Crystalline Site 2: The Topos

Last time we defined crystalline cohomology by defining the crystalline site. Our setup was to fix {(S, \mathcal{I}, \gamma)} a PD-scheme and look at an {S}-scheme {X} to which {\gamma} extends. Remember that the objects of the site {\mathrm{Crys}(X/S)} were triples {(U, T, \delta)} that we short-handed with {T}. Today we’ll start by re-examining these objects. We can give a little better description which will help us figure out what a sheaf on this site is. Recall that the full subcategory of the category {\mathrm{Psh}(Crys(X/S))} of presheaves (of sets) on the site is a topos which we’ll denote {(X/S)_{Crys}}.

Suppose {\mathcal{F}} is an object of {(X/S)_{Crys}}, i.e. a sheaf. This just means it is a functor {\mathcal{F}: \mathrm{Cyrs}(X/S)^{op}\rightarrow \mathrm{Set}} that also satisfies for any covering {\{T_i\rightarrow T\}} the following sequence being exact {\mathcal{F}(T)\rightarrow \prod \mathcal{F}(T_i)\stackrel{\rightarrow}{\rightarrow} \mathcal{F}(T_i\cap T_j)}.

Let’s fix some object {(U,T,\delta)} and some sheaf {\mathcal{F}}. If {T'\hookrightarrow T} is a Zariski open set then define {U'=U\cap T'} and {\delta'=\delta|_{T'}}. There is an inclusion {(U', T', \delta')\rightarrow (U, T, \delta)}, which by construction is a map in the category {\mathrm{Crys}(X/S)}, so by virtue of being a contravariant functor we get {\mathcal{F}(T)\rightarrow \mathcal{F}(T')}. The contravariant functor {T'\mapsto \mathcal{F}(T')} is a sheaf on the Zariski site of {T} which we denote {\mathcal{F}_{(U, T, \delta)}}.

We have a proposition that the data of a sheaf {\mathcal{F}} on {\mathrm{Crys}(X/S)} is equivalent to the data that for ever {S}-PD thickening {(U, T, \delta)} we assign a Zariski sheaf on {T} denoted conspicuously by {\mathcal{F}_T} and for every morphism {u:(U_1, T_1, \delta _1)\rightarrow (U, T, \delta)} we assign a map {\rho_u: u^{-1}(\mathcal{F}_T)\rightarrow \mathcal{F}_{T_1}} satisfying two compatibility conditions.

1) If {v:(U_2, T_2, \delta_2)\rightarrow (U_1, T_1, \delta_1)} is another map then we have a commutative diagram

{\begin{matrix} v^{-1}u^{-1}\mathcal{F}_T & \rightarrow & u^{-1}\mathcal{F}_{T_1} & \rightarrow & \mathcal{F}_{T_2} \\ \| & & & & \| \\ (u\circ v)^{-1}\mathcal{F}_T & - & - & \rightarrow & \mathcal{F}_{T_2} \end{matrix}}


2) If {u: T_1\rightarrow T} is an open immersion, the map {\rho_u^{-1}: u^{-1}(\mathcal{F}_T)\rightarrow \mathcal{F}_{T_1}} is an isomorphism.

Using this new characterization it is much easier to describe what the structure sheaf {\mathcal{O}_{X/S}} in this site should be, since to give a sheaf I need to give on every thickening {T} a Zariski sheaf and we have a natural Zariski structure sheaf {\mathcal{O}_T}. This assignment can be seen to satisfy the conditions and hence we have a crystalline sheaf. Note that this was not the only choice that works, though. We’ve already pointed out that {|U|\simeq |T|}, so we could also assign {T\mapsto \mathcal{O}_U} which is another crystalline sheaf that we’ll denote {\mathcal{O}_X}.

For our last example, let’s just create a new sheaf from our two above by taking {T\mapsto \mathrm{Ker}(\mathcal{O}_T\rightarrow \mathcal{O}_U)}. This is actually a sheaf of P.D. ideals in {\mathcal{O}_{X/S}} and we’ll call it {\mathcal{J}_{X/S}}. By construction it sits in an exact sequence {0\rightarrow \mathcal{J}_{X/S}\rightarrow \mathcal{O}_{X/S}\rightarrow \mathcal{O}_X \rightarrow 0}.

The last thing to point out for today is that this alternate way of thinking about crystalline sheaves is that it really gives us a way of thinking in Zariski terms which we’re already familiar with. In particular, if {v:\mathcal{F}\rightarrow \mathcal{G}} is a map of crystalline sheaves we can check whether it is an isomorphism (or surjective or injective) on stalks by checking for each {x\in X} and each {S}-P.D. thickening {T} of a neighborhood if the map {(\mathcal{F}_T)_x\rightarrow (\mathcal{G}_T)_x} is an isomorphism.


Crystalline Site 1

I’ve decided on pulling the motivation back into the picture. Recall way back when we were thinking about the shortcomings of trying to replicate a de Rham type cohomology theory in positive characteristic. One of our motivations is that we want a theory that has no problem being done in positive characteristic, but actually gives us the de Rham cohomology if there is some lift to characteristic {0}. We even tried to just define it this way. Take a lift, do de Rham, and then check that the result is independent of lift. The problem is that there are things that don’t lift to characteristic {0}, and the lifting process is definitely not an efficient process for computing.

This is where crystalline cohomology enters the picture. We’ll make this more precise later on, but if we have a smooth lifting {X\rightarrow S} to characteristic {0}, then we’d like to have a canonical isomorphism {H^*_{crys}(X_0/S)\rightarrow \mathbf{H}^*(X_{zar}, \Omega^\cdot)}. Since we’ve already talked about what it means to do cohomology of a sheaf on a site, we can actually state pretty easily what crystalline cohomology is. Suppose {X} is a variety over {S}. There is the crystalline site {\mathrm{Crys}(X/S)}, and {H^n_{crys}(X/S)=H^n(X_{cyrs}, \mathcal{O}_{X/S})}, so the crystalline cohomology is just sheaf cohomology on the crystalline site. The work is going to be in figuring out how to think about this new site.

First, we define a P.D. scheme. This is exactly what it sounds like. There is no problem in extending all the definitions done for rings so far into definitions on the sections of sheaves. For instance, if {X} is a space and {\mathcal{A}} and {\mathcal{I}} are sheaves of rings on {X}, then we say {(\mathcal{A}, \mathcal{I}, \gamma)} is a sheaf of P.D. rings if {(\mathcal{A}(U), \mathcal{I}(U), \gamma)} is a P.D. ring for all {U\subset X} open. A P.D. ringed space is just a ringed space with a sheaf of P.D. rings on it {(X, (\mathcal{A}, \mathcal{I}, \gamma))}. Inverse image and pushforwards of sheaves under maps {f:X\rightarrow Y} preserve the property of being a sheaf of P.D. rings.

Given a P.D. ring {(A, I, \gamma)} we can define {\mathrm{Spec}(A, I, \gamma)} to be the locally ringed space {(|\mathrm{Spec}(A)|, \mathcal{O})} where {\mathcal{O}} is the sheaf of P.D. rings obtained under the canonical extensions we get of {\gamma} since localization is flat and we checked previously that {\gamma} extends to any flat {A}-algebra. A P.D. scheme is a locally ringed space locally isomorphic to {\mathrm{Spec}(A, I, \gamma)}. Morphisms in this category are morphisms of locally ringed spaces that are P.D. morphisms on sections.

We’ll spend a little more time with these definitions and related issues next time. The goal of this post is to define the crystalline site. Now we’ll want to fix a base, so let {(S, I, \gamma)} be a P.D. scheme. If {X} is an {S}-scheme, then by looking locally it makes sense to ask whether or not {\gamma} extends to {X}. If it does, then we can define the crystalline site {\mathrm{Crys}(X/S)} as follows: The objects are pairs {(U\hookrightarrow T, \delta)} where {U\subset X} is Zariski open and {U\hookrightarrow T} is a closed {S}-immersion defined by the quasi-coherent sheaf of ideals {\mathcal{J}} where {\delta} is a P.D. structure on {\mathcal{J}} compatible with {\gamma}. We abuse notation and call {(U\hookrightarrow T, \delta)} just {T}.

The morphisms {u:T\rightarrow T'} are commutative diagrams

{\begin{matrix} U & \hookrightarrow & T \\ \downarrow & & \downarrow \\ U' & \hookrightarrow & T' \end{matrix}}

where {U\rightarrow U'} is a Zariski inclusion of open sets of {X} and {T\rightarrow T'} is a P.D. map over {S}. A covering is just a collection of maps {\{u_i: T_i\rightarrow T\}} such that {T_i\rightarrow T} are open immersions and {T=\cup T_i}. We’ll just end this post by making several remarks and giving an example that we’re aiming at.

First, the term for an object {(U\hookrightarrow T, \delta)} is an “S-PD thickening of {U}“. One of the consequences of requiring {\gamma} to extend to {X} is that it makes {(U\rightarrow U, 0)} an object of our site for any Zariski open {U\subset X}. Another consequence of our definitions is that all our thickenings {U\rightarrow T} are topological homeomorphisms since they are defined by nilpotent ideals, {\mathcal{J}}. The last remark is that if {\{T_i\rightarrow T\}} is a covering, it comes with a collection of P.D. structures: {\delta_i}. By compatibility, the collection {\delta_i} completely determines {\delta} and conversely, given {\delta}, we can restrict and find out what the {\delta_i} must be.

The example that we want to think about is when we have some lifting of {X} over a postive characteristic field {k} to {W_n(k)}. In this situation {S=\mathrm{Spec}(W_n)} with {\mathcal{I}=(p)} with the canoncial P.D. structure inherited from {W}. We’ll look at this more closely when we are working with actual examples of lifted schemes.

Divided Powers 4

Today we’ll look at the P.D. envelope of an ideal. To do this properly would take many pages of gory calculations, so we’ll be a little sketchy in order to get the idea out there. Before we do that we need to look at a construction I’ve been avoiding on purpose. Suppose {M} is an {A}-module. Then there is a P.D. algebra {(\Gamma_A(M), \Gamma_A^+(M), \gamma)} and an {A}-linear map {\phi:M\rightarrow \Gamma_A^+(M)} satisfying the universal property that given any {(B, J, \delta)} an {A}-P.D. algeba and {\psi: M\rightarrow J} an {A}-linear map there is a unique P.D. map {\overline{\psi}:(\Gamma_A(M), \Gamma_A^+(M), \gamma)\rightarrow (B, J, \delta)} with the property {\overline{\psi}\circ \phi=\psi}.

Let’s be a little more explicit what this is now. First, {\Gamma_A(M)} is a graded algebra with {\Gamma_0(M)=A}, {\Gamma_1(M)=M} and {\Gamma^+(M)=\bigoplus_{i\geq 1} \Gamma_i(M)}. Let’s denote {x^{[1]}} for {\phi(x)} and {x^{[n]}} for {\gamma_n(\phi(x))}. In fact, by abusing notation we often just write {[ \ ]} in place of {\gamma} for the P.D. structure. This is because {\Gamma_n(M)} is generated as an {A}-module by {\{x^{[q]}=x_1^{[q_1]}\cdots x_k^{[q_k]} : \sum q_i=n, x_i\in M\}}. This should just be thought of as a “generalized P.D. polynomial algebra”. We’ll soon see its importance. Now back to the regularly planned post.

Let {(A, I, \gamma)} be a P.D. algebra and {J} an ideal of {B} which is an {A}-algebra. There exists a P.D. envelope of {J} which is a {B}-algebra denoted {\frak{D}_{B,\gamma}(J)} (or sometimes just {\frak{D}} when the rest is understood) with a P.D. ideal {(\overline{J}, [ \ ])} such that {J\frak{D}_{B,\gamma}(J)\subset \overline{J}} compatible with {\gamma} and satisfying a universal property:

For any {B}-algebra, {C}, containing an ideal {K} containing {JC} with a compatible (with {\gamma}) P.D. structure {\delta}, there is a unique P.D. morphism {(\frak{D}_{B, \gamma}(J), \overline{J}, [ \ ])\rightarrow (C, K, \delta)} which makes the diagram commute:

{\begin{matrix} (B, J) & \rightarrow & (\frak{D}, \overline{J}) \\ \uparrow & \searrow & \downarrow \\ (A, I) & \rightarrow & (C, K) \end{matrix}}

Where that vertical right arrow should be dotted. Like I said, the existence of this thing takes a lot of tedious calculations. You basically show by brute force that it exists in a few special cases and then reduce the general case to these. These calculations actually bear out a few interesting things that we’ll just list:

First, {\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma}(J+IB)}. Next, if our structure map actually factors {A\rightarrow A'\rightarrow B} and we have an extension to {\gamma'} on {A'}, then {\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma'}(J)}, which essentially tells us that it really is an “envelope” or some sort of minimal construction.

More importantly, we have a more concrete way to think about the construction. Namely, {\frak{D}} is generated as a {B}-algebra by {\{x^{[n]}: n\geq 0, x\in J\}} and any set of generators of {J} gives a set of P.D. generators for {\overline{J}}.

Let’s do two examples to get a feel for what this thing is. Let {B=A[x_1, \ldots, x_n]} and {J=(x_1, \ldots, x_n)}, then we’ll consider the trivial P.D. structure on {A} given by the {0} ideal. This gives us {\frak{D}(J)=A\langle x_1, \ldots, x_n\rangle} the P.D. polynomial algebra. Along the same lines but slightly more generally, suppose {M} is an {A}-module and {B=\mathrm{Sym}(M)}. Let {J} be the ideal of generated by the postive graded part {J=\mathrm{Sym}^+(M)} and do everything with respect to the trivial P.D. structure on {A} again. We get {\frak{D}=\Gamma_A(M)}.

The other example is that when we have {\gamma} extending to {B/J} with a section {B/J\rightarrow B}, then the compatibility condition is irrelevant. This just means that {\frak{D}_{B, \gamma}(J)\simeq \frak{D}_{B, 0}(J)}. This is just because the section gives us that {\frak{D}_{B, 0}(J)=B/J\bigoplus \overline{J}} and an application of the universal property.

I’m not sure what to do next. I’m sure I’ve alienated all my readers with all this. At this point I could shift over and at least define crystalline cohomology. We have enough to cover some of the basic definitions and actually show that all this has a purpose. On the other hand, we definitely have a ton more properties we should do if we want to get anywhere with crystalline cohomology.

Divided Power Structures 1

At some point in the distant future we may want to work with Divided Power structures if I ever get around to crystalline cohomology, so why not start writing about it now? Basically this is how we are going to be able to talk about things that require division when working in positive characteristic. Today we’ll just quickly give the definition and then a bunch of easy examples.

Suppose {A} is a commutative ring and {I} an ideal. Divided powers on {I} are a collection of maps {\gamma_i: I\rightarrow A} for all integers {i\geq 0} for which we have the following five properties:

1. For all {x\in I}, {\gamma_0(x)=1}, {\gamma_1(x)=x} and {\gamma_i(x)\in I}
2. For {x, y\in I} we have {\gamma_k(x+y)=\sum_{i+j=k}\gamma_i(x)\gamma_j(y)}
3. For {\lambda\in A}, {x\in I}, we have {\gamma_k(\lambda x)=\lambda^k\gamma_k(x)}
4. For {x\in I} we have {\gamma_i(x)\gamma_j(x)=((i,j))\gamma_{i+j}(x)}, where {((i,j))=\frac{(i+j)!}{(i!)(j!)}}
5. {\gamma_p(\gamma_q(x))=C_{p,q}\gamma_{pq}(x)} where {C_{p,q}=\frac{(pq)!}{p!(q!)^p}}, the number of partitions of a set with {pq} elements into {p} subsets with {q} each.

You may have noticed that these conditions seem to just be a formal encoding of the power map {\gamma_n(x)=x^n} from characteristic {0}. You’d be wrong, but basically correct. A close examination of the fourth condition actually gives that the map must be {\gamma_n(x)=\frac{x^n}{n!}} when dividing makes sense (i.e. {A} is a {\mathbb{Q}}-algebra).

We say {(I, \gamma)} is a P.D. ideal, {(A, I, \gamma)} is a P.D. ring and {\gamma} is a P.D. structure on {I}. Of course, P.D. stands for “Divided Power”. OK, not really, it stands for “Puissances Divisees” which is just french for divided powers. We may want to form this into a category, so we’ll say a P.D. morphism {f:(A, I, \gamma)\rightarrow (B, J, \delta)} is a ring map {f:A\rightarrow B} with the property {f(I)\subset J} and it commutes with the divided powers {f\circ \gamma_n(x)=\delta_n\circ f(x)} for all {x\in I}.

We already gave the {\mathbb{Q}}-algebra example. In fact, in that case every ideal has that as its unique P.D. structure. For any ring, {\{0\}} together with {\gamma_0(0)=1} and {\gamma_i(0)=0} is a P.D. structure. The first interesting example is to let {V} be a DVR of unequal characteristic {(p,0)} with uniformizing parameter {\pi}. Recall that if {p=u\pi^e}, then {e} is the absolute ramification index of {V}. We get that {\frak{m}} has a P.D. structure if and only if {e\leq p-1}. In particular, if {k} is perfect, then since {W(k)} is absolutely unramified {pW(k)} has a unique P.D. structure on it.

We can also define subobjects in the obvious way. If {(I, \gamma)} is a P.D. ideal, then another ideal {J\subset I} is a sub P.D. ideal if {\gamma_i(x)\in J} for any {x\in J}. In other words, the P.D. structure restricts to be a P.D. structure on the smaller ideal.

Let’s end with a nice little lemma for how find sub P.D. ideals. Suppose {(A, I, \gamma)} is a P.D. algebra, {S\subset I} a subset and {J} the ideal generated by {S}. We have that {J} is a sub P.D. ideal if and only if {\gamma_n(s)\in J} for all {s\in S}.

Here is the proof. By definition if it is a sub P.D. ideal, then {\gamma_n(s)\in J}. That direction is done. Now suppose {\gamma_n(s)\in J} for all {s\in S}. Let {J'} be the subset of {J} of the {x} for which {\gamma_n(x)\in J} for all {n\geq 1}. By assumption and construction {S\subset J' \subset J}. By definition of generation, if {J'} is an ideal, then {J'=J} and we are done. Choose {x,y \in J'} and fix {n\geq 1}. Now {\gamma_n(x+y)=\sum_{i+j=n}\gamma_i(x)\gamma_j(y)\in J} since {J} is an ideal. Thus {x+y\in J'}. Lastly, suppose {\lambda\in A}, then using {J} an ideal again {\gamma_n(\lambda x)=\lambda^n\gamma_n(x)\in J}, so {\lambda x\in J'}. Thus {J'} is an ideal which proves the lemma.

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 ├ętale 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.