# Generically a quotient

In this post I want to outline an argument that proves “most” algebraic stacks are generically “global” quotient stacks. I don’t have the time to add this to the stacks project now, but I hope to return to it in the not too distant future.

To fix ideas suppose that X is a Noetherian, reduced, irreducible algebraic stack whose geometric generic stabilizer is affine. Then I would like to show there exists a dense open substack U ⊂ X such that U ≅ [W/GL_n] for some Noetherian scheme W endowed with action of GL_n. The proof consists in repeatedly replacing X by dense open substacks each of which has some additional property:

1. We may assume that X is a gerbe, i.e., that there exists an algebraic space Y and a morphism X —> Y such that X is a gerbe over Y. This follows from Proposition Tag 06RC.
2. We may assume Y is an affine Noetherian integral scheme. This holds because X —> Y is surjective, flat, and locally of finite presentation, so Y is reduced, irreducible, and locally Noetherian by descent. Thus we get what we want by replacing Y be a nonempty affine open.
3. We may assume there exists a surjective finite locally free morphism Z —> Y such that there exists a morphism s : Z —> X over Y. Namely, pick a finite type point of the generic fibre of X —> Y and do a limit argument.
4. We may assume the projections R = Z ×_X Z —> Z are affine. Namely, the geometric generic fibres of R —> Z ×_Y Z are torsors under the geometric generic stabilizer which we assumed to be affine. A limit argument does the rest (note that we may shrink Z and Z ×_Y Z by shrinking Y).
5. We may assume the projections s, t : R —> Z are free, i.e., s_*O_R and t_*O_R are free O_Z-modules. This follows from generic freeness.
6. General principle. Suppose that (U, R, s, t, c) is a groupoid scheme with U, R affine and s, t free and of finite presentation. Consider the morphism p : U —> [U/R]. Then p_*O_U is a filtered colimit of finite free modules V_i on the algebraic stack [U/R]. This follows from a well known trick with basis elements.
7. General principle, continued. For sufficiently large i the stabilizer groups of [U/R] act faithfully on the fibres of the vector bundle V_i.
8. General principle, continued. [U/R] ≅ [W/GL_n] for some algebraic space W and integer n. Namely W is the quotient by R of the frame bundle of the vector bundle V_i.
9. We conclude that X = [W/GL_n] for some Noetherian, reduced irreducible algebraic space W.
10. The set of points where W is not a scheme is GL_n-invariant and not dense, hence we may assume W is a scheme by shrinking. (I think this works — there should be something easier you can do here, but I don’t see it right now.)

Note that we can’t assume that W is affine (a counter example is X = [Spec(k)/B] where B is the Borel subgroup of SL_{2, k} and k is a field). But with a bit more work it should be possible to get W quasi-affine as in the paper by Totaro (which talks about the harder question of when the entire stack X is of the form [W/GL_n] and relates it to the resolution property).

# Compact and perfect objects

Let R be a ring. Let D(R) be the derived category of R-modules. An object K of D(R) is perfect if it is quasi-isomorphic to a finite complex of finite projective R-modules. An object K of D(R) is called compact if and only of the functor Hom_{D(R)}(K, – ) commutes with arbitrary direct sums. In the previous post I mentioned two results on perfect complexes which I added to the stacks project today. Both are currently in the second chapter on algebra of the stacks project. Here are the statements with corresponding tags:

1. An object K of D(R) is perfect if and only if it is compact. This is Proposition Tag 07LT.
2. If I ⊂ R is an ideal of square zero and K ⊗^L R/I is a perfect object of D(R/I), then K is a perfect object of D(R). This is Lemma Tag 07LU.

Enjoy! If anybody knows a reference for the first result which predates the paper “Morita theory for derived categories” by Rickard I’d love to hear about it. Thanks.

# Updated list of contributors

So I’ve now updated the list of contributors to include literally everybody who I know has ever contributed a typo, error, made a suggestion, etc. In particular, please let me know if your name should be there and it isn’t.

The reason for doing this is that this is the only maintainable choice. Up till now I tried to make some choice as to when to put somebody on the list based on number and importance of contributions. However, looking at the list of people I added today I now see this was completely arbitrary and not at all objective! I hope I didn’t offend anybody!

You can look for what people have contributed by searching in the logs (except as with everything in life it isn’t perfect). To do this, first clone the project using

git clone git://github.com/stacks/stacks-project.git

then cd stacks-project and finally use something like

git log --grep=Lastname git log --grep=lastname git log --grep=Lastname --color -p git log --grep=lastname --color -p 

Future contributors who send new material in the form of latex will show up as authors as I described in this post a while back and those who point out typos, errors, suggestions, etc without sending latex will show up in the commit messages.

Please let me know if you have an idea for a more efficient way to keep track of contributors.

# Finiteness of Crystalline Cohomology

In my lectures on crystalline cohomology I have worked through the basic comparison theorem of crystalline cohomology with de Rham cohomology. This comparison allows one to prove that crystalline cohomology has some good properties exactly as is done by Berthelot in his thesis. To formulate these properties we introduce some notation

• p is a prime number,
• (A, I, γ) is a divided power ring over Z_{(p)},
• S = Spec(A) and S_0 = Spec(A/I),
• f : X —> Y is a quasi-compact, quasi-separated smooth morphism of schemes over S_0,
• E = Rf_{cris, *}O_{X/S} on Cris(Y/S).

The comparison theorem is the main ingredient in showing that E has the following properties:

1. the cohomology sheaves H^i(E) are locally quasi-coherent, i.e., for every object (V, T, δ) of Cris(Y/S) the restriction H^i(E)_T = H^i(E_T) of H^i(E) to the Zariski site of T is a quasi-coherent O_T-module,
2. E is a “crystal”, i.e., for every morphism h : (V, T, δ) —> (V’, T’, δ’) of Cris(Y/S) the comparison map Lh^*E_{T’} —> E_T is a quasi-isomorphism,
3. if (V, T, δ) is an object of Cris(Y/S) and if f is PROPER and T is a NOETHERIAN scheme, then E_T is a perfect complex of O_T-modules.

In case f is proper, I did not find a reference in the literature proving that E is a perfect complex of modules on Cris(Y/S), i.e., that E_T is a perfect complex of O_T-modules for every object T of the small crystalline site of Y. For those of you who are quickly scanning this web-page let me make the statement explicit as follows:

Let (A, I, γ) be a divided power ring with p nilpotent in A. Let X be proper smooth over A/I. Then RΓ(Cris(X/A), O_{X/A}) is a perfect complex of A-modules.

Again, far as I know this isn’t in the literature, but let me know if you have a reference [Edit: see comment by Bhargav below]. Here is an argument which I think works, but I haven’t written out all the details. By the base change theorem (part 2 above) it is enough if we can find a divided power ring (B, J, δ) with p nilpotent in B and a homomorphism of divided power rings B —> A such that X is the base change of a quasi-compact smooth scheme over B/J and such that the result holds over B. Arguing in this way and using standard limit arguments in algebraic geometry we reduce the question to the case where A/I is a finitely generated Z-algebra. Writing A/I = Z/p^NZ[x_1, …, x_n]/(f_1, …, f_m) we reduce to the case where A is the divided power envelope of (f_1, …, f_m) in Z/p^NZ[x_1, …, x_n]. In this case we see that we can lift X to a smooth scheme over A/(f_1, …, f_m)A because A/I maps to A/(f_1, …, f_m)A! Moreover, the ideal (f_1, …, f_m) is a nilpotent ideal in A! Now suppose we have a modified comparison theorem which reads as follows:

Modified comparsion. Let (A, I, γ) be a divided power ring with p nilpotent in A. Let J ⊂ I be an ideal. Let X’ be proper smooth over A’ = A/J and set X = X’ ⊗ A/I. Then RΓ(Cris(X/A), O_{X/A}) ⊗^L_A A’ = RΓ(X’, Ω^*_{X’/A’}).

The tricky bit is that J needn’t be a divided power ideal, but I think a Cech cover argument will work (this is a bit shaky). The usual arguments show that RΓ(X’, Ω^*_{X’/A’}) is a perfect complex of A’-modules. Proof of the statement above is finished by observing that a complex of A-modules K^* such that K^* ⊗^L_A A’ is perfect is perfect. (My proof of this uses that the kernel J of A —> A’ is a nilpotent ideal and the characterization of perfect complexes as those complexes such that hom in D(A) out of them commutes with direct sums.)

I’ll think this through more carefully in the next few days and if I find something wrong with this argument I’ll edit this post later. Let me know if you think there is a problem with this idea.

# Proper hypercoverings and cohomology

Fix a field k. Let X be a variety. Let ε : X_* —> X be a proper hypercovering (see earlier post). Let Λ be a finite ring. Then we have H^*(X, Λ) = H^*(X_*, Λ) for H^* = etale cohomology. This follows from the proper base change theorem combined with the case where X is a (geometric) point, in which case it follows from the fact that there is a section X —> X_*. (This is a very rough explanation.)

Now let k be a perfect field of characteristic p > 0. For a smooth proper variety X over k denote H^*(X/W) the crystalline cohomology of X with W coefficients. So for example H^0(Spec(k)/W) = W. Similarly H^*(X/k) is crystalline cohomology with coefficients in k, so that H^*(X/k) = H^*(X, Ω_{X/k}).

It turns out that crystalline cohomology does not satisfy descent for proper hypercoverings. What I mean is this: If X_* —> X is a proper hypercovering and all the varieties X, X_n are smooth and projective, then it is not the case that H^*(X/W) = H^*(X_*/W). (Note: It takes some work to even define H^*(X_*/W)……….) Descent fails for two reasons.

Nobuo Tsuzuki has shown that descent for proper hypercoverings does hold for rigid cohomology, and hence for H^*(X/W)[1/p] in the situation above. The point of this post is to consider coefficients where p is not invertible.

The first is purely inseparable morphisms. Namely, if X_0 —> X is a finite universal homeomorphism, then the constant simplicial scheme X_* with X_n = X_0 is a proper hypercovering of X. In particular, if k is a perfect field of characteristic p,  X_* is the constant simplicial scheme with value P^1_k, X = P^1_k and the augmentation is given by π : X_0 —> X which raises the coordinate on P^1 to the pth power. Hence, if H^* is a cohomology theory which satisfies descent for proper hypercoverings, then π^* : H^*(P^1) —> H^*(P^1) should be an isomorphism, which isn’t the case for crystalline cohomology because π^* : H^2(P^1) —> H^2(P^1) is multiplication by p which is not an automorphism of W.

Let X_* —> X be a proper hypercovering (of smooth proper varieties over k) with the following additional property: For every separably algebraically closed field K/k the map of simplicial sets X_*(K) —> X(K) is an equivalence (not sure what the correct language is here and I am too lazy to look it up; what I mean is that it is a hypercovering in the category of sets with the canonical topology). This does not hold for our example above because the generic point of X does not lift to X_0 even after any separable extension. It turns out that crystalline cohomology does not satisfy descent for such proper hypercoverings either.

To construct an example, note that if the desent is true with W coefficients, then it is true with k-coefficients because RΓ(X/k) = RΓ(X/W) \otimes_W k (again details on general theory have to be filled in here, but this is just a blog…….). OK, now go back to the hypercovering I described in this post. Namely, assume char(k) = 2, let X = P^1, let X_0 —> X be an Artin-Schreier covering ramified only above infinity, let X_1 the disjoint union of 2 copies of X_0, let X_2 be the disjoint union of 4 copies of X_0 plus 4 extra points over infinity, and so on. Then, I claim, you get one of these proper hypercoverings described above. I claim that H^1(X_*/k) is not zero which will prove that the descent for crystalline cohomology fails. To see this note that there is a spectral sequence H^p(X_q/k) => H^{p + q}(X_*/k). Look at the term H^1(X_0/k). This has dimension 2g where g is the genus of X_0. The kernel of the differential to H^1(X_0/k) —> H^1(X_1/k) is the subspace of invariants under the involution on X_0. Hence it has dimension at least g (because of the structure of actions of groups of order 2 on vector spaces in characteristic 2). The next differential maps into a subquotient of H^0(X_2/k). But since I needed only to add 4 points to construct my proper hypercovering, it follows that dim H^0(X_2/k) ≤ 8. Hence we see that dim H^1(X_*/k) is at least g – 8. As we can make the genus of a Artin-Schreyer covering arbitrarily large, we find that this is nonzero in general.

This is exactly the kind of negative result nobody would ever put in an article. I think in stead of a journal publishing papers that were rejected by journals, it might be fun to have a place where we collect arguments that do not work, or even just things that aren’t true. What do you think?

# Rlim

Let N be the natural numbers. Think of N as a category with a unique morphisms n —> m whenever m ≥ n and endow it with the chaotic topology to get a site. Then a sheaf of abelian groups on N is an inverse system (M_e) and H^0 corresponds to the limit lim M_e of the system. The higher cohomology groups H^i correspond to the right dervided functors R^i lim. A good exercise everybody should do is prove directly that R^i lim is zero when i > 1.

Consider D(Ab N). This is the derived category of the abelian category of inverse systems. An object of D(Ab N) is a complex of inverse systems (and not an inverse system of complexes — we will get back to this). The functor RΓ(-) corresponds to a functor Rlim. Given a bounded below complex of inverse systems where all the transition maps are surjective, then Rlim is computed by simply taking the lim in each degree.

What if we have an inverse system with values in D(Ab)? In other words, we have objects K_e in D(Ab) and transition maps K_{e + 1} —> K_e in D(Ab). By choosing suitable complexes K^*_e representing each K_e we can assume that there are actual maps of complexes K^*_{e + 1} —> K^*_e representing the transition maps in D(Ab). Thus we obtain an object K of D(Ab N) lifting the inverse system (K_e). Having made this choice, we can compute Rlim K.

During the lecture on crystalline cohomology yesterday morning I asked the following question: Is Rlim K independent of choices? The reason for this question is that there are a priori many isomorphism classes of objects K in D(Ab N) which give rise to the inverse system (K_e) in D(Ab). It turns out that Rlim K is somewhat independent of choices, as Bhargav explained to me after the lecture. Namely, you can identify Rlim K with the homotopy limit, i.e., Rlim K sits in a distinguished triangle

Rlim K —–> Π K_e —–> Π K_e ——> Rlim K

in D(Ab) where the second map is given by 1 – transition maps. And this homotopy limit depends, up to non-unique isomorphism, only on the inverse system in D(Ab).

One of the things I enjoy about derived categories is how things don’t work, but how in the end it sort of works anyway. The above is a nice illustration of this phenomenon.