Etale cohomology of toric varieties defined by infinite fans

Abstract

Toric varieties are a special class of normal rational varieties defined by means of a combinatorial object called a fan. The fan defining the variety gives a designated open cover and a dictionary describing how the open sets in the cover intersect. When all the open sets in the designated cover and their intersections have trivial cohomology, the cohomology of the whole variety is determined by the fan. In this article we calculate the low degree cohomology on the Zariski and Etale sites of toric varieties over an algebraically closed field of characteristic = 0. These computations extend (with different proofs) most of the results in DeMeyer et al., (1993) where the fan was assumed to be finite. As a result, we show any countable direct product of finite cyclic groups is isomorphic to the cohomological Brauer group of some toric variety. All the definitions and basic facts about toric varieties which we use can be found in Oda (1988).