Topology in prime characteristic

This semester in Berkeley, we have been running a learning seminar on Weil conjecture and perverse sheaves following the book by Kiehl and Weissauer. All of this business with “weights” used to scare me a great deal, but after this semester, I feel I understand a bit. I thought I would try to share with everyone a bit of what we have been learning. It will also give me an opportunity to make sure that I understand this stuff.

Let $X_0$ denote a variety over a finite field $\mathbb{F}_q$ and let $X$ denote its base change to the algebraic closure.

The first important point is that if X is such a variety, then there is a well behaved etale cohomology of X, $H^*(X, \mathbb{C})$. This cohomology behaves like you would expect it too — in particular, if there is an obvious version of X over the complex numbers, then typically this etale cohomology will agree with the ordinary cohomology of the complex points of X. The development of etale cohomology is actually quite complicated and was not the subject of our seminar and will not be the subject of this post.

Since $X$ comes from $X_0$ by base change, $X$ comes with a Frobenius map $F : X \rightarrow X$. For example if $X_0 = \mathbb{A}_1$, then $F(x) = x^q$. Now, this Frobenius map gives us a endomorphism of the etale cohomology $F : H^*(X, \mathbb{C}) \rightarrow H^*(X, \mathbb{C})$.

If $X$ is smooth and proper, then the following amazing theorem is true: The eigenvalues of $F$ acting on $H^i(X, \mathbb{C})$ are all algebraic integers of size $q^{i/2}$ (this is the “Riemann hypothesis” part of the Weil conjecture).

(There are some notes online of a talk by Beilinson which give some complex geometry motivation for this statement.)

Let us abstract this notion. If $V$ is a vector space with an endomorphism $F$, then we say that $V$ is pure of weight w if all eigenvalues of $F$ have size $q^{w/2}$. We say that V has weight $\le w$ if all eigenvalues of F have size $\le q^{w/2}$. So if X is proper, then $H^i(X, \mathbb{C})$ is pure of weight i.

Now, we consider more general constructible sheaves on $X_0$, which you can think of as coming from gluing local system on subvarieties. If A is such a sheaf, and $x \in X_0(\mathbb{F}_q)$, then F acts on the stalk $A_x$. We say that A has weight $\le w$, if all stalks of $A_x$ have weight $\le w$. Then the above theorem generalizes as follows: If $f : X \rightarrow Y$ is a morphism, and the weight of A is less than w, then the weight of $R^if_!(A)$ is also less than $w +i$.

(Actually I’ve overly simplified this definition of weight. You have to check not just the $\mathbb{F}_q$ points, but also the points over all $\mathbb{F}_{q^n}$ and the corresponding Frobenii.)

Let us now generalize some more. We consider complexes of sheaves A on $X_0$. We say that A has weight $\le w$, if each $\mathcal{H}^n(A)$ has weight $\le w +n$. We say that A has weight $\ge w$, if $D(A)$ has weight $\le -w$ (here D denote Verdier duality). We say that A is pure if it has weight $\le w$ and weight $\ge w$.

The Riemann hypothesis part of the Weil conjecture generalizes as: if $f : X \rightarrow Y$ is a morphism and A has weight $\le w$, then $Rf_! A$ has weight $\le w$. In particular, using duality we see that if f is proper and A is pure of weight w, then $Rf_* A$ is pure of weight w.

Now we move to perverse sheaves, which form an interesting abelian subcategory of the derived category of constructible sheaves, closed under Verdier dual. Perverse sheaves interact nicely with weights. Here are two lemmas which illustrate this interaction and then the two main theorems of this post.

Lemma Every simple perverse sheaf is pure.

Lemma If A and B are pure perverse sheaves with wt(A) < wt(B), then there are no non-trivial extensions of A by B.

Theorem (Weight filtration) Any perverse sheaf has a canonical weight filtration. This is a filtration by subobjects such that all subquotients are pure of increasing weight.

Theorem (Decomposition) Any pure perverse sheaf on X is isomorphic to a direct sum of simple perverse sheaves.

Let me illustrate the weight filtration and decomposition theorems with a couple of examples.

Example 1
If f is the inclusion of $\mathbb{A}^1 \setminus 0$ into $\mathbb{A}^1$, then $Rf_* \mathbb{C}[1]$ has the following weight filtration $0 \subset \mathbb{C}[1] \subset Rf_* \mathbb{C}[1]$. We have that $\mathbb{C}[1]$ is pure of weight 1 and the quotient $Rf_* \mathbb{C}[1]/ \mathbb{C}[1] = \mathbb{C}_0(-1)$ is pure of weight 2 (here $\mathbb{C}_0$ is the skyscraper sheaf at $0$, and $(-1)$ denotes Tate twist).

Generalizing this example, we may more generally consider the inclusion of an open subset U into a smooth variety whose complement is a divisor. I’ve heard that the corresponding weight filtration on $Rf_* \mathbb{C}_U$ can be “seen” for varieties over $\mathbb{C}$ using Deligne’s log complex. Does anyone have a reference for that?

Example 2
Consider the map $f :T^*(\mathbb{P}^1) \rightarrow \mathcal{N}$. Here, $\mathcal{N}$ denotes the nilpotent cone for the Lie algebra sl(2). Then f is proper and semismall and so $Rf_*(\mathbb{C}[1])$ is a pure perverse sheaf on $\mathcal{N}$ and hence by the decomposition theorem it is a direct sum of simple perverse sheaves. In fact $Rf_*(\mathbb{C}[1]) = IC_{\mathcal{N}} \oplus \mathbb{C}_0$.

Note for experts: to simplify things, throughout this post I’ve fixed an isomorphism between $\overline{\mathbb{Q}}_l$ and $\mathbb{C}$ and I’ve assumed that all sheaves are mixed.

17 thoughts on “Topology in prime characteristic”

1. Toby says:

2. bb says:

(Working over the complex numbers) If j:U -> X is an open immersion with X proper smooth and Y = X\U a normal crossings divisor, then $K = j_*(\Omega^\cdot_U)$ computing the de Rham cohomology of U is quasi-isomorphic to the log de Rham complex $L = \Omega^\cdot_U (\mathrm{log} Y)$ thanks to Grothendieck’s algebraic de Rham theorem. The complex L admits two filtrations: the stupid one (brutally truncate the complex) and the weight one (filter the terms by order of poles that show up). The former gives rise to a spectral sequence defining the Hodge filtration on the cohomology of U, while the latter gives rise to the weight spectral sequence which goes (essentially) from the cohomology of repeated fibre powers of Y over X to the cohomology of U. The filtration it defines is the weight filtration on U. If you’re working instead with a family over a DVR, then I think this goes under the name of the Rapoport-Zink spectral sequence (and is cool because it tells you the etale cohomology of a smooth variety with *bad* reduction over Q_p essentially in terms of varieties over finite fields). A reference for the Hodge theoretic weight spectral sequence is section 3 of Deligne’s “Theorie de Hodge II” paper.

3. bb says:

whoops, can someone correct my latex screwup? thanks.

4. Great. That is the kind of result I was alluding to above. Now, my question is:
Why is this weight filtration (for varieties over C) defined by the order of poles related to the weight filtration (for varieties over finite characteristic) defined using the eigenvalues of the Frobenius?

Perhaps I should read Hodge II.

5. bb says:

Hmm, I don’t know a reference for the weight filtrations matching up. Here’s how I’d try to say it (except one detail which I haven’t checked):

Let $j:U \to X$ be an open immersion and let $i:Y \to X$ the complement of U. Assume that X is smooth projective of equidimension n, and Y is a sncd (so repeated intersections of irreducible components of Y are smooth projective).

First off, $R \Gamma (Y,\mathbb{Q})$ has a natural mixed Hodge structure coming from the obvious proper hypercover $Y_\bullet \to Y$ — take $Y_0$ to be the disjoint union of components of Y, and take $Y_p$ to be the p-fold fibre product of $Y_0$ over Y. This describes the cohomology of Y in terms of the cohomology of the smooth projective varieties $Y_p$ which are simply p-fold intersections of components of Y.

Secondly, we have an exact sequence $0 \to j_!(\mathbb{Q}) \to \mathbb{Q} \to i_*(\mathbb{Q}) \to 0$ of (constructible) sheaves on X. Therefore, we obtain $R\Gamma_c(U,\mathbb{Q}) = \mathrm{ker} (R\Gamma (X,\mathbb{Q}) \to R\Gamma (Y,\mathbb{Q}))$ (the homotopy kernel) which defines $R\Gamma_c(U,\mathbb{Q})$ as an object of the derived category of mixed Hodge structures (the subscript indicates compactly supported cohomology). Duality tells us we should define $R\Gamma (U,\mathbb{Q})$ as $R\mathrm{Hom}(R\Gamma_c(U,\mathbb{Q}(n)[2n]), \mathbb{Q})$ to get a mixed Hodge structure on the cohomology of U given one on the compactly supported version. The weight filtration on $H^i(U,\mathbb{Q})$ thus defined agrees with the one defined using the de Rham complex because of the Poincare residue formula (3.1.5.2 of Hodge II) describing the associated graded of the weight filtration on the log de Rham complex, and the explicit description of the cohomology of Y in terms of that of the $Y_p$ — I haven’t checked this last statement.
The upshot of this is that we have described $R\Gamma_c(U,\mathbb{Q}_\ell)$ as the kernel of $R\Gamma(X,\mathbb{Q}_\ell) \to R\Gamma(Y,\mathbb{Q}_\ell)$ in Hodge land. The same description is equally valid in Galois land (i.e.: take a spread to everything in sight, reduce mod a suitably chosen prime and work with etale cohomology). Therefore, the weight filtration on $R\Gamma_c(U,\mathbb{Q}_\ell)$ is the same in the Hodge and Galois worlds provided the same statement is true for $R\Gamma(X,\mathbb{Q}_\ell)$ and $R\Gamma(Y,\mathbb{Q}_\ell)$. But this latter statement is clearly true because we know that the i-th cohomology group of a smooth projective variety has weight i in both worlds (for Y use the same argument on the hypercover).
In short, the requirement that $H^i(X,\mathbb{Q}_\ell)$ have weight i and geometry force uniqueness of the weight filtration while resolution of singularities guarantees existence.

6. bb says:

another latex screwup on my part: can someone make wordpress parse what it thinks is unparseable (an R\Gamma is the common feature of all mistakes, i think)? also, can “latex” be inserted before the two R\Gamma’s in the second to last sentence of the second to last paragraph? thanks. to avoid future trouble, i’ll avoid latex from now. sorry!

7. ninguem says:

8. I fixed the latex fixed in comment 6. Actually the problem was that you were writing \Q, not \mathbb{Q}. No shortcuts allowed in blog latex!

9. bb says:

Joel: whoops! thanks for fixing it.

ninguem: i used rational coeffecients for mixed Hodge structures (where it’s okay!), but was careful to use l-adic coeffecients for etale cohomology.

10. Ninguem: I wasn’t careful to use l-adic coefficients in my post. As I said in the last line of my post, I’ve fixed an isomorphism between $\overline{\mathbb{Q}}_\ell$ and $\mathbb{C}$.

11. bb: Very nice argument in 6. Essentially you are relating both weight filtrations to the one coming from the hypercover, and this one works well in both worlds.

12. It looks pretty good. I don’t particularly recommend the book by Kiehl and Weissauer by the way. I think that it (the book) has an excellent selection of topics but the material isn’t always presented in the most comprehensible way.

13. travis says:

are you missing some +i’s (e.g., weight of R^i f_!(A) is <= w+i rather than just w; similarly for complexes we say that having
weight <= w means having weight of H^i(A) <= w+i …?)

14. Anonymous says:

Hi. What is a filtration by subobjects?