Central separable algebras with purely inseparable splitting rings of exponent one

Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic. Given a commutative ring extension C over A of prime characteristic p, there are three variants for the Brauer group B(C/A) of central separable A-algebras split by C: the Amitsur cohomology group H2(C/A, Gm), the Chase-Rosenberg group PV(C/A), and Hochschild's group 4(C, g) of regular restricted Lie algebra extensions of C by the Lie algebra g of all A-derivations on C. In this paper we show that if C is finitely generated projective as an A-module and C [g] = EndA (C), then H2(C/A, Gm) , C) ,l(C/A). As a corollary we show that Hi(C/A, Gm) is zero for all i > 2. When C is a field, these are the results of Berkson, Hochschild and Rosenberg and Zelinsky [4], [11], [12]. As in [11] we show that the Lie algebra extensions which arise from central separable algebras are trivial extensions when regarded as ordinary extensions so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. We also show that if R is a commutative C-algebra which is finitely generated, projective as a C-module, then the Brauer group B(R/A) is mapped onto the Brauer group B(R/C). The last result is also due to Hochschild when C is a field [10]. ?1 contains the background on projective modules which came into the picture. Due to their peculiar behavior all relevant automorphisms turn out to be inner which explains why instead of some exact sequences we get two isomorphism theorems. In ?2 the isomorphism of e(g, C) with 91(C/A) is proved. ?3 and ?4 provide the preliminary materials for ?5. ?3 contains an exposition on the theory of differentials in rings of prime characteristic. Its application to Amitsur cohomology is given in ?4. The main results are given in ?5. Throughout this paper C over A always denotes a commutative ring extension of prime characteristic p such that C is finitely generated projective as an A-module Received by the editors October 30, 1969. AMS 1970 subject classifications. Primary 13A20.