Witt Cohomology Caution

Hopefully I’ll start posting more now that last week is over. Today we’ll look at a counterexample to see that the Witt cohomology we’ve been looking at is not always a a finite type {\Lambda}-module. Just to recall a bit, we’re working over a perfect field of characteristic {p}, and {\Lambda=W_{p^\infty}(k)}. Given a variety {X} over {k} we can use the structure sheaf {\mathcal{O}_X} to form {\mathcal{W}_n}, which is the sheaf of length {n} Witt vectors over {\mathcal{O}_X}. This is just {\mathcal{O}_X^n} with a special ring structure that on stalks has the property of being a complete DVR with residue field {k} and fraction field of characteristic {0}.

The restriction map given by chopping off the last coordinate {R: \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}} gives us a projective system of sheaves and using standard abelian sheaf cohomology we can define {H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)}.

This brings us to the purpose of today. It is possible that in very nice (projective even) cases we have {H^q(X, \mathcal{W}_n)} a finite type {\Lambda}-module, yet have that {H^q(X, \mathcal{W})} is not. Let {X} be a genus zero cuspidal curve with cusp {P}. Let {X'\rightarrow X} be the normalization of {X}. We will shorthand {\mathcal{O}} and {\mathcal{O}'} as the structure sheaves of {X} and {X'} respectively.

We have that {\mathcal{O}_x=\mathcal{O}_x'} when {x\neq P}. We have {\mathcal{O}_P} is the subring of {\mathcal{O}_P'} formed from functions {f} where the differential {df} vanishes at {P}.

Let’s use the standard exact sequence we get from normalizing a curve: {0\rightarrow \mathcal{O}\rightarrow \mathcal{O}'\rightarrow \mathcal{F}\rightarrow 0} where {\mathcal{F}} is concentrated at {P} with the property {\mathcal{F}_P=k}. If we take the long exact sequence in cohomology we see that { H^0(X, \mathcal{F})\hookrightarrow H^1(X, \mathcal{O})\rightarrow H^1(X, \mathcal{O}')\rightarrow}. Note that {X'} is non-singular of genus {0}, so {H^1(X', \mathcal{O}')=H^1(X, \mathcal{O})=0}. Also, {H^0(X, \mathcal{F})=\mathcal{F}_p=k}. So {\mathrm{dim}_k H^1(X, \mathcal{O})=1}.

Now we can use the standard sequence of restriction {0\rightarrow \mathcal{O}\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}\rightarrow 0} and induction to get that the length of the module {H^1(X, \mathcal{W}_n)} is {n}. Now let’s use the normalization sequence above and take Witt sheaves associated to all of them. We’ll denote this by {0\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_n'\rightarrow \mathcal{F}_n\rightarrow 0}.

Note that we still have a bijection with the coboundary map {\delta: H^0(X, \mathcal{F}_n)\rightarrow H^1(X, \mathcal{W}_n)}. Let’s now think about the Frobenius map {F}. Since our field is perfect, we get a bijection {\mathcal{W}_n'\rightarrow \mathcal{W}_n'} and also between {\mathcal{W}_n\rightarrow \mathcal{W}_n}. On {\mathcal{O}_P'} we get that {F(f)=f^p} and hence the differential is {0}, which means it is in {\mathcal{O}_P}.

Applying Frobenius to our exact sequence we get the square

\displaystyle \begin{matrix} H^0(X, \mathcal{F}_n) & \rightarrow & H^1(X, \mathcal{W}_n) \\ F \downarrow & & \downarrow F \\ H^0(X, \mathcal{F}_n) & \rightarrow & H^1(X, \mathcal{W}_n) \end{matrix}

Here we see that {F: H^1(X, \mathcal{W}_n)\rightarrow H^1(X, \mathcal{W}_n)} is identically {0}. This means that {p} annihilates {H^1(X, \mathcal{W}_n)} which means that it is not only a length {n} {\Lambda}-module, but is a vector space over {k} of dimension {n}. Thus the projective limit {H^1(X, \mathcal{W})} is an infinite dimensional vector space over {k} and hence is not a finite type {\Lambda}-module.

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