Sheaf of Witt Vectors

I was going to go on to prove a bunch of purely algebraic properties of the Witt vectors, but honestly this is probably only interesting to you if you are a pure algebraist. From that point of view, this ring we’ve constructed should be really cool. We already have the ring of {p}-adic integers, and clearly {W_{p^\infty}} directly generalizes it. They have some nice ring theoretic properties, especially {W_{p^\infty}(k)} where {k} is a perfect field of characteristic {p}.

Unfortunately it would take awhile to go through and prove these things, and it would just be tedious algebra. Let’s actually see why algebraic geometers and number theorists care about the Witt vectors. First, we’ll need a few algebraic facts that we haven’t talked about. For today, we’re going to fix a prime {p} and we have an {\mathbf{important}} notational change: when I write {W(A)} I mean {W_{p^\infty}(A)}, which means I’ll also write {(a_0, a_1, \ldots)} when I mean {(a_{p^0}, a_{p^1}, \ldots)} and I’ll write {W_n(A)} when I mean {W_{p^n}(A)}. This shouldn’t cause confusion as it is really just a different way of thinking about the same thing, and it is good to get used to since this is the typical way they appear in the literature (on the topics I’ll be discussing).

There is a cool application by thinking about these functors as representable by group schemes or ring schemes, but we’ll delay that for now in order to think about cohomology of varieties in characteristic {p} and hopefully relate it back to de Rham stuff from a month or so ago.

In addition to the fixed {p}, we will assume that {A} is a commutative ring with {1} and of characteristic {p}.

We have a shift operator {V: W_n(A)\rightarrow W_{n+1}(A)} that is given on elements by {(a_0, \ldots, a_{n-1})\mapsto (0, a_0, \ldots, a_{n-1})}. The V stands for Verschiebung which is German for “shift”. Note that this map is additive, but is not a ring map.

We have the restriction map {R: W_{n+1}(A)\rightarrow W_n(A)} given by {(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})}. This one is a ring map as was mentioned last time.

Lastly, we have the Frobenius endomorphism {F: W_n(A)\rightarrow W_n(A)} given by {(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)}. This is also a ring map, but only because of our necessary assumption that {A} is of characteristic {p}.

Just by brute force checking on elements we see a few relations between these operations, namely that {V(x)y=V(x F(R(y)))} and {RVF=FRV=RFV=p} the multiplication by {p} map.

Now on to the algebraic geometry part of all of this. Suppose {X} is a variety defined over an algebraically closed field of characteristic {p}, say {k}. Then we can form the sheaf of Witt vectors on {X} as follows. Notice that all the stalks of the structure sheaf {\mathcal{O}_x} are local rings of characteristic {p}, so it makes sense to define the Witt rings {W_n(\mathcal{O}_x)} for any postive {n}. Now just form the natural sheaf {\mathcal{W}_n} that has as its stalks {(\mathcal{W}_{n})_x=W_n(\mathcal{O}_x)}.

Note that forgetting ring structure and thinking only as a sheaf of sets we have that {\mathcal{W}_n} is just {\mathcal{O}^n}, and when {n=1} it is actually isomorphic as a sheaf of rings. For larger {n} the addition and multiplication is defined in that strange way, so we no longer get an isomorphism of rings. Using our earlier operations and the isomorphism for {n=1}, we can use the following sequences to extract information.

When {n\geq m} we have the exact sequence {0\rightarrow \mathcal{W}_m\stackrel{V}{\rightarrow} \mathcal{W}_n\stackrel{R}{\rightarrow}\mathcal{W}_{n-m}\rightarrow 0}. If we take {m=1}, then we get the sequence {0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}\rightarrow 0}. This will be useful later when trying to convert cohomological facts about {\mathcal{O}_X} to {\mathcal{W}}.

We could also define {H^q(X, \mathcal{W}_n)} as sheaf cohomology because we can think of {\mathcal{W}_n} just as a sheaf of abelian groups. Let {\Lambda=W(k)}, then since {\mathcal{W}_n} are {\Lambda}-modules annihilated by {p^n\Lambda}, we get that {H^q(X, \mathcal{W}_n)} are also {\Lambda}-modules annihilated by {p^n\Lambda}. Next time we’ll talk about some other fundamental properties of the cohomology of these sheaves.


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