Zoran Skoda

Somebody (H.K.) will teach a course “D-modules and representations” in Zagreb 2011 following

  • R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser

Thus I will modify plan from hom10connDmod and I am considering to teach instead a complementary course on geometric aspects of D-modules, connections etc. on algebraic and formal schemes and generalizations. But the scope, title etc. are not finally decided yet.

In any case, I would like not to teach differential operators in mechanical way as it is possible in some special setups, by providing the generators from the start. Such explicit descriptions are important in many applications but do not show the general story and do not show fully the origins of calculus on structured spaces as it was pioneered by Grothendieck. One benefit is that one can then develop the calculus in prime characteristic.

Grothendieck considered the differential operators by looking at linearizations of maps of quasicoherent sheaves of 𝒪\mathcal{O}-modules. Cf.

  • P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243

This leads immediately to the descriptions via the descent data, so called Grothendieck connection). The descent data in abelian situations, correspond to flat connections. This is true in rather general setups like in algebra (cf. connection for a coring). This geometric picture involves infinitesimal thickenings of diagonal; they induce the differential filtrations on the bimodules, and of the notion of a differential part of a bimodule. In particular the differential part of a Hom-bimodules yields differential operators.

One can also closely related and equivalent language of infinitesimally close generalized points of schemes. Cf.

  • Lurie’s notes “Crystals and D-modules” in Gaitsgory’s seminar pdf

Note that here one has also the nonlinear version: not only crystals of quasicoherent modules but also crystals of schemes. There are also noncommutative versions of thickenings, like Kapranov's noncommutative geometry and so on, and Lunts-Rosenberg theory of differential operators and D-modules on noncommutative rings and noncommutative schemes.

Last revised on February 24, 2011 at 17:23:37. See the history of this page for a list of all contributions to it.