geometry of triangulated category and D-modules theory

Question

This question was motivated by the answers On noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence space. So,there might be no topological space and notions of sheaf in this settings.

My question might be related to this observation but for triangulated category. It seems that Beilinson-Bernstein take the derived category of coherent D-modules as a non-existence space, right? They used various adjoint triangle functor for these derived categories of D-modules.

So, is there a geometric space(topological space)in this framework? Is there notion of sheaf for derived category?

More general, for derived category of D-module on a scheme(not necessarily smooth), can we define topological space and sheaf for this derived category? Is there a global section functor which recover this derived category?

Answer 1

To me it seems as if the most conceptual way to think about this is as follows:

• the deRham space $dR(X)$ of X is (as described there) the decategorification of what i like to call the infinitesimal path oo-groupoid $\Pi^{inf}(X)$ of $X$, which is modeled by the infintiesimal singluar simplicial complex $X^{(\Delta^\bullet_{inf})}$ of $X$.

• a D-module on X, hence a quasicoherent sheaf on $dR(X)$ is therefore a decategorification of a morpism $\Pi^{inf}(X) \to \infty Mod$ that encodes a (generalized) flat oo-vector bundle: over each "point" of $X$ a fiber, over each infinitesimal path a morphism of fibers, over each infintiesimal homotopy a homotopy of morphisms of fibers, etc.

So the category of D-modules on a space is something like its category of flat (generalized) vector bundles.

As currently discussed for instance here on the nCafe, the way to think about this in terms of notions of space is the following:

in the oo-context, the plain overcategory of our ambient oo-tops over a space X is just this space X regarded dually in terms of the oo-topos of oo-stacks over it. This is a change of perspective (space to things on the space) that essentially does not lose information.

But then we can instead first form the overcategory and then stabilize it. This does lose information. And in fact, this turns out to form the category not of oo-stacks but of quasicoherent sheaves over $X$. These are to be thought of as a "linearization of all oo-stacks". I try to talk about that here.

So this gives a description of the original space which is a little more indirect. Now, with D-modules, it becomes yet a bit more indirect: instead of all stabilized oo-stacks, we just retain those that have a flat connection, in a way.

The original underlying space need not be fully reconstructible from this. But then one take the perspective that we are just interested in the linearized and flat situation, and take a stable oo-ocategory to be a formal dual of a possibly fictitious space.

Answer 2

As Scott implies, there are probably lots of sensible answers to this question, but one perspective is to view the category of $\mathcal D$-modules on a (smooth) variety $X$ as a category of quasi-coherent sheaves on a "noncommutative deformation" of the cotangent bundle $T^*X$. This is essentially just another way of viewing the existence of the symbol map, but for exactly that reason it's a good point of view -- you can lift $\mathcal D$-modules to modules for the microdifferential operators (or pseudodifferential operators) on the cotangent bundle, and working locally on the cotangent bundle is then a useful technique. This process gets called "microlocalization", and I think is at least mentioned in the discussion of noncommutative geometry.

Answer 3

I can't tell what sort of answer you want, but I can say some words connected to D-modules and geometry.

The derived category of D-modules on a scheme $X$ is equivalent to the derived category of quasicoherent sheaves on the De Rham stack $X^{DR}$. It is probably best to think of $X^{DR}$ as a functor from commutative rings to sets rather than as a topological space. In particular, we define $X^{DR}(A) = X(A/nil (A))$ for a commutative ring $A$ with nilradical $nil(A)$, and do the natural thing for morphisms.

Another way to consider $D$-modules is as quasicoherent sheaves on the infinitesimal site of $X$. Objects in the category are pairs $(U,T)$, where $U \to X$ is etale, and $U \to T$ is a nilimmersion. Morphisms are the evident commutative diagrams. Covers come from etale surjections on collections of $U$. You might want to think of the infinitesimal topos as a proxy for a topological space.