Other Attempts at Cohomology Theories

I should first point out that I’m basically sketching out Grothendieck’s article on crystals in Dix Expose, so if you want to see more that’s where you should look. Let’s first answer those questions from last time and explain exactly what it is that we are looking for in a cohomology theory.

If ${X/k}$ is of finite type and ${k}$ a perfect field of positive characteristic, then we want to keep all of those properties from our earlier theories. The important ones are that the cohomologie groups are modules over an integral domain that has the property of having characteristic ${0}$ fraction field. We also want to keep the formal properties that we checked for the earlier ones like functoriality, being finite dimensional when ${X}$ is proper, having some sort of duality, having a Kunneth formula, having a flat or smooth base change theorem, and the list continues.

We already have theories that do these things, so remember the key thing we need to be able to add in is that we want information about the ${p}$-torsion of the singular cohomology. For reasons we won’t go into we can’t take our coefficients to be ${\mathbb{Z}_p}$ or ${\mathbb{Q}_p}$. But we’ve already put in the work to see that ${W(k)}$ is a nice choice since it has residue field ${k}$ and fraction field of characteristic ${0}$.

There have been a few failed attempts. Without giving rigorous definitions of the attempts, I’ll just point them out. One might try to build an analogue of the ${\ell}$-adic attmept but using a different site. It was attempted to do this using the fppf site (it was a bit more complicated than just using the fppf site, though). What happens is you get a theory that works great in dimension ${1}$, but then you lose things like Poincare duality for ${\dim(X)\geq 2}$. This theory should still give some nice connections with our original one via Dieudonne modules. If we talk about this later it will be defined more rigorously.

The next attempt by Monsky and Washnitzer was quite beautiful. The idea is that everything works if ${k}$ is of characteristic ${0}$, so let’s just put ourselves in that situation. Let ${S= \mathrm{Spec}(W)}$ then we can consider a lifting of ${X}$ to characteristic ${0}$ by say ${M\rightarrow S}$. Since we are now in characteristic ${0}$, we may as well use de Rham, so the theory should just be ${H^i_{dR}(M/S)}$. Of course, one needs to check several things, the first of which is that ${H^i_{dR}(M/S)}$ is independent of the choice of lift.

With this approach we do get lots of nice things like the correct Betti numbers and finite dimensionality when ${M/S}$ is proper. There is however a very huge problem with the approach. There exists schemes with no lift to characteristic ${0}$. What do we do about this? Well, one can try to get around it by trying to construct a formal lift ${\frak{X}\rightarrow S}$ and then considering the hypercohomology ${\mathbf{H}^*(\frak{X}, \Omega_{\frak{X}/S}^\cdot)}$ using limits, but some problems arise. Again, there isn’t always even a formal lift, and even if there was you can check that this doesn’t give finite dimensional answers.

They actually carefully constructed a way to not need the lift, and when one exists they get ${H^i_{MW}(X)=H^i_{dR}(M/S)}$. Unfortunately this theory still uses differential forms and hence requires some nicenesss hypothesis (maybe even smooth) to make sure everything works. Also, many of the properties we want are unknown to be true like being finite ${W}$-modules when ${X}$ is proper. Admittedly, this theory is the best so far that we’ve looked at, and by the fact that ${H^i_{MW}(X)}$ is defined without the lift, it proves that ${H^i_{dR}(M/S)}$ is independent of lift when one exists.