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.

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