A Mind for Madness

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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.


Problems with de Rham and l-adic

Let’s begin by reviewing the fact that when we work over {\mathbb{C}} we have a nice comparison theorem. Last time it was briefly mentioned that if {X} is smooth over {\mathbb{C}}, then we can consider the analytic topology on the {\mathbb{C}}-valued points which we’ll call {X^{an}}. Using GAGA and the degeneration of the Hodge-de Rham spectral sequence from last time we immediately get that {\mathbf{H}^*(X, \Omega_{X/\mathbb{C}}^\cdot)\stackrel{\sim}{\rightarrow} \mathbf{H}^*(X^{an}, \Omega_{X^{an}}^\cdot)\simeq H^*(X, \mathbb{C})}.

Thus we have a comparison theorem that allows us to use purely algebraic methods to recover the topological information of the singular cohomology. But what happens if we work over a field that is not {\mathbb{C}}? Well let’s assume that {X} is smooth and of finite type over {k}. If {k} has characteristic {0}, then the standard idea of the Lefschetz principle gives us that everything works out again and we can recover the singular cohomology.

Obviously lots can go wrong in positive characteristic. Say {\mathrm{char}(k)=p}, then we’ll throw out the case of affine things because for instance if {X} is an affine curve the de Rham cohomology is not even finitely generated. But if we assume {X/k} is proper and have finite generatedness, we still have problems with de Rham. It doesn’t recover the correct topological information. For instance, we’d like a cohomology that gives us “correct” Betti numbers.

If we define the Hodge Betti numbers to be {\displaystyle b^i_H=\sum_{p+q=i}\dim_k H^q(X, \Omega^p)} and the de Rham Betti numbers {b^i_{dR}=\dim_k H^i_{dR}(X/k)} we can just look at the HdR SS and see that {b^i_H\leq b^i_{dR}} and we have equality if and only if the HdR SS degenerates at {E_1}. Degeneration of this spectral sequence is related to having some sort of analogue of {X} in characteristic {0}, also known as a “lift” of {X}. Even if this spectral sequence degenerates, it merely relates these two Betti numbers, but doesn’t fully relate back to the topological Betti numbers.

Also, despite the fact that there is still a Lefschetz fixed point formula for de Rham cohomology in positive characteristic, it only gives the right answer mod {p}. We’ve seen that de Rham cohomology is a nice attempt at coming up with an algebraic way of defining singular cohomology (and is quite useful for lots and lots of things), but it seems very tied to smoothness and characteristic {0} issues.

Anyway, it was good to point that out, but many of you might be frustrated because we already knew that de Rham was going to be a failure in characteristic {p}. If you’ve been introduced to {\ell}-adic cohomology, then you probably already realized that this was basically invented to solve some of these problems. Now we’ll impose the extra condition that {k} be algebraically closed. Recall that we can just pick a prime {\ell\neq p} and define {H^i_\ell (X)=H^i(X_{et}, \mathbb{Z}_\ell)=\lim H^i(X_{et}, \mathbb{Z}/\ell^n)}. This is a {\mathbb{Z}_\ell}-module.

Using the comparison theorem we see that if {k=\mathbb{C}}, then {H^i_\ell(X)\simeq H^i(X^{an}, \mathbb{Z})\otimes \mathbb{Z}_\ell}. This is quite nice. {H^i_\ell(X)} tells us exactly the topological information we sought, since we recover the rank of {H^i(X, \mathbb{Z})} and the {\ell}-primary torsion. Thus if we run through all primes {\ell} we can completely recover {H^i(X, \mathbb{Z})} up to isomorphism. Also, we see that the rank of {H^i_\ell(X)} is finite and independent of {\ell}.

Here is the first of many problems. Although for a smooth projective scheme in positive characteristic we do get that the rank of {H^i_\ell(X)} is finite, it is unknown if this is a number independent of {\ell}. The second much more severe problem is that we only have the reasonable properties listed in the previous paragraph for {\ell\neq p}. So even though we can get a bunch of topological information about {H^i(X, \mathbb{Z})}, we are unfortunately left clueless about the {p}-torsion. We also lose a bunch of classical theorems (even when {\ell\neq p}) such as the Lefschetz hyperplane theorem. Or if we consider the induced map from {X\rightarrow X} on {H^i_\ell(X)\rightarrow H^i_\ell(X)} there may be powers of {p} in the denominator of the characteristic polynomial that we won’t ever be able to get information about.

Clearly, {\ell}-adic cohomology isn’t going to quite do the trick. All I seem to be doing is complaining about how these cohomology theories are failing what I want, but I haven’t ever told you exactly what it is that I do want from a cohomology theory. So next time we’ll start working on stating those properties and seeing if anything we’ve come across satisfies them and if not how we’re going to construct something that does.


Hodge and de Rham Cohomology Revisited

I was going to talk about how the moduli of K3 surfaces is stratified by height in positive characteristic and some of the cool properties of this (for instance, “most” K3 surfaces have height 1). Instead I’m going to shift gears a little. We’ve talked about {\ell}-adic étale cohomology, Witt cohomology, cohomology on any site you want to put on {X}, de Rham cohomology, and we’ve implicitly used Hodge theory in places. Secretly we’ve been heading straight towards cyrstalline cohomology. I think it might be neat to start a series of posts on how each of these relate to eachother and then really motivate the need for crystalline stuff.

Away from this blog I’ve been thinking about degeneration of the Hodge-de Rham Spectral Sequence a lot. Suppose for a minute we’re in the nicest situation possible. We have a smooth variety {X} over {\mathbb{C}}. This means we can look at the {\mathbb{C}}-points and get an actual complex manifold. We defined the algebraic de Rham cohomology awhile ago to be {H^i_{dR}(X/\mathbb{C}):=\mathbf{H}^i(\Omega_{X/\mathbb{C}}^\cdot)} the hypercohomology of the complex {0\rightarrow \mathcal{O}_X\rightarrow \Omega^1\rightarrow \Omega^2\rightarrow \cdots}. Since we’re in this nice case, this actually agrees perfectly with the standard singular cohomology on the manifold with coefficients in {\mathbb{C}} (and by the de Rham theorem, the standard de Rham cohomology).

On a complex manifold we also have a nice working notion of Hodge theory. The Hodge numbers are h^{ij}=\dim_{\mathbb{C}}H^j(X, \Omega^i) which we would normally derive through the Dolbeault resolution. We also have a Hodge decomposition {H_{dR}^j(X/\mathbb{C})=\bigoplus_{p+q=j} H^q(X, \Omega^p)}.

How do we see this using fancy language? Well, merely from the fact that de Rham cohomology is defined as the hypercohomology of a complex, we get the spectral sequence arising from hypercohomology. Without doing any work we can just check what this spectral sequence is and we find {E_1^{ij}=H^j(X, \Omega^i)\Rightarrow H_{dR}^{i+j}(X/\mathbb{C})}. This is because the {E_1^{ij}} terms come from resolving each individual part of the complex which by definition just gives sheaf cohomology of the {\Omega^i}.

Of course, there was nothing special about {X} being over {\mathbb{C}}, we could just as easily be over an arbitrary field and all of this still works. There is a great theorem that says that this spectral sequence degenerates at {E_1} if {X} is smooth over a characteristic {0} field. There are several known proofs, some more analytic and some more algebraic. The coolest one is certainly by Deligne and Illusie.

They prove this preliminary result that if {X} is smooth over a field {k} of characteristic {p} where {p>\dim(X)} and {X} has a lift to {W_2(k)}, then the Hodge-de Rham spectral sequence degenerates at {E_1}. Maybe we’ll talk about how this is done some other day, but if you know about the Cartier isomorphism then it is related to that. Using this side result that seems to be about as unrelated to the characteristic {0} case as possible they then amazingly prove the characteristic {0} case by reducing to positive characteristic and using a Lefschetz principle type argument.

Now despite the fact that H-dR degenerating being the norm in characteristic {0}, it turns out to be not so much the case in positive characteristic, so it is really shocking that to prove the characteristic {0} case they moved themselves to this situation where it was likely not to degenerate. But we’ll get a better intuition later for why this wasn’t as risky as it sounds. Namely that since it came from characteristic {0}, there wasn’t going to be a problem lifting it back so the lifting to {W_2(k)} was not a problem. It seems that the obstruction to being able to do this is almost exactly the failure of degeneracy. Recall that every K3 surface lifts to characteristic {0}, so (if you don’t know the proof of this) you’d expect the H-dR SS to degenerate at {E_1}. It might be a fun exercise for you to try to figure out why this is (very important hint: there are no global vector fields on a K3 so {h^{1,0}=0}).

Before ending this post it should be pointed out that all of this can be done in the relative setting as well. We actually originally defined de Rham cohomology purely in the relative setting without thinking about it over a field like we did today. Suppose {\pi: X\rightarrow S} is a smooth scheme. The relative H-dR SS is given by {E_1^{ij}=H^j(X, \Omega^i_{X/S})\Rightarrow \mathbf{R}^{i+j}\pi_*(\Omega_{X/S}^\cdot)=H_{dR}^{i+j}(X/S)}.

We’ll continue with this next time, but I’ll just leave you with the thought that you can basically formulate for any class of schemes you want a large open problem by asking yourself whether or not the HdR SS degenerates at {E_1} or at all.


Heights of Varieties













Heights of p-divisible Groups












The Dieudonné Module

I’m not sure how much of this Witt sheaf stuff to keep talking about. There is this beautiful invariant associated to any variety in positive characteristic that doesn’t come up in characteristic {0}. It is called the height, and the way it is defined is by attaching a {p}-divisible (formal) group to your variety and looking at the height of that. This will tie together all these things we’ve been talking about since it turns out that the Dieudonné module of this formal group is exactly {H^n(X, \mathcal{W})}, and the non-finite generatedness of this module corresponds to the variety being “supersingular” which just means it has infinite height.

So anyway, this means at some point I should talk about formal groups, and {p}-divisible groups, and height, and Dieudonné modules if I ever want to get there, which means we should cut off the discussion on Witt sheaf cohomology soon. I’ve essentially been giving you the highlights of Serre’s paper on the topic and we’re only about half-way through, so it seems a shame to stop now.

One thing that we are consistently interested in is the projective system {W_n(A)\rightarrow W_{n-1}(A)} formed by restriction or truncating. We seem to consistently ignore that we also have an inductive system formed by shifting {W_n(A)\rightarrow W_{n+1}(A)}, i.e. {V(a_0, \ldots, a_{n-1})=(0, a_0, \ldots, a_{n-1})}. Recall that we’ve already thought about several properties related to this like {RFV=p}, and it is how we know that {W(k)/pW(k)\simeq k}, since this is essentially just quotient by the shift.

We should be a bit more careful, though. Now let {\Lambda=W(k)}. We normally treat {W(A)} as a {\Lambda}-module by taking {\lambda\in \Lambda} and considering it as an element in {W(A)} via the natural map {W(k)\rightarrow W(A)} induced from the algebra map and then using Witt multiplication. When we do this, recall that {V} is only a semi-linear map. It has the property that {V(\lambda a)=\lambda^{1/p}V(a)}. So it is not a map of {\Lambda}-modules, and hence the inductive system is not a system of {\Lambda}-modules.

We can alter the {\Lambda}-structure to make all of this work. Awhile ago we made the blanket assumption that {k} was perfect. We will still need this here. For notation, we’ll say that {\overline{\lambda}} is the image of {\lambda} via {W(k)\rightarrow W(A)\rightarrow W_n(A)=W(A)/V^n(W(A))}. Let {W_n(A)} be a {\Lambda}-module by the altered action {\lambda \star x= \overline{\lambda}^{p^{1-n}}x}, where two elements next to eachother means Witt multiplication.

Let’s check that this makes {V} a {\Lambda}-linear map. {\displaystyle V(\lambda\star a)=T(\overline{\lambda}^{p^{1-n}}a)=V(F(\overline{\lambda}^{p^{-n}})a)=\overline{\lambda}^{p^{-n}}V(a)=\lambda\star V(a)}

Thus we have an inductive system of {\Lambda}-modules. Let’s shift categories for a second. If we have an affine commutative unitary group scheme over {k}, then we use this inductive system to define {D(G)}, the Dieudonne module of {G} by taking {\displaystyle D(G)=\lim_{\rightarrow} Hom(G, W_n(k))}, and similarly in the Ind category so we have a Dieudonne module for formal group schemes and {p}-divisible groups.

Since all the {V} operators are are monomorphisms, we get that {Hom(G, W_n(k))\rightarrow Hom(G, W_{n+1}(k))} are all injective and hence we can identify {Hom(G, W_n(k))} with a submodule of {D(G)} or explicitly we know that {Hom(G, W_n(k))=\{m \in D(G): V^n(m)=0\}}. Thus every element of {D(G)} is killed by a power of {V}.

If we introduce one more strange bit of abstraction we can see the beauty of all this. Let {\mathbf{D}_k=\Lambda\{F, V\}} be the noncommutative polynomial ring over the Witt vectors on two indeterminates that satisfy the commutation laws {Fw=w^pF}, {w^pV=Vw}, and {FV=VF=p}. This is called the Dieudonné ring. We have a canonical way to consider {D(G)} as a left {\mathbf{D}_k}-module. Thus {G\mapsto D(G)} is a contravariant functor from affine unitary group schemes to the category of {\mathbf{D}_k}-modules with {V} torsion. This turns out to be an anti-equivalence of categories.

We can get tons of information about these group schemes by studying their corresponding properties as {\mathbf{D}_k}-modules (which we probably won’t do). Maybe we’ll start thinking about {p}-divisible groups next time to try to work our way towards defining the height of a variety.

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Witt Cohomology Caution

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 {\Lambda}-module. Just to recall a bit, we’re working over a perfect field of characteristic {p}, and {\Lambda=W_{p^\infty}(k)}. Given a variety {X} over {k} we can use the structure sheaf {\mathcal{O}_X} to form {\mathcal{W}_n}, which is the sheaf of length {n} Witt vectors over {\mathcal{O}_X}. This is just {\mathcal{O}_X^n} with a special ring structure that on stalks has the property of being a complete DVR with residue field {k} and fraction field of characteristic {0}.

The restriction map given by chopping off the last coordinate {R: \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}} gives us a projective system of sheaves and using standard abelian sheaf cohomology we can define {H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)}.

This brings us to the purpose of today. It is possible that in very nice (projective even) cases we have {H^q(X, \mathcal{W}_n)} a finite type {\Lambda}-module, yet have that {H^q(X, \mathcal{W})} is not. Let {X} be a genus zero cuspidal curve with cusp {P}. Let {X'\rightarrow X} be the normalization of {X}. We will shorthand {\mathcal{O}} and {\mathcal{O}'} as the structure sheaves of {X} and {X'} respectively.

We have that {\mathcal{O}_x=\mathcal{O}_x'} when {x\neq P}. We have {\mathcal{O}_P} is the subring of {\mathcal{O}_P'} formed from functions {f} where the differential {df} vanishes at {P}.

Let’s use the standard exact sequence we get from normalizing a curve: {0\rightarrow \mathcal{O}\rightarrow \mathcal{O}'\rightarrow \mathcal{F}\rightarrow 0} where {\mathcal{F}} is concentrated at {P} with the property {\mathcal{F}_P=k}. If we take the long exact sequence in cohomology we see that { H^0(X, \mathcal{F})\hookrightarrow H^1(X, \mathcal{O})\rightarrow H^1(X, \mathcal{O}')\rightarrow}. Note that {X'} is non-singular of genus {0}, so {H^1(X', \mathcal{O}')=H^1(X, \mathcal{O})=0}. Also, {H^0(X, \mathcal{F})=\mathcal{F}_p=k}. So {\mathrm{dim}_k H^1(X, \mathcal{O})=1}.

Now we can use the standard sequence of restriction {0\rightarrow \mathcal{O}\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}\rightarrow 0} and induction to get that the length of the module {H^1(X, \mathcal{W}_n)} is {n}. Now let’s use the normalization sequence above and take Witt sheaves associated to all of them. We’ll denote this by {0\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_n'\rightarrow \mathcal{F}_n\rightarrow 0}.

Note that we still have a bijection with the coboundary map {\delta: H^0(X, \mathcal{F}_n)\rightarrow H^1(X, \mathcal{W}_n)}. Let’s now think about the Frobenius map {F}. Since our field is perfect, we get a bijection {\mathcal{W}_n'\rightarrow \mathcal{W}_n'} and also between {\mathcal{W}_n\rightarrow \mathcal{W}_n}. On {\mathcal{O}_P'} we get that {F(f)=f^p} and hence the differential is {0}, which means it is in {\mathcal{O}_P}.

Applying Frobenius to our exact sequence we get the square

\displaystyle \begin{matrix} H^0(X, \mathcal{F}_n) & \rightarrow & H^1(X, \mathcal{W}_n) \\ F \downarrow & & \downarrow F \\ H^0(X, \mathcal{F}_n) & \rightarrow & H^1(X, \mathcal{W}_n) \end{matrix}

Here we see that {F: H^1(X, \mathcal{W}_n)\rightarrow H^1(X, \mathcal{W}_n)} is identically {0}. This means that {p} annihilates {H^1(X, \mathcal{W}_n)} which means that it is not only a length {n} {\Lambda}-module, but is a vector space over {k} of dimension {n}. Thus the projective limit {H^1(X, \mathcal{W})} is an infinite dimensional vector space over {k} and hence is not a finite type {\Lambda}-module.


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