Here is just a quick post on why one might want to know what the ring of Witt vectors is. I won’t tell you what they are, but here are some interesting ways in which they are used. It is hard to find any resources on their construction, so we’ll try to get some information out there.
Given a ring, you can construct the Witt ring from it. For fields of positive characteristic, this ring has some very nice properties. It is a DVR with residue field and fraction field of characteristic . I’m very interested in a class of problems in algebraic geometry known as “lifting problems”. One wants to know if a particular variety defined over a positive characteristic field has a lift to characteristic .
What this means is that you have a deformation of the variety where the special fiber is the variety itself, but another fiber is of characteristic . This probably hurts your brain if you are used to thinking of deformations as “continuously” changing a variety, but recall all it really means is that you have a flat family.
Here is where the Witt vectors shine. Suppose you are trying to lift a variety to characteristic . Then you might try to find an and a flat map with the property that the fiber over the closed point is . Then you’ve lifted it, since the generic fiber is a deformation and is defined over which is of characteristic . Note that finding such an is usually very difficult and often requires constructing a formal scheme one step at a time and proving that this is algebraizable, but now we’re getting ahead of ourselves.
Now I’ll just list some other applications that we won’t focus on, but hopefully something catches your interest so that you’ll want to find out what they are. The last few posts were about de Rham cohomology in arbitrary characteristic, and we have our eye towards crystalline cohomology. Number theorists care a lot about crystalline cohomology since it is central in all of this Langland’s stuff going on. The tie in with Witt vectors is in the de Rham-Witt complex.
Witt rings show up in K-theory in the form of of the category of endomorphisms of projective modules over a commutative ring. Lastly, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.
I’m sure there are lots of other examples where these things come up. “If there so important why has no one really heard of them,” you may be asking? I have no idea. I wish there was more out there on them so that it was easier to learn what they are. I think it has to do with the fact that for the most part you have to write down a really gross formula for the multiplication.