Vol. 42, No. 3, 1972

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Cohomology in the finite topology and Brauer groups

Raymond Taylor Hoobler

Vol. 42 (1972), No. 3, 667–679

An exact sequence relating Br(X), the Brauer group of a regular scheme of dimension 2, and Amitsur cohomology (obtained as the cohomology of the sheaf of units on an appropriate Grothendieck topology) is derived by functorial methods. In order to do this we first show that any torsion element of H1(Xet,Gm), i.e., Pic(X), and H2(Xet,Gm), i.e., Br (X), is split by a finite, faithfully flat covering Y X. After proving a divisibility result for Pic (X) under such coverings and some preliminary investigation of cohomology in the topology defined from such coverings, the exact sequence which is analogous to that of Chase and Rosenberg is obtained.

Mathematical Subject Classification 2000
Primary: 14F20
Secondary: 13A20
Received: 2 April 1971
Revised: 24 March 1972
Published: 1 September 1972
Raymond Taylor Hoobler