Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem

Abstract: This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes, and a foliated index formula for weakly Fredholm operators.