Topos Theory

topos theory



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In terms of differential operators

A D-module (introduced by Mikio Sato) is a sheaf of modules over the sheaf D XD_X of regular differential operators on a ‘variety’ XX (the latter notion depends on whether we work over a scheme, manifold, analytic complex manifold etc.), which is quasicoherent as O XO_X-module. As O XO_X is a subsheaf of D XD_X consisting of the zeroth-order differential operators (multiplications by the sections of structure sheaf), every D XD_X-module is an O XO_X-module. Moreover, the (quasi)coherence of D XD_X-modules implies the (quasi)coherence of a D XD_X-module regarded as an O XO_X-module (but not vice versa).

In terms of sheaves on the deRham space

The category of 𝒟\mathcal{D}-modules on a smooth scheme XX may equivalently be identified with the category of quasicoherent sheaves on its deRham space dR(X)dR(X) (in non-smooth case one needs to work in derived setting, with de Rham stack instead).

(Lurie, above theorem 0.4, Gaitsgory-Rozenblyum 11, 2.1.1)

Remembering, from this discussion there, that

this shows pretty manifestly how D-modules are “sheaves of modules with flat connection”, as described more below.

Meaning and usage

DD-modules are useful as a means of applying the methods of homological algebra and sheaf theory to the study of analytic systems of partial differential equations.

Insofar as an OO-module on a ringed site (X,O)(X, O) can be interpreted as a generalization of the sheaf of sections of a vector bundle on XX, a D-module can be interpreted as a generalization of the sheaf of sections of a vector bundle on XX with flat connection \nabla. The idea is that the action of the differential operation given by a vector field vv on XX on a section σ\sigma of the sheaf (over some patch UU) is to be thought of as the covariant derivative σ vσ\sigma \mapsto \nabla_v \sigma with respect to the flat connection \nabla.

In fact when XX is a complex analytic manifold, any D XD_X-module which is coherent as O XO_X-module is isomorphic to the sheaf of sections of some holomorphic vector bundle with flat connection. Furthermore, the subcategory of nonsingular D XD_X-modules coherent as D XD_X-modules is equivalent to the category of local systems.

If XX is a variety over a field of positive characteristic pp, the terms “O XO_X-coherent coherent D XD_X-module” and “vector bundle with flat connection” are not interchangeable, since D XD_X no longer is the enveloping algebra of O XO_X and Der X(O X,O X)\text{Der}_X(O_X,O_X). A theorem by Katz states that for smooth XX the category of O XO_X-coherent D XD_X-modules is equivalent to the category with objects sequences (E 0,E 1,)(E_0, E_1,\ldots) of locally free O XO_X-modules together with O XO_X-isomorphisms σ i:E iF *E i+1\sigma_i: E_i\rightarrow F^* E_{i+1}, where FF is the Frobenius endomorphism of XX.

John Baez: it would be nice to have a little more explanation about how not every DD-module that is coherent as an OO-module is coherent as a DD-module. If I understand correctly, this may be the same question as how not every holomorphic vector bundle with flat connection is a local system. Perhaps the answer can be found under local system? Apparently not. Perhaps the point is that not every flat connection on a holomorphic vector bundle is locally holomorphically trivializable? If so, this is different than how it works in the C C^\infty category, which might explain my puzzlement.


Six operations yoga

Discussion of six operations yoga for pull-push of (coherent, holonomic) D-modules is in (Bernstein, around p. 18). This is reviewed for instance in (Etingof, Ben-Zvi & Nadler 09).

The most efficient and intuitive way to define the six operations on D-modules is to transfer them from Ω-modules? (i.e., modules over the differential graded algebra of differential forms) using Koszul duality. The six operations on Ω-modules? can be defined in the standard way using the fact that differential forms can be pulled back, unlike differential operators. See the article Koszul duality for more information.

Relation to geometric representation theory

For the moment see at Harish Chandra transform.


A comprehensive account is in chapter 2 of

Discussion in derived algebraic geometry is in

Lecture notes include

See also

Review of six operations yoga for D-modules is in

See also

Blog discussion