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The Cartier-Yuan exact sequence is used to calculate Picard groups and Brauer groups of Zariski surfaces and their generalizations. A result of Blass-Deligne on the factoriality of general affine Zariski surfaces is extended to all higher dimensional Zariski schemes. In this note we use the Cartier-Yuan exact sequence to calculate Picard groups and Brauer groups of Zariski surfaces and their generalizations. We also extend a result of Blass-Deligne on the factoriality of general affine Zariski surfaces to all higher dimensional Zariski schemes. All calculations are carried out using etale cohomology but at several points we use the identification of etale and Zariski cohomology if the coefficient sheaf is a coherent Os-module. Since we are interested in both the affine and projective cases we begin by estab- lishing some notation. Let S be a regular, quasi-projective scheme in characteristic p > 0 with very ample bundle 0(1). If e is a positive integer, (p, e) = 1, and G G T(S, 0s{pe)), we let A be the graded, quasi-coherent sheaf of Os-algebras de- fined by A = ©i°^0 0{l)(Z)/{Zp — G) where Z is an indeterminate of degree e. The Zariski scheme associated to (S, G) is X = Proj(.A). It comes with a structure map 7r: X —► S which makes 7r,0x into a locally free Os-module of rank p. Note that if S is affine, then X = Spec(0s(Z)/(Zp - G)). For the remainder of this note we assume that X is a normal scheme. This will be the case, for instance, if S is a smooth, quasi-projective variety and the locus of zeros of the Jacobian matrix of G/x^e has codimension > 1 in D+{xj) C S and all Xj G T{S, 0(1)). In this case, p-i

Picard and Brauer groups of Zariski schemes