Connections and Curvature … on Schemes! … in Characteristic p?!

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 {p>0} lifts to characteristic {0}. 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 {S} over a field {k} (no assumptions here!). Let {\mathcal{E}} be a quasi-coherent sheaf of {\mathcal{O}_S}-modules. We’ll write {\Omega} for {\Omega^1_{S/k}} and unless otherwise noted, all tensor products will be over {\mathcal{O}_S}. We say that {\nabla} is a connection on {\mathcal{E}} if it is a homomorphism {\nabla: \mathcal{E}\rightarrow \Omega\otimes \mathcal{E}} that satisfies the “Leibniz rule”.

In other words, {\nabla(fg)=f\nabla(g)+df\otimes g}. This is the standard shorthand meaning {\nabla(U): \mathcal{E}(U)\rightarrow \Omega(U)\otimes \mathcal{E}(U)} satisfies the rule where {f\in \mathcal{O}_S(U)}, {g\in \mathcal{E}(U)} and {df} is the image of {f} under the universal map {\mathcal{O}_S\rightarrow \Omega}.

Given a connection {\rho}, we get homomorphisms for all {i}, {\rho_i: \Omega^i\otimes\mathcal{E}\rightarrow \Omega^{i+1}\otimes \mathcal{E}}. These are given by {\rho_i(\omega\otimes e)=d\omega \otimes e+(-1)^i \omega\wedge \rho(e)}.

The notation is just the one that makes sense: {\rho(e)\in \Omega\otimes \mathcal{E}}, so it looks like {\tau\otimes \epsilon}. So we define {\omega\wedge \rho(e)=\omega\wedge(\tau\otimes \epsilon)} to be {(\omega\wedge \tau)\otimes \epsilon\in \Omega^{i+1}\otimes \mathcal{E}}.

Now we define the curvature of the connection {K:\mathcal{E}\rightarrow \Omega^2\otimes \mathcal{E}} to be the map {\rho_1\circ \rho}. The curvature is related to the other {\rho_i} by an easy check {\rho_{i+1}\circ \rho_i(\omega\otimes e)=\omega\wedge K(e)}.

This gives some sort of meaning to the curvature now. If the curvature is {0}, then the natural de Rham-like sequence we get from a connection by stringing together the {\rho_i} as follows {0\rightarrow \mathcal{E}\stackrel{\rho}{\rightarrow} \Omega\otimes\mathcal{E}\stackrel{\rho_1}{\rightarrow} \Omega^2\otimes \mathcal{E}\rightarrow \cdots} is an honest complex that we can take cohomology with respect to, since {\rho_{i+1}\circ \rho_i=0}.

When this happens we call the connection {\rho} integrable. Now let {\mathcal{D}er_k(\mathcal{O}_S)} be the sheaf of germs of {k}-derivations of {\mathcal{O}_S} into itself. From the fact that the module of differentials is a representing object, we get that as a sheaf of {\mathcal{O}_S}-modules, {\mathcal{D}er_k(\mathcal{O}_S)\simeq \mathcal{H}om_{\mathcal{O}_S}(\Omega, \mathcal{O}_S)}.

Let {\mathcal{E}nd_k(\mathcal{E})} be the sheaf of germs of {k}-linear endomorphisms of {\mathcal{E}}. Given any connection {\rho} on {\mathcal{E}} we get an induced {\mathcal{O}_S}-linear map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} as follows. Let {\delta} be a derivation, then it corresponds to a map {D: \Omega\rightarrow \mathcal{O}_S}.

So consider the composition {\overline{D}:\mathcal{E}\rightarrow \Omega\otimes \mathcal{E}\rightarrow \mathcal{O}_S\otimes \mathcal{E}\simeq \mathcal{E}}, where the first is the connection and the second is {D\otimes Id}. This gives the map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} as {\delta\mapsto \overline{D}}.

Lastly for today, note that we get a nice relation between {D} and {\overline{D}} as follows {\overline{D}(fe)=D(f)e+f\overline{D}(e)} and that any map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} satisfying this relation comes from a unique connection on {\mathcal{E}}.

Today was just a bunch of notation and definitions, but next time it should get more interesting.


5 thoughts on “Connections and Curvature … on Schemes! … in Characteristic p?!

  1. To my knowledge there is basically only one (“introductory”) reference, and it is out of print. It is Notes on Crystalline Cohomology by Berthelot and Ogus. I found the first (non-introduction) section to be horrifying. The parts afterwards don’t seem so bad, but I want to get a handle on what is happening in the first section before moving on.

    There is Bloch’s Algebraic K-theory and Crystalline Cohomology for free, but every couple of paragraphs he refers you to other sources to actually learn what’s going on.

    It seems probably the canonical reference is Illusie’s Complexe de de Rham-Witt et Cohomologie Cristalline. I haven’t actually looked this one up yet to see how it is. It is 160 pages of French, though.

    In all honesty, the best thing I’ve found so far is Pete Clark’s Note’s on the de Rham Cohomology of Varieties (it is third from the bottom). It is a nice read so far, and it is putting into place where lots of these definitions and maps come from in the first section of Berthelot. So I think just a quick read-through of that will put me in a position to start tackling the harder references that assume you already know what’s going on.

    Oh. Also, if you find anything useful not listed here I’d love to hear about it as well.

  2. Thanks. These look good. Right now I’m working through Tamme’s book on etale cohomology, which is very nice and one of the most thorough things I’ve seen yet, after which my plan is to read Milne and SGA 4-4.5. I have not yet learned anything about other Weil cohomologies (though my understanding is that Dwork’s proof of the rationality of the zeta-function is based on crystalline theory).

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