# Crystalline Site 1

I’ve decided on pulling the motivation back into the picture. Recall way back when we were thinking about the shortcomings of trying to replicate a de Rham type cohomology theory in positive characteristic. One of our motivations is that we want a theory that has no problem being done in positive characteristic, but actually gives us the de Rham cohomology if there is some lift to characteristic ${0}$. We even tried to just define it this way. Take a lift, do de Rham, and then check that the result is independent of lift. The problem is that there are things that don’t lift to characteristic ${0}$, and the lifting process is definitely not an efficient process for computing.

This is where crystalline cohomology enters the picture. We’ll make this more precise later on, but if we have a smooth lifting ${X\rightarrow S}$ to characteristic ${0}$, then we’d like to have a canonical isomorphism ${H^*_{crys}(X_0/S)\rightarrow \mathbf{H}^*(X_{zar}, \Omega^\cdot)}$. Since we’ve already talked about what it means to do cohomology of a sheaf on a site, we can actually state pretty easily what crystalline cohomology is. Suppose ${X}$ is a variety over ${S}$. There is the crystalline site ${\mathrm{Crys}(X/S)}$, and ${H^n_{crys}(X/S)=H^n(X_{cyrs}, \mathcal{O}_{X/S})}$, so the crystalline cohomology is just sheaf cohomology on the crystalline site. The work is going to be in figuring out how to think about this new site.

First, we define a P.D. scheme. This is exactly what it sounds like. There is no problem in extending all the definitions done for rings so far into definitions on the sections of sheaves. For instance, if ${X}$ is a space and ${\mathcal{A}}$ and ${\mathcal{I}}$ are sheaves of rings on ${X}$, then we say ${(\mathcal{A}, \mathcal{I}, \gamma)}$ is a sheaf of P.D. rings if ${(\mathcal{A}(U), \mathcal{I}(U), \gamma)}$ is a P.D. ring for all ${U\subset X}$ open. A P.D. ringed space is just a ringed space with a sheaf of P.D. rings on it ${(X, (\mathcal{A}, \mathcal{I}, \gamma))}$. Inverse image and pushforwards of sheaves under maps ${f:X\rightarrow Y}$ preserve the property of being a sheaf of P.D. rings.

Given a P.D. ring ${(A, I, \gamma)}$ we can define ${\mathrm{Spec}(A, I, \gamma)}$ to be the locally ringed space ${(|\mathrm{Spec}(A)|, \mathcal{O})}$ where ${\mathcal{O}}$ is the sheaf of P.D. rings obtained under the canonical extensions we get of ${\gamma}$ since localization is flat and we checked previously that ${\gamma}$ extends to any flat ${A}$-algebra. A P.D. scheme is a locally ringed space locally isomorphic to ${\mathrm{Spec}(A, I, \gamma)}$. Morphisms in this category are morphisms of locally ringed spaces that are P.D. morphisms on sections.

We’ll spend a little more time with these definitions and related issues next time. The goal of this post is to define the crystalline site. Now we’ll want to fix a base, so let ${(S, I, \gamma)}$ be a P.D. scheme. If ${X}$ is an ${S}$-scheme, then by looking locally it makes sense to ask whether or not ${\gamma}$ extends to ${X}$. If it does, then we can define the crystalline site ${\mathrm{Crys}(X/S)}$ as follows: The objects are pairs ${(U\hookrightarrow T, \delta)}$ where ${U\subset X}$ is Zariski open and ${U\hookrightarrow T}$ is a closed ${S}$-immersion defined by the quasi-coherent sheaf of ideals ${\mathcal{J}}$ where ${\delta}$ is a P.D. structure on ${\mathcal{J}}$ compatible with ${\gamma}$. We abuse notation and call ${(U\hookrightarrow T, \delta)}$ just ${T}$.

The morphisms ${u:T\rightarrow T'}$ are commutative diagrams

${\begin{matrix} U & \hookrightarrow & T \\ \downarrow & & \downarrow \\ U' & \hookrightarrow & T' \end{matrix}}$

where ${U\rightarrow U'}$ is a Zariski inclusion of open sets of ${X}$ and ${T\rightarrow T'}$ is a P.D. map over ${S}$. A covering is just a collection of maps ${\{u_i: T_i\rightarrow T\}}$ such that ${T_i\rightarrow T}$ are open immersions and ${T=\cup T_i}$. We’ll just end this post by making several remarks and giving an example that we’re aiming at.

First, the term for an object ${(U\hookrightarrow T, \delta)}$ is an “S-PD thickening of ${U}$“. One of the consequences of requiring ${\gamma}$ to extend to ${X}$ is that it makes ${(U\rightarrow U, 0)}$ an object of our site for any Zariski open ${U\subset X}$. Another consequence of our definitions is that all our thickenings ${U\rightarrow T}$ are topological homeomorphisms since they are defined by nilpotent ideals, ${\mathcal{J}}$. The last remark is that if ${\{T_i\rightarrow T\}}$ is a covering, it comes with a collection of P.D. structures: ${\delta_i}$. By compatibility, the collection ${\delta_i}$ completely determines ${\delta}$ and conversely, given ${\delta}$, we can restrict and find out what the ${\delta_i}$ must be.

The example that we want to think about is when we have some lifting of ${X}$ over a postive characteristic field ${k}$ to ${W_n(k)}$. In this situation ${S=\mathrm{Spec}(W_n)}$ with ${\mathcal{I}=(p)}$ with the canoncial P.D. structure inherited from ${W}$. We’ll look at this more closely when we are working with actual examples of lifted schemes.