Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over a sheaf of differential operators, but for spaces that aren't smooth in some sense, this definition doesn't work that well, and you want to use a different definition. My overall question is how to reinterpret microlocalization in this alternative definition.
deRham spaces
This definition is that a D-module on $X$ is a quasi-coherent sheaf on a new space $X_{dR}$, the deRham space of $X$. It's easiest to define this is in terms of its functor of points: a map of Spec R to $X_{dR}$ is by definition a map of Spec $R/J_R$ to $X$ where $J_R$ is the nilpotent radical of $R$. So this is not a topological space, but it is a sheaf on the big Zariski site, and I can make sense of a quasi-0-coherent sheaf on one of those. For more details, you can see the notes of Jacob Lurie on these.
More informally $X_{dR}$ is $X$ "with all infinitesimally close points identified." A sheaf on this space is like a D-module in that a D-module is a sheaf with a connection, i.e. where the fibers of infinitesimally close points are identified. You'll note, I say "space" here, since I want to be vague about what this object is. It's very hard from being a scheme, but I believe it is a (EDIT: not actually algebraic!) stack.
microlocalization
Now, one of the lovely things about D-modules is that they have a secret life on the cotangent bundle of X. You might think a D-module is a sheaf on X, but this is not the whole picture: there is also a microlocal version of things.
The sheaf of functions on $T^*X$ has a quantization $\mathcal{O}^h$; this is a non-commutative algebra over $\mathbb{C}[[h]]$ such that $\mathcal{O}^h/h\mathcal{O}^h\cong \mathcal{O}_{T^*X}$, defined using Moyal quantization.
There's a ring map $p^{-1}\mathcal{D}\to \mathcal{O}^h[h^{-1}]$, and thus a functor from D-modules to sheaves of $\mathcal{O}^h[h^{-1}]$-modules on $T^*X$ given by $\mathcal{O}^h[h^{-1}]\otimes_{p^{-1}\mathcal{D}}\mathcal{M}$, called microlocalization, because it makes D-modules even more local than they were before. This is an equivalence between D-modules and $\mathbb{C}^*$-equivariant $\mathcal{O}^h[h^{-1}]$-modules.
Given an $\mathbb{C}^*$ invariant open subset $U$ of $T^*X$, one can look at $\mathcal{O}^h[h^{-1}]$-modules on $U$, and obtain a microlocalized category of D-modules, which has all kinds of interesting geometry one couldn't see before. I'm particularly interested in the semi-stable points for the action of some group $G$ on $X$ (extended to $T^*X$).
my question:
Now, I'm something of a convert to derived algebraic geometry, so it feels intuitive to me that anything one has to say about D-modules should be sayable using deRham spaces. On the other hand, I have no idea how microlocalization can be phrased in this way. Do any of you out in MathOverflowLand?
Best Answer
The first thing to say is that the abelian category of sheaves on the de Rham space is only a good model for D-modules if you're in the smooth setting, or very close to it (see for example arXiv:math/0212094 for a setting where all the different notions agree).. so unless you're fully derived you need to be careful with this identification. In any case, rather than talk about the de Rham space you can talk about crystals, which give the right notion in general -- see for example chapter 7 (section 10 or 11) of Beilinson-Drinfeld on Quantization of Hitchin for an excellent discussion. Another picture you might like better is as modules over the enveloping algebroid of the tangent complex -- ie everything is fine if you replace naive differential operators by their correct derived analog. Yet another picture is as dg modules for the de Rham complex. My favorite is as S^1-equivariant sheaves on the derived loop space of X. (I assume for all of these we're in characteristic zero, otherwise there are many different analogs of D-modules..)
In any case micrlocalization can be said in terms of the deformation to the normal cone of the de Rham groupoid - or the Hodge filtration on nonabelian cohomology, after Simpson. (A good reference for this is Simpson's paper with that title and the awesome preprint on D-modules on stacks by Simpson and Teleman available on the latter's webpage.) Namely there's a canonical deformation (the Hodge stack) from the de Rham space to a stack (not a space this time!) which is the quotient of X by the tangent bundle (acting trivially) -- sheaves on which are the same as quasicoherent sheaves on the cotangent bundle. On the level of sheaves this is just the deformation quantization from sheaves on the cotangent bundle to D-modules with a parameter h - ie we consider modules over the Rees algebra of D rather than over D itself. The category of Rees modules sheafifies over the projectivized cotangent bundle, and so you can define microlocal categories by restricting to your favorite open subsets.
This is not so special to the de Rham stack: we're just saying jets of sections of a formal deformation of the category of sheaves on a variety sheafify over this variety. Once you know in general that D-modules degenerate to sheaves on the cotangent bundle, and this is true in arbitrary generality once you define both sides correctly, you can microlocalize (if you take into account correctly the C^* equivariance of the deformation).