. Then {X(k[\epsilon])} are points plus a choice of tangent vector. But if you are used to thinking in terms of smooth manifolds or something, then you probably don’t think of {(0, v)} and {(0, -v)} where {v} is a tangent vector at {0} as different points because they are literally the same point with the only difference being some infinitesimal information. We call them being “infinitely close”. Since {k[\epsilon]/\mathcal{N}\simeq k}, we see that {X_{dR}(k[\epsilon])=X(k)}. So the infinitesimal information is killed off.

In the case of {X} being a smooth variety the natural map {X\rightarrow X_{dR}} is surjective and hence we can honestly think of {X_{dR}} as a quotient of {X} where our relation is to identify the infinitely close points. Now if we go back to a {X} being a scheme, we get the following: The big site {\mathrm{Ring}^{op}/X_{dR}} is just the crystalline site of {X}. When we are over characteristic {0} this is often called the infinitesimal site. We now have in some sense a categorical way to think of the construction.

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