# D-modules, deRham spaces and microlocalization

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?

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Technically, $X_{dR}$ is not an algebraic stack. If $X$ is smooth, the fibers of the morphism $X\to X_{dR}$ looks like formal schemes rather than schemes (the morphism is not representable). In fact, for smooth $X$ people sometimes describe $X_{dR}$ as the quotient of $X$ by its formal groupoid acting formally transitively (the groupoid is the infinitesimal neighborhood of the diagonal in $X\times X$). – t3suji Jan 3 '10 at 2:24
How's that for proof by intimidation! – Ben Webster Jan 3 '10 at 3:39
Shizhuo- that perspective is hard to make rigorous, since there's no good notion of a D-module on an arbitrary scheme. Instead, X_dR is a "space" in a certainly funny sense algebraic geometers like; it's not a topological space, but it is a space in the "functor of points" formalism. You can see more details at the notes I linked to above. – Ben Webster Jan 3 '10 at 3:44
Oh, if $X$ is a stack, then $X_{dR}$ is a stack. And a sheaf of spaces is a stack too, of course. Maybe Jacob just forgot that he was talking about schemes and not stacks! – Reid Barton Jan 3 '10 at 4:11
The de Rham stack (and its name I believe) go back to Carlos Simpson. It's a deformation of the Higgs stack (relative classifying stack of the tangent bundle), the whole deformation being called the Hodge stack. Yes it's a space (sheaf of sets) but I believe it was called a stack partly since the terminology [non-algebraic] "space" is less universal. – David Ben-Zvi Jan 4 '10 at 14:53