Local-to-global extensions of D-modules in positive characteristic

Abstract

In "On the calculation of some differential Galois groups" (Invent. Math. 87 (1987), no. 1), Katz defines the notion of a special flat connection on the complex affine line minus the origin, and he shows that the functor which restricts a flat connection to the punctured disc around the point at infinity gives rise to an equivalence between the category of special flat connections and the category of differential modules on the field of Laurent series over the complex numbers. In this article, we prove the corresponding statement over an algebraically closed field k of positive characteristic. The role of flat connections is played by vector bundles carrying an action of the (full) ring of differential operators. We call such objects stratified bundles. The formal local variant on the field of Laurent series k((t)) is called iterated differential module. We define the notion of a special stratified bundle on the affine line over k minus the origin, and show that restriction to the punctured disc around the point at infinity induces an equivalence between the category of special stratified bundles and iterated differential modules. This extends one of the main results of "Local-to-global extensions of representations of fundamental groups" (N. Katz, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4), and has several interesting consequences which extend well-known statements about étale fundamental groups to higher dimensional fundamental groups.