Sheaf of Witt Vectors 2


Recall last time we talked about how we can form the sheaf of Witt vectors over a variety {X} that is defined over an algebraically closed field {k} of characteristic {p}. The sections of the structure sheaf form rings and we can take {W_n} of those rings. The functoriality of {W_n} gives us that this is a sheaf that we denote {\mathcal{W}_n}. For today we’ll be define {\Lambda} to be {W(k)}.

Recall that we also noted that {H^q(X, \mathcal{W}_n)} makes sense and is a {\Lambda}-module annihilated by {p^n\Lambda} (recall that we noted that Frobenius followed by the shift operator is the same as multiplying by {p}, and since Frobenius is surjective, multiplying by {p} is just replacing the first entry by {0} and shifting, so multiplying by {p^n} is the same as shifting over {n} entries and putting {0}‘s in, since the action is component-wise, {p^n\Lambda} is just multiplying by {0} everywhere and hence annihilates the module).

In fact, all of our old operators {F}, {V}, and {R} still act on {H^q(X, \mathcal{W}_n)}. They are easily seen to satisfy the formulas {F(\lambda w)=F(\lambda)F(w)}, {V(\lambda w)=F^{-1}(\lambda)V(w)}, and {R(\lambda w)=\lambda R(w)} for {\lambda\in \Lambda}. Just by using basic cohomological facts we can get a bunch of standard properties of {H^q(X, \mathcal{W}_n)}. We won’t write them all down, but the two most interesting (of the very basic) ones are that if {X} is projective then {H^q(X, \mathcal{W}_n)} is a finite {\Lambda}-module, and from the short exact sequence we looked at last time {0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n \rightarrow \mathcal{W}_{n-1}\rightarrow 0}, we can take the long exact sequence associated to it to get {\cdots \rightarrow H^q(X, \mathcal{O}_X)\rightarrow H^q(X, \mathcal{W}_n)\rightarrow H^q(X, \mathcal{W}_{n-1})\rightarrow \cdots}

If you’re like me, you might be interested in studying Calabi-Yau manifolds in positive characteristic. If you’re not like me, then you might just be interested in positive characteristic K3 surfaces, either way these cohomology groups give some very good information as we’ll see later, and for a Calabi-Yau’s (including K3’s) we have {H^i(X, \mathcal{O}_X)=0} for {i=1, \ldots , n-1} where {n} is the dimension of {X}. Using this long exact sequence, we can extrapolate that for Calabi-Yau’s we get {H^i(X, \mathcal{W}_n)=0} for all {n>0} and {i=1, \ldots, n-1}. In particular, we get that {H^1(X, \mathcal{W})=0} for {X} a K3 surface where we just define {H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)} in the usual way.

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