Stratification 2

Before defining stratification, we’ll look at what this notion is in what is hopefully a more familiar context. Let’s forget about all the PD stuff for today (but keep it in mind for later). Suppose {S} is a scheme and {X} is smooth of finite type over {S}. The diagonal is a closed immersion {\Delta: X\rightarrow X\times_S X}. Suppose it is defined by the quasi-coherent ideal sheaf {\mathcal{I}} (it is generated by things of the form {t\otimes 1 - 1 \otimes t}). So far all this should feel very familiar from our setup in the last post.

Rather than worry about PD things, we’ll define the {n}-th infinitesimal neighborhood of {\Delta} to be {X^{(n)}}, the subscheme defined by {\mathcal{I}^{n+1}}. There are natural inclusions {X\hookrightarrow X^{(2)}\hookrightarrow X^{(3)}\rightarrow \cdots}. and all of these sit inside {X\times X}. Define {p_1} and {p_2} to be the first and second projections {X\times X\rightarrow X}. Given a quasi-coherent sheaf {\mathcal{E}}, we already have a notion of connection. Recall that it is just linear map {\nabla: \mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1} that satsifies the Leibniz rule.

Grothendieck defined a connection on {\mathcal{E}} to be an isomorphism {(p_1^*\mathcal{E})|_{X^{(2)}}\stackrel{\sim}{\rightarrow} (p_2^*\mathcal{E})|_{X^{(2)}}} which restricts to the identity on {\Delta}. I’ll give you the punchline up front. This definition is equivalent to the other one! If you look at the last post, it should be clear that this is the definition that will extend easier in the PD case. Let’s try to understand what this definition is saying.

This definition is somehow related to parallel transport. How could we think about parallel transport? Suppose you have some pointed {S}-scheme, say {T} with {t}. Consider a closed immersion {f:(T,t)\rightarrow (X, x)}. Maybe this is like a path and you remember the starting point (but it doesn’t have to be). There is always the constant map {f_x: (T,t)\rightarrow (X,x)} which just sends all of {T} to {x}. Parallel transport along {f(T)} from {x} of {\mathcal{E}} should be an isomorphism {\mathcal{E}|_{f(T)}\stackrel{\sim}{\rightarrow} \mathcal{E}|_{f_x(T)}} which restricts to be the identity at {x}.

Strictly speaking what I just wrote is nonsense, so what do I mean? If we restrict {\mathcal{E}} to the image of {T} I want to be able to choose a trivialization to give a linear isomorphism with the trivial bundle having fiber {\mathcal{E}|_{x}}. If you are thinking in terms of vector bundles, you could also think of it as a compatible choice of isomorphisms of the fiber at all points of {F(T)} with the fiber at {x} and the isomorphism at {x} must be the identity.

Now the definition of connection we gave should probably more accurately be described as “first-order” parallel transport where for us we should be thinking that {T=\mathrm{Spec}(k[\epsilon])}. Let’s check that our new notion of connection gives parallel transport with this {T}. Choose a {k}-point on {x\in\Delta}, then {f: T\rightarrow X\times X} is just going to {x} and choosing a tangent vector there. Since we have by definition an isomorphism {p_1^*\mathcal{E}|_{X^{(2)}}\rightarrow p_2^*\mathcal{E}|_{X^{(2)}}} we can just restrict this further to {f(T)} which lies in {X^{(2)}} to get the parallel transport isomorphisms. This shows that connections give parallel transport along tangent vectors (i.e. along first-order infinitesimally short paths).

Now nothing is stopping us from continuing in this fashion. What happens if we take {X_3^{(2)}} as the first-order infinitesimal neighborhood of the diagonal in {X\times X\times X}. Then we have three projections {X\times X\times X\rightarrow X\times X} which we’ll call {p_{i,j}} for projecting onto the {i} and {j} factors. Using the definition of connection we can now obtain {\epsilon_{i,j}: p_i^*\mathcal{E}|_{X_3^{(2)}}\stackrel{\sim}{\rightarrow} p_j^*\mathcal{E}|_{X_3^{(2)}}} which just comes from first pulling back using {p_{i,j}} then restricting.

Let’s think back to when we talked about connections months ago. The next major definition was what it meant to be integrable. We defined the curvature {K(\nabla)} to be the composition {\mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1\rightarrow \mathcal{E}\otimes \Omega^2} and {\nabla} was integrable if the curvature was {0}. Or in other words, if the associated sequence was actually a complex.

Using our new definition we get that a connection is integrable if {\epsilon_{1,3}=\epsilon_{2,3}\circ \epsilon_{1,2}}. In other words, if the isomorphisms coming from the associated {X\times X\times X} satisfy some sort of cocycle condition. In characteristic {0} being integrable actually guarantees that you can lift the isomorphisms to all {n}-th order neighborhoods. This gives {n}-th order parallel transport, meaning we get parallel transport using {T=\mathrm{Spec}(k[x]/(x^n))}.

Now everything is compatible here (we’re lifting the isomorphisms, meaning they restrict to the previous one). Thus we actually have directed systems to take a limit. This gives us what could maybe be called formal local parallel transport. Believe it or not, this is exactly the type of thing we are after in the PD case. It is basically the purpose of building a notion of “{x^n/n!}” so that we can get some sort of formal power series. I think that is a good enough reminder of this “classical” case. Maybe if you are super motivated you can go to the previous post and work out how these definitions extend. The setup is all there.

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