# 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.