Last time we looked at the characteristic 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 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 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 product and called it . A theorem that we won’t prove is that a connection on can be lifted compatibly to for all if and only if it is integrable. This finally brings us to our definition of stratification.
If is an -module, a PD stratification on is a collection of isomorphisms such that each is -linear, the ‘s are compatible (they restrict to the previous one and the is the identity) and they satisfy the standard cocycle condition at all levels.
Essentially, we took the intuition from the characteristic 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 -divisible groups. Since we already have some groundwork on -divisible groups done, I hope we can actually prove that the deformation functor over Artin -algebras is formally smooth and prorepresentable by where is the dimension of the group times the dimension of its dual.