# Stratification 3: The Definition

Last time we looked at the characteristic ${0}$ case to figure out how our old definition of a connection on a sheaf could be rephrased in terms of a “parallel transport” rule. This took the form of giving an isomorphism ${(p_1^*\mathcal{E})|_{X^{(2)}}\rightarrow (p_2^*\mathcal{E})|_{X^{(2)}}}$ that restricted to the identity on the diagonal. Moreover, if the connection is integrable you can lift these isomorphisms to all infinitesimal neighborhoods of the diagonal ${X^{(n)}}$ so that the restrictions are all the previous ones (they are compatible).

Two times ago we looked at the n-th infinitesimal neighborhood of the PD-envelope of the diagonal in the ${(\nu+1)}$ product and called it ${D_{X/S}^n(\nu)}$. A theorem that we won’t prove is that a connection on ${\mathcal{E}}$ can be lifted compatibly to ${D_{X/S}^n(1)}$ for all ${n}$ if and only if it is integrable. This finally brings us to our definition of stratification.

If ${\mathcal{E}}$ is an ${\mathcal{O}_X}$-module, a PD stratification on ${\mathcal{E}}$ is a collection of isomorphisms ${\epsilon_n: \mathcal{D}_{X/S}^n(1)\otimes \mathcal{E}\rightarrow \mathcal{E}\otimes \mathcal{D}_{X/S}^n(1)}$ such that each ${\epsilon_n}$ is ${\mathcal{D}_{X/S}^n(1)}$-linear, the ${\epsilon_n}$‘s are compatible (they restrict to the previous one and the ${\epsilon_0}$ is the identity) and they satisfy the standard cocycle condition at all levels.

Essentially, we took the intuition from the characteristic ${0}$ case and just encoded it into a definition. A stratification is just a compatible choice of infinitesimal parallel transport at all levels. I don’t want to go too far down this road which will involve differential operators and things. I plan to come back to these ideas in the not-to-distant future, but for the next few weeks I want to change gears.

One thing that keeps coming up for me and I keep using is the deformation theory of ${p}$-divisible groups. Since we already have some groundwork on ${p}$-divisible groups done, I hope we can actually prove that the deformation functor over Artin ${W}$-algebras is formally smooth and prorepresentable by ${W[[t_1, \ldots, t_d]]}$ where ${d}$ is the dimension of the group times the dimension of its dual.

# Stratification 1

I’m back. I have several trains of thought started at the nlab, so I’ll probably be jumping back and forth for awhile, but I consider this blog to be of more importance at this time since I need to understand this material sooner. It’s been awhile, so I’ll recall that we talked about divided powers on ideals of rings for several posts. This was a way to get something that looked “power series-like” in positive characterstic where we can’t actually divide by some numbers.

Then we noticed that all these concepts sheafified, so we could define the category of sheaves of P.D. rings. This allowed us to define a PD scheme, as something locally isomorphic to ${\mathrm{Spec}(A, I, \gamma)}$ where the spectrum of a PD ring is defined to be the locally ringed space ${(|\mathrm{Spec}(A)|, \mathcal{O})}$ where ${\mathcal{O}}$ is the sheaf of PD rings inherited from ${\gamma}$.

Then we defined crystalline cohomology, but instead of moving forward with more concepts along these lines, we’ll go backwards for a post or two to figure out what is happening with PD schemes better. Recall the setup for the crystalline site, we fix some PD scheme ${(S, \mathcal{I}, \gamma)}$ and consider ${X}$ an ${S}$-scheme. In general, when ${i:X\rightarrow Y}$ is a closed immersion we can define the quasi-coherent ${\mathcal{O}_X}$-algebra ${\mathcal{D}_{\mathcal{O}_Y, \gamma}(\mathcal{J})}$ as the sheaffified version of taking the PD envelope of ${\mathcal{J}}$ where ${\mathcal{J}}$ is the defining ideal. We’ll just use the shorthand ${D_{X, \gamma}(Y)}$ for this. As a scheme we have ${D_{X, \gamma}(Y)=\underline{Spec}_Y(\mathcal{D}_{X,\gamma}(Y))}$.

If ${\gamma}$ extends to ${X}$, then ${\mathcal{D}_{X, \gamma}(Y)/\overline{\mathcal{J}}\simeq \mathcal{O}_X}$. I.e. ${i}$ factors through a PD immersion ${j: X\rightarrow D_{X, \gamma}(Y)}$ defined by ${\overline{\mathcal{J}}}$. We define ${D^n_{X, \gamma}(Y):=\mathcal{D}_{X, \gamma}(Y)/\overline{\mathcal{J}}^{[n+1]}}$ and it is called the ${n}$-th order divided power neighborhood of ${X}$ in ${Y}$. Caution: This is NOT in general a subscheme of ${Y}$. This ${n}$-th order divided power neighborhood can be formed for the locally closed immersion case as well.

Example: If ${X\rightarrow Y}$ is an immersion of smooth ${S}$-schemes and ${m\mathcal{O}_Y=0}$, then ${\mathcal{D}_{X, \gamma}(Y)}$ is locally isomorphic to a PD polynomial algebra over ${\mathcal{O}_X}$.

One important thing we can look at is ${X/S^{\nu+1}}$ defined to be the ${(\nu+1)}$-fold product of ${X}$ with itself over ${S}$. Let ${\Delta: X\rightarrow X/S^{\nu+1}}$ be the diagonal map. This is a locally closed immersion so we can apply the above construction. Suppose ${\gamma}$ extends to ${X}$ and ${m\mathcal{O}_X=0}$ or ${X/S}$ is separated. This allows us to define the divide power envelope ${D_{X/S}(\nu)}$ of ${X}$ in ${X/S^{\nu+1}}$ and the ${n}$-th order divided power neighborhood ${D^n_{X/S}(\nu)}$.

By the example if ${X/S}$ is smooth, ${m\mathcal{O}_X=0}$ and we pick ${x_1, \ldots, x_n}$ local coordinates on ${X}$, then the structure sheaf ${\mathcal{D}_{X/S}(1)}$ of ${D_{X/S}(1)}$ (the envelope of ${\Delta: X\rightarrow X\times_S X}$) is isomorphic to ${\mathcal{O}_X\langle \xi_1, \ldots, \xi_n\rangle}$ where ${\xi_i=1\otimes x_i- x_i\otimes 1}$. Well, that was like a paragraph from Berthelot and Ogus. Next time we’ll move on to what a stratification is.

# 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 2

Today we’ll do a short post on some P.D. algebra properties and constructions. Let’s start with properties of P.D. ideals. Our first proposition is that given ${(I, \gamma)}$ and ${(J, \delta)}$ as two P.D. ideals in ${A}$, then ${IJ}$ is a sub P.D. ideal of both ${I}$ and ${J}$. This is very straightforward to check using the criterion from last time, since ${IJ}$ is generated by the set of products ${xy}$ where ${x\in I}$ and ${y\in J}$. This proposition immediately gives us that powers of P.D. ideals are sub P.D. ideals and there is a natural choice for P.D. structure on them.

Another proposition is that given two P.D. ideals as above with the additional property that ${I\cap J}$ is a P.D. ideal of ${I}$ and ${J}$ and that ${\gamma}$ and ${\delta}$ restrict to the same thing on the intersection, then there is a unique P.D. structure on ${I+J}$ such that ${I}$ and ${J}$ are sub P.D. ideals. Proving this would require developing some techniques that would lead us too far astray. We probably won’t use this one anyway. It just gives a sense of the types of constructions that are compatible with P.D. structures.

Another construction that requires no extra effort are direct limits. If ${\{A_i, I_i, \gamma_i\}}$ is a directed system of P.D. algebras, then ${\displaystyle \left(\lim_{\rightarrow} A_i, \lim_{\rightarrow} I_i\right)}$ has a unique P.D. structure ${\gamma}$ such that each natural map ${(A_i, I_i, \gamma_i)\rightarrow (A, I, \gamma)}$ is a P.D. morphism.

Unfortunately, one common construction that doesn’t work automatically is the tensor product. It works in the following specific case. If ${B}$ and ${C}$ are ${A}$-algebras, and ${I\subset B}$ and ${J\subset C}$ are augmentation ideals with P.D. structures ${\gamma}$ and ${\delta}$ respectively, then form the ideal ${K=\mathrm{ker}(B\otimes C\rightarrow B/I \otimes B/J)}$. We then get that ${K}$ has a P.D. structure ${\epsilon}$ such that ${(B, I, \gamma)\rightarrow (B\otimes C, K, \epsilon)}$ and ${(C, J, \delta)\rightarrow (B\otimes C, K, \epsilon)}$ are P.D. Morphisms.

Next time we’ll start thinking about how to construct compatible P.D. structures over thickenings. Since we’ll be thinking a lot about ${W_m(k)}$ I’ll just end this post by pointing out that ${(p)\subset W_m}$ actually has many choices of P.D. structure. But last time we said that ${(p)\subset W(k)}$ actually has a unique one, so our convention is going to be to choice the “canonical” P.D. structure on ${(p)\subset W_m}$ induced from the unique one in ${W(k)}$.

# 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.