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.

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?

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[1]

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.

Consider the topology τ on the category of schemes where a covering is a finite family of proper morphisms which are jointly surjective. (Dear reader: does this topology have a name?) For the purpose of this post proper hypercoverings will be τ-hypercoverings as defined in the chapter on hypercoverings. Proper hypercoverings are discussed specifically in Brian Conrad’s write up. In this post I wanted to explain an example which I was recently discussing with Bhargav on email. I’d love to hear about other “explicit” examples that you know about; please leave a comment.

The example is an example of proper hypercovering for curves. Namely, consider a separable degree 2 map X —> Y of projective nonsingular curves over an algebraically closed field and let y be a ramification point. The simplicial scheme X_* with X_i = normalization of (i + 1)st fibre product of X over Y is NOT a proper hypercovering of Y. Namely, consider the fibre above y (recall that the base change of a proper hypercovering is a proper hypercovering). Then we see that X_0 has one point above y, X_1 has 2 points above y, and X_2 has 4 points above y. But if X_2 is supposed to surject onto the degree 2 part of cosk_1(X_1 => X_0) then the fibre of X_2 over y has to have at least 8 points!!!!

Namely cosk_1(S —> *) where S is a set and * is a singleton set is the simplicial set with S^3 in degree 2, S^6 in degree 3, etc because an n-simplex should exist for any collection of (n + 1 choose 2) 1-simplices since each of the 1-simplices bounds the unique 0-simplex on both sides, see for example Remark 0189. So I think that to construct the proper hypercovering we have to throw in some extra points in simplicial degree 2 which sort of glue the two components of X_1.

Now, as X_* does work over the complement of the ramification locus in Y, I think you can argue that it really does suffice to add finite sets of points to X_* (over ramification points) to get a proper hypercovering!

PS: Proper hypercoverings are interesting since they can be used to express the cohomology of a (singular) variety in terms of cohomologies of smooth varieties. But that’s for another post.

Theorem. Let f : X —> Y be a proper morphism of varieties and let y ∈ Y with f^{-1}(y) finite. Then there exists a neighborhood V of y in Y such that f^{-1}(V) —> V is finite.

If X is quasi-projective, then there is a simple proof: Choose an affine open U of X containing f^{-1}(y); this uses X quasi-projective. Using properness of f, find an affine open V ⊂ Y such that f^{-1}V ⊂ U. Then f^{-1}V = V x_Y U is affine as Y is separated. Hence f^{-1}V —> V is a proper morphism of affines varieties. Such a morphism is finite, see Lemma Tag 01WM for an elementary argument.

I do not know a truly simple proof for the general case. (Ravi explained a proof to me that avoids most cohomological machinery, but unfortunately I forgot what the exact method was; it may even be one of the arguments I list below.) Here are some different approaches.

(A) One can give a proof using cohomology and the theorem on formal functions, see Lemma Tag 020H.

Let ZMT be Grothendieck’s algebraic version of Zariski’s main theorem, see Theorem Tag 00Q9.

(B) One can prove the result using ZMT and etale localization. Namely, one proves that given any finite type morphism X —> Y with finite fibre over y, there is after etale localization on Y, a decomposition X = U ∐ W with U finite over Y and the fibre W_y empty (see Section Tag 04HF). In the proper case it follows that W is empty after shrinking Y. Finally, etale descent of the property “being finite” finishes the argument. This method proves a general version of the result, see Lemma Tag 02LS.

(C) A mixture of the above two arguments using ZMT and a characterization of affines:

1. Show that after replacing Y by a neighborhood of y we may assume that all fibers of f are finite. This requires showing that dimensions of fibres go up under specialization. You can prove this using generic flatness and the dimension formula (as in Eisenbud for example) or using ZMT.
2. Let X’ —> Y be the normalization of Y in the function field of X. Then X’ —> Y is finite and X’ and X are birational over Y. Finiteness of X’ over Y requires finiteness of integral closure of finite type domains over fields, which follows from Noether normalization + epsilon.
3. Let W ⊂ X x_Y X’ be the closure of the graph of the birational rational map from X to X’. Then W —> X is finite and birational and W —> X’ is proper with finite fibres and birational.
4. Using ZMT one shows that W —> X’ is an isomorphism. Namely, a corollary of ZMT is that separated quasi-finite birational morphisms towards normal varieties are open immersions.
5. Now we have X’ —> X —> Y with the first arrow finite birational and the composition finite too. After shrinking Y we may assume Y and X’ are affine. If X is affine, then we win as O(X) would be a subalgebra ofa finite O(Y)-algebra.
6. Show that X is affine because it is the target of a finite surjective morphism from an affine. Usually one proves this using cohomology. The Noetherian case is Lemma Tag 01YQ (this uses less of the cohomological machinery but still uses the devissage of coherent modules on Noetherian schemes). In fact, the target of a surjective integral morphism from an affine is affine, see Lemma Tag 05YU.

This post is another attempt to explain how incredibly useful the notion of a cocontinuous functor of sites really is. I already tried once here.

Let u : C —> D be a functor between sites. We say u is cocontinuous if for every object U of C and every covering {V_j —> u(U)} in D there exists a covering {U_i —> U} in C such that {u(U_i) —> U} refines {V_j —> u(U)}. This is the direct translation of SGA 4, II, Defintion 2.1 into the language of sites as used in the stacks project and in Artin’s notes on Grothendieck topologies. Note that we do not require that u transforms coverings into coverings, i.e., we do not assume u is continuous. Often the condition of cocontinuity is trivial to check.

Lemma Tag 00XO A cocontinuous functor defines a morphism of topoi g : Sh(C) —> Sh(D) such that g^{-1}G is the sheaf associated to U |—> G(u(U)).

The reader should contrast this with the “default” which is morphisms of topoi associated to continuous functors (where one has to check the exactness of the pull back functor explicitly in each case!). Let’s discuss some examples where the lemma applies.

The standard example is the functor Sch/X —> Sch/Y associated to a morphism of schemes X —> Y for any of the topologies Zariski, etale, smooth, syntomic, fppf. This defines functoriality for the big topoi. This also works to give functoriality for big topoi of algebraic spaces and algebraic stacks. In exactly the same way we get functoriality of the big crystalline topoi.

Another example is any functor u : C —> D between categories endowed with the chaotic topology, i.e., such that sheaves = presheaves. Then u is cocontinuous and we get a morphism of topoi Sh(C) —> Sh(D).

Finally, an important example is localization. Let C be a site and let K be a sheaf of sets. Let C/K be the category of pairs (U, s) where U is an object of C and s ∈ K(U). Endow C/K with the induced topology, i.e., such that {(U_i, s_i) —> (U, s)} is a covering in C/K if and only if {U_i —> U} is a covering in C. Then C/K —> C is cocontinuous (and continuous too) and we obtain a morphism of topoi Sh(C/K) —> Sh(C) whose pullback functor is restriction.

What I am absolutely not saying is that the lemma above is a “great” result. What I am saying is that, in algebraic geometry, the lemma is easy to use (no additional conditions to check) and situations where it applies come up frequently and naturally.

PS: Warning: In some references a cocontinuous functor is a functor between categories (not sites) is defined as a functor that commutes with colimits. This is a different notion. Too bad!

Since the last update we have added a new chapter. This chapter explains the Popescu-Ogoma-Andre-Swan proof of general Neron desingularization (GND). As explained here there is a way to reduce to the case of a base field. This does simplify the rest of the arguments somewhat, but not as much as I’d have liked.

The heart of the proof of GND is in the proof of Lemma Tag 07FJ. For some reason working through this proof made me think of playing chess, in that you have to think ahead several moves and the steps you take early in the proof almost don’t seem to make sense. I have a hard time explaining it, even to myself. But then, I was never any good at chess.

Well actually 3011 pages at this very moment. Also

• 290489 lines of tex,
• 9300 tags, and
• 2226 commits since we started using git on May 20, 2008.

Enjoy!

Since the last updateon October 12 we have added the following material

1. Gabber’s argument that categories of quasi-coherent modules form a Grothendieck abelian category (for schemes, spaces, and algebraic stacks),
2. an example of an fpqc space which is not an algebraic space,
3. an example of a quasi-compact non-quasi-separated morphism of schemes such that pushforward does not preserve quasi-coherency,
4. some material related to my course on commutative algebra: exercise, lemmas, shorten proof of ZMT, etc
5. introduced lisse-etale (and flat-fppf) sites,
6. functoriality of lisse-etale topos for smooth morphisms (and flat-fppf for flat morphisms),
7. material on Grothendieck abelian categories, incuding existence of injectives and existence of enough K-injective complexes (following Spaltenstein and Serp\’e),
8. cohomology of unbounded complexes and adjointness of Lf^* and Rf_*,
9. a lot of material on D_{QCoh}(X) for an algebraic stack X, including Rf_* (on bounded below for quasi-compact and quasi-separated morphisms) and Lf^* (unbounded for general f).

In particular my suggestion in this post worked out exactly as advertised. The existence of Rf_* is straightforward. It turns out that once you prove that the category D_{QCoh}(X) as defined in the blog post is equivalent to the version of D_{QCoh}(X) in L-MB or Martin Olsson’s paper (i.e. defined using the lisse-etale site), then you immediately obtain the existence of Lf^*. Namely, the existence of the lisse-etale site is used to prove that the Verdier quotient used to define D_{QCoh}(X) is a Bousfield colocalization (technically it is easier to use the flat-fppf site to do this, because we use the fppf topology as our default topology, but one can use either).

A bit of care is needed when working with the lisse-etale site and the lisse-etale topos. As discussed elsewhere, one reason is that the lisse-etale topos isn’t functorial for morphisms of algebraic stacks. Here is a another. There is a comparison morphism of topoi

g : Sh(X_{lisse,etale}) —-> Sh(X_{etale})

The functor g^{-1} has a left adjoint denoted g_! (on sheaves of sets) and we have g^{-1}g_! = g^{-1}g_* = id. This means that Sh(X_{lisse,etale}) is an essential subtopos of Sh(X_{etale}), see SGA 4, IV, 7.6 and 9.1.1. Let K be a sheaf of sets on X_{lisse,etale}. Let I be an injective abelian sheaf on X_{etale}. Question: H^p(K, g^{-1}I) = 0? In other words, is the pullback by g of an injective abelian sheaf limp? If true this would be a convenient way to compare cohomology of sheaves on X_{etale} with cohomology of sheaves on the lisse-etale site. Unfortunately, we think this isn’t true (Bhargav made what is likely a counter example — but we haven’t fully written out all the details).

Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated.

Let k be a field of characteristic p > 0. Let G = Z/pZ be the cyclic group of order p. Set S = Spec(k[x]) and let X = [S/G] be the stacky quotient where G acts trivially on S. Consider the morphism f : X —> S. Then Rf_*O_X is a complex with cohomology sheaves isomorphic to O_S for all p >= 0. In fact Rf_*O_X is quasi-isomorphic to ⊕ O_S[-n] where n runs over nonnegative integers.

Now consider the complex K = ⊕ O_X[m] where m runs over the nonnegative integers. This is an object of D_{QCoh}(X) but it isn’t bounded below. So we have to pay attention if we want to compute Rf_*K. Namely, in D(O_X) the complex K is also K = ∏ O_X[m]. Since cohomology commutes with products, we see that

Rf_*K = ∏ Rf_*O_X[m] = ∏ (⊕ O_S[m – n]).

In degree 0 we get an infinite product of copies of O_S which isn’t quasi-coherent.

Conclusion: Rf_* does not map D_{QCoh}(X) into D_{QCoh}(S).

Of course if f is a quasi-compact and quasi-separated morphism between algebraic spaces, then this kind of thing doesn’t happen.