REFLEXIVE MODULES OF RANK ONE OVER WEYL ALGEBRAS OF NON-ZERO CHARACTERISTICS

Abstract

Many of the properties of a Weyl algebra A_n over a base field of non-zero characteristic are explained in terms of connections and curvatures on a vector bundle on an affine space X = A²ⁿ. In particular, it is known that an algebra endomorphism φ of A_n gives rise to a symplectic endomorphism f of X with a gauge transformation g. In this paper we study the converse problem of finding φ from an arbitrary symplectic endomorphism f of X = A²ⁿ. It is shown that, given such f , we may construct a projective left A_n-module (which corresponds to ‘the sheaf of local gauge transformations' ) such that its triviality is equivalent to the existence of the ‘lift' φ. Some properties of such a module will be discussed using the theory of reflexive sheaves.