Frobenius Semi-linear Algebra: 1

Today I want to explain some “well-known” facts in semilinear algebra. Here’s the setup. For safety we’ll assume {k} is algebraically closed of characteristic {p>0} (but merely being perfect should suffice for the main point later). Let {V} be a finite dimensional vector space over {k}. Consider some {p}-semilinear operator on {V} say {\phi: V\rightarrow V}. The fact that we are working with {p} instead of {p^{-1}} is mostly to not scare people. I think {p^{-1}} actually appears more often in the literature and the theory is equivalent by “dualizing.”

All this means is that it is a linear operator satisfying the usual properties {\phi(v+w)=\phi(v)+\phi(w)}, etc, except for the scalar rule in which we scale by a factor of {p}, so {\phi(av)=a^p\phi(v)}. This situation comes up surprisingly often in positive characteristic geometry, because often you want to analyze some long exact sequence in cohomology associated to a short exact sequence which involves the Frobenius map or the Cartier operator. The former will induce a {p}-linear map of vector spaces and the latter induces a {p^{-1}}-linear map.

The facts we’re going to look at I’ve found in three or so papers just saying “from a well-known fact about {p^{-1}}-linear operators…” I wish there was a book out there that developed this theory like a standard linear algebra text so that people could actually give references. The proof today is a modification of that given in Dieudonne’s Lie Groups and Lie Hyperalgebras over a Field of Characteristic {p>0} II (section 10).

Let’s start with an example. In the one-dimensional case we have the following {\phi: k\rightarrow k}. If the map is non-trivial, then it is bijective. More importantly we can just write down every one of these because if {\phi(1)=a}, then

\displaystyle \begin{array}{rcl} \phi(x) & = & \phi(x\cdot 1) \\ & = & x^p\phi(1) \\ & = & ax^p \end{array}

In fact, we can always find some non-zero fixed element, because this amounts to solving {ax^p-x=x(ax^{p-1}-1)=0}, i.e. finding a solution to {ax^{p-1}-1=0} which we can do by being algebraically closed. This element {b} obviously serves as a basis for {k}, but to set up an analogy we also see that {Span_{\mathbb{F}_p}(b)} are all of the fixed points of {\phi}. In general {V} will breakup into parts. The part that {\phi} acts bijectively on will always have a basis of fixed elements whose {\mathbb{F}_p}-span consists of exactly the fixed points of {\phi}. Of course, this could never happen in linear algebra because finding a fixed basis implies the operator is the identity.

Let’s start by proving this statement. Suppose {\phi: V\rightarrow V} is a {p}-semilinear automorphism. We want to find a basis of fixed elements. We essentially mimic what we did before in a more complicated way. We induct on the dimension of {V}. If we can find a single {v_1} fixed by {\phi}, then we would be done for the following reason. We kill off the span of {v_1}, then by the inductive hypothesis we can find {v_2, \ldots, v_n} a fixed basis for the quotient. Together these make a fixed basis for all of {V}.

Now we need to find a single fixed {v_1} by brute force. Consider any non-zero {w\in V}. We start taking iterates of {w} under {\phi}. Eventually they will become linearly dependent, so we consider {w, \phi(w), \ldots, \phi^k(w)} for the minimal {k} such that this is a linearly dependent set. This means we can find some coefficients that are not all {0} for which {\sum a_j \phi^j(w)=0}.

Let’s just see what must be true of some fictional {v_1} in the span of these elements such that {\phi(v_1)=v_1}. Well, {v_1=\sum b_j \phi^j(w)} must satisfy {v_1=\phi(v_1)=\sum b_j^p \phi^{j+1}(w)}.

To make this easier to parse, let’s specialize to the case that {k=3}. This means that {a_0 w+a_1\phi(w)+a_2\phi^2(w)=0} and by assumption the coefficient on this top power can’t be zero, so we rewrite the top power {\phi^2(w)=-(a_0/a_2)w - (a_1/a_2)\phi(w)}.

The other equation is

\displaystyle \begin{array}{rcl} b_0w+b_1\phi(w) & = & b_0^p\phi(w)+b_1^p\phi^2(w)\\ & = & -(a_0/a_2)b_1^pw +(b_0^p-(a_1/a_2)b_1^p)\phi(w) \end{array}

Comparing coefficients {b_0=-(a_0/a_2)b_1^p} and then forward substituting {b_1=-(a_0/a_2)^pb_1^{p^2}-(a_1/a_2)b_1^p}. Ah, but we know the {a_j} and this only involves the unknown {b_1}. So since {k} is algebraically closed we can solve to find such a {b_1}. Then since we wrote all our other coefficients in terms of {b_1} we actually can produce a fixed {v_1} by brute force determining the coefficients of the vector in terms of our linear dependence coefficients.

There was nothing special about {k=3} here. In general, this trick will work because it only involves the fact that applying {\phi} cycled the vectors forward by one which allows us to keep forward substituting all the equations from the comparison of coefficients to get everything in terms of the highest one including the highest one which transformed the problem into solving a single polynomial equation over our algebraically closed field.

This completes the proof that if {\phi} is bijective, then there is a basis of fixed vectors. The fact that {V^\phi=Span_{\mathbb{F}_p}(v_1, \ldots, v_n)} is pretty easy after that. Of course, the {\mathbb{F}_p}-span is contained in the fixed points because by definition the prime subfield of {k} is exactly the fixed elements of {x\mapsto x^p}. On the other hand, if {c=\sum a_jv_j} is fixed, then {c=\phi(c)=\sum a_j^p \phi(v_j)=\sum a_j^p v_j} shows that all the coefficients must be fixed by Frobenius and hence in {\mathbb{F}_p}.

Here’s how this is useful. Recall the post on the fppf site. We said that if we wanted to understand the {p}-torsion of certain cohomology with coefficients in {\mathbb{G}_m} (Picard group, Brauer group, etc), then we should look at the flat cohomology with coefficients in {\mu_p}. If we specialize to the case of curves we get an isomorphism {H^1_{fl}(X, \mu_p)\simeq Pic(X)[p]}.

Recall the exact sequence at the end of that post. It told us that via the {d\log} map {H^1_{fl}(X, \mu_p)=ker(C-I)=H^0(X, \Omega^1)^C}. Now we have a ridiculously complicated way to prove the following well-known fact. If {E} is an ordinary elliptic curve over an algebraically closed field of characteristic {p>0}, then {E[p]\simeq \mathbb{Z}/p}. In fact, we can prove something slightly more general.

By definition, a curve is of genus {g} if {H^0(X, \Omega^1)} is {g}-dimensional. We’ll say {X} is ordinary if the Cartier operator {C} is a {p^{-1}}-linear automorphism (I’m already sweeping something under the rug, because to even think of the Cartier operator acting on this cohomology group we need a hypothesis like ordinary to naturally identify some cohomology groups).

By the results in this post we know that the structure of {H^0(X, \Omega^1)^C} as an abelian group is {\mathbb{Z}/p\oplus \cdots \oplus \mathbb{Z}/p} where there are {g} copies. Thus in more generality this tells us that {Jac(X)[p]\simeq Pic(X)[p]\simeq H^0(X, \Omega^1)^C\simeq \mathbb{Z}/p\oplus \cdots \oplus \mathbb{Z}/p}. In particular, since for an elliptic curve (genus 1) we have {Jac(E)=E}, this statement is exactly {E[p]\simeq \mathbb{Z}/p}.

This point is a little silly, because Silverman seems to just use this as the definition of an ordinary elliptic curve. Hartshorne uses the Hasse invariant in which case it is quite easy to derive that the Cartier operator is an automorphism (proof: it is Serre dual to the Frobenius which by the Hasse invariant definition is an automorphism). Using this definition, I’m actually not sure I’ve ever seen a derivation that {E[p]\simeq \mathbb{Z}/p}. I’d be interested if there is a lower level way of seeing it than going through this flat cohomology argument (Silverman cites a paper of Duering, but it’s in German).

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What’s up with the fppf site?

I’ve been thinking a lot about something called Serre-Tate theory lately. I want to do some posts on the “classical” case of elliptic curves. Before starting though we’ll go through some preliminaries on why one would ever want to use the fppf site and how to compute with it. It seems that today’s post is extremely well known, but not really spelled out anywhere.

Let’s say you’ve been reading stuff having to do with arithmetic geometry for awhile. Then without a doubt you’ve encountered étale cohomology. In fact, I’ve used it tons on this blog already. Here’s a standard way in which it comes up. Suppose you have some (smooth, projective) variety {X/k}. You want to understand the {\ell^n}-torsion in the Picard group or the (cohomological) Brauer group where {\ell} is a prime not equal to the characteristic of the field.

What you do is take the Kummer sequence:

\displaystyle 0\rightarrow \mu_{\ell^n}\rightarrow \mathbb{G}_m\stackrel{\ell^n}{\rightarrow} \mathbb{G}_m\rightarrow 0.

This is an exact sequence of sheaves in the étale topology. Thus it gives you a long exact sequence of cohomology. But since {H^1_{et}(X, \mathbb{G}_m)=Pic(X)} and {H^2_{et}(X, \mathbb{G}_m)=Br(X)}. Just writing down the long exact sequence you get that the image of {H^1_{et}(X, \mu_{\ell^n})\rightarrow Pic(X)} is exactly {Pic(X)[\ell^n]}, and similarly with the Brauer group. In fact, people usually work with the truncated short exact sequence:

\displaystyle 0\rightarrow Pic(X)/\ell^n Pic(X) \rightarrow H^2_{et}(X, \mu_{\ell^n})\rightarrow Br(X)[\ell^n]\rightarrow 0

Fiddling around with other related things can help you figure out what is happening with the {\ell^n}-torsion. That isn’t the point of this post though. The point is what do you do when you want to figure out the {p^n}-torsion where {p} is the characteristic of the ground field? It looks like you’re in big trouble, because the above Kummer sequence is not exact in the étale topology.

It turns out that you can switch to a finer topology called the fppf topology (or site). This is similar to the étale site, except instead of making your covering families using étale maps you make them with faithfully flat and locally of finite presentation maps (i.e. fppf for short when translated to french). When using this finer topology the sequence of sheaves actually becomes exact again.

A proof is here, and a quick read through will show you exactly why you can’t use the étale site. You need to extract {p}-th roots for the {p}-th power map to be surjective which will give you some sort of infinitesimal cover (for example if {X=Spec(k)}) that looks like {Spec(k[t]/(t-a)^p)\rightarrow Spec(k)}.

Thus you can try to figure out the {p^n}-torsion again now using “flat cohomology” which will be denoted {H^i_{fl}(X, -)}. We get the same long exact sequences to try to fiddle with:

\displaystyle 0\rightarrow Pic(X)/p^n Pic(X) \rightarrow H^2_{fl}(X, \mu_{p^n})\rightarrow Br(X)[p^n]\rightarrow 0

But what the heck is {H^2_{fl}(X, \mu_{p^n})}? I mean, how do you compute this? We have tons of books and things to compute with the étale topology. But this fppf thing is weird. So secretly we really want to translate this flat cohomology back to some étale cohomology. I saw the following claimed in several places without really explaining it, so we’ll prove it here:

\displaystyle H^2_{fl}(X, \mu_p)=H^1_{et}(X, \mathbb{G}_m/\mathbb{G}_m^p).

Actually, let’s just prove something much more general. We actually get that

\displaystyle H^i_{fl}(X, \mu_p)=H^{i-1}_{et}(X, \mathbb{G}_m/\mathbb{G}_m^p).

The proof is really just a silly “trick” once you see it. Since the Kummer sequence is exact on the fppf site, by definition this just means that the complex {\mu_p} thought of as concentrated in degree {0} is quasi-isomorphic to the complex {\mathbb{G}_m\stackrel{p}{\rightarrow} \mathbb{G}_m}. It looks like this is a useless and more complicated thing to say, but this means that the hypercohomology (still fppf) is isomorphic:

\displaystyle \mathbf{H}^i_{fl}(X, \mu_p)=\mathbf{H}^i_{fl}(X, \mathbb{G}_m\stackrel{p}{\rightarrow} \mathbb{G}_m).

Now here’s the trick. The left side is the group we want to compute. The right hand side only involves smooth group schemes, so a theorem of Grothendieck tells us that we can compute this hypercohomology using fpqc, fppf, étale, Zariski … it doesn’t matter. We’ll get the same answer. Thus we can switch to the étale site. But of course, just by definition we now extend the {p}-th power map (injective on the etale site) to an exact sequence

\displaystyle 0\rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m\rightarrow \mathbb{G}_m/\mathbb{G}_m^p\rightarrow 0.

Thus we get another quasi-isomorphism of complexes. This time to {\mathbb{G}_m/\mathbb{G}_m^p[-1]}. This is a complex concentrated in a single degree, so the hypercohomology is just the etale cohomology. The shift by {-1} decreases the cohomology by one and we get the desired isomorphism {H^i_{fl}(X, \mu_p)=H^{i-1}_{et}(X, \mathbb{G}_m/\mathbb{G}_m^p)}. In particular, we were curious about {H^2_{fl}(X, \mu_p)}, so we want to figure out {H^1_{et}(X, \mathbb{G}_m/\mathbb{G}_m^p)}.

Alright. You’re now probably wondering what in the world to I do with the étale cohomology of {\mathbb{G}_m/\mathbb{G}_m^p}? It might be on the étale site, but it is a weird sheaf. Ah. But here’s something great, and not used all that much to my knowledge. There is something called the multiplicative de Rham complex. On the étale site we actually have an exact sequence of sheaves via the “dlog” map:

\displaystyle 0\rightarrow \mathbb{G}_m/\mathbb{G}_m^p\stackrel{d\log}{\rightarrow} Z^1\stackrel{C-i}{\rightarrow} \Omega^1\rightarrow 0.

This now gives us something nice because if we understand the Cartier operator (which is Serre dual to the Frobenius!) and know things how many global {1}-forms are on the variety (maybe none?) we have a hope of computing our original flat cohomology!

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.