Algebras of p-symbols, noncommutative p-residue, and the Brauer group

Importance of the pseudodifferential symbol calculus extends far beyond the fundamental role it is known to play in Global and Microlocal Analysis. In this article, we demonstrate that algebras of symbols contribute to subtle phenomena in characteristic p > 0. A perfect fit between Smooth Geometry and de Rham Theory in characteristic zero leads many to interpret the situation in characteristic p > 0 as an apparent failure of de Rham Theory in positive characteristic. Smoothness, equated with the existence of local coordinates, i.e., of an étale map from a neighborhood of an arbitrary point to the affine space An, is a concept independent of the ground ring. What however is very much dependent on the ground ring k and its characteristic is the geometry of the affine space itself which provides a local model for Smooth Geometry after all. Local calculations in Smooth Geometry rely on the fact that the affine spaces are objects of the category of commutative unipotent algebraic groups. When the ground ring is a field of characteristic zero, this category is equivalent to the category of finite-dimensional vector spaces, all objects are semisimple, and the additive group Ga, which corresponds to the one-dimensional affine space A1, is the sole simple object. In contrast, over a ring of characteristic p > 0, the line is not even semisimple: Ga fits for example into the nontrivial extension of algebraic group schemes (0.1) Ga Ga Ga,1 u u F u x where F : Ga → Ga, the Frobenius morphism, corresponds to the k-algebra endomorphism of O(Ga) = k[z] which sends z to zp. 1991 Mathematics Subject Classification. Primary 58J42, 16H05, 16K50; Secondary 14G17, 47L80, 58B34.