I feel bad about my absence. I lasted posted during winter break, and now winter quarter is completely over. I kept meaning to do a series on “well-known” algebraic geometry results and constructions that don’t appear with any amount of thoroughness in the references. I thought it would be good to get that information out there. Unfortunately, I had already written these things down into a notebook and just couldn’t motivate myself to type something up that I already had. Anyway, one thing led to another and I didn’t do any posts. I’m not sure why I’m trying to justify my absence with an excuse.
Recently I’ve been typing up a translation of Deligne’s argument (written down by Illusie) that every K3 surface in characteristic lifts to characteristic . I’m not to the point of trying to understand it, but I wanted a typed version, so that when I get the background material (namely crystalline cohomology!) and go to understand it, I can just fill in the details into my typed notes quickly and easily. I also was curioius as to the overall format of the argument.
This led me to the 1968 paper by Katz and Oda called On the Differentiation of de Rham Cohomology Classes with Respect to Parameters. The next few posts will be about the main result from this paper. It is really quite amazing.
First, some definitions. We’ll always be working with a smooth scheme over a field (no assumptions here!). Let be a quasi-coherent sheaf of -modules. We’ll write for and unless otherwise noted, all tensor products will be over . We say that is a connection on if it is a homomorphism that satisfies the “Leibniz rule”.
In other words, . This is the standard shorthand meaning satisfies the rule where , and is the image of under the universal map .
Given a connection , we get homomorphisms for all , . These are given by .
The notation is just the one that makes sense: , so it looks like . So we define to be .
Now we define the curvature of the connection to be the map . The curvature is related to the other by an easy check .
This gives some sort of meaning to the curvature now. If the curvature is , then the natural de Rham-like sequence we get from a connection by stringing together the as follows is an honest complex that we can take cohomology with respect to, since .
When this happens we call the connection integrable. Now let be the sheaf of germs of -derivations of into itself. From the fact that the module of differentials is a representing object, we get that as a sheaf of -modules, .
Let be the sheaf of germs of -linear endomorphisms of . Given any connection on we get an induced -linear map as follows. Let be a derivation, then it corresponds to a map .
So consider the composition , where the first is the connection and the second is . This gives the map as .
Lastly for today, note that we get a nice relation between and as follows and that any map satisfying this relation comes from a unique connection on .
Today was just a bunch of notation and definitions, but next time it should get more interesting.