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 -module. Just to recall a bit, we’re working over a perfect field of characteristic , and . Given a variety over we can use the structure sheaf to form , which is the sheaf of length Witt vectors over . This is just with a special ring structure that on stalks has the property of being a complete DVR with residue field and fraction field of characteristic .
The restriction map given by chopping off the last coordinate gives us a projective system of sheaves and using standard abelian sheaf cohomology we can define .
This brings us to the purpose of today. It is possible that in very nice (projective even) cases we have a finite type -module, yet have that is not. Let be a genus zero cuspidal curve with cusp . Let be the normalization of . We will shorthand and as the structure sheaves of and respectively.
We have that when . We have is the subring of formed from functions where the differential vanishes at .
Let’s use the standard exact sequence we get from normalizing a curve: where is concentrated at with the property . If we take the long exact sequence in cohomology we see that . Note that is non-singular of genus , so . Also, . So .
Now we can use the standard sequence of restriction and induction to get that the length of the module is . Now let’s use the normalization sequence above and take Witt sheaves associated to all of them. We’ll denote this by .
Note that we still have a bijection with the coboundary map . Let’s now think about the Frobenius map . Since our field is perfect, we get a bijection and also between . On we get that and hence the differential is , which means it is in .
Applying Frobenius to our exact sequence we get the square
Here we see that is identically . This means that annihilates which means that it is not only a length -module, but is a vector space over of dimension . Thus the projective limit is an infinite dimensional vector space over and hence is not a finite type -module.