A unified Kummer-Artin-Schreier sequence

Abstract

Let p be a prime, R a commutative ring. There are two familiar group scheme sequences yielding computations of the flat (or etale) cohomology group Hi(R, Z/pZ): the Kummer exact sequence I~7Z/pZ~G,,~G,,~I, applicable where p is invertible and a primitive p-th root of unity ( is in R, and the ArtinSchreier sequence 1 ~71/pZ~ G , ~ Ga ~ 1 that is valid when p is zero in R (see e.g. [6, pp. 126-127]). In this paper I shall show that these are both special cases of a single sequence of the same kind defined over Z [(]. The two groups involved (besides Z/pZ) have H 1-cohomology computable by familiar K-theory, and hence we have a way of computing H 1 (R, 7Z/pZ) for all Z [(]-algebras R. In particular, the computation works for all rings when p = 2, and we recover the description of etale quadratic algebras derived previously by ad hoc methods (see e.g. [7]). The group schemes occurring in the exact sequence are smoothed versions of congruence subgroups, objects which (I believe) were first described in general in my joint work with Boris Weisfeiler classifying smooth connected group schemes of dimension one [8].