The Cohomological Brauer Group of a Toric Variety

Toric varieties are a special class of rational varieties defined by equations of the form monomial = monomial. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety X contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in Rr. The coordinate rings of each of these affine open sets is a graded ring generated over the ground field by monomials. As a consequence, toric varieties provide a good context in which cohomology can be calculated. The purpose of this article is to describe the second étale cohomology group with coefficients in the sheaf of units of any toric variety X. This is the so-called cohomological Brauer group of X. Let X = TNemb(∆) be a toric variety which determines and is determined by the fan ∆. To say ∆ is a fan means ∆ consists of finitely many elements, each of which is a convex cone in Rr, satisfying the following conditions: 1. If σ, τ ∈ ∆, then σ ∩ τ is a face of both σ and τ . 2. If σ ∈ ∆, then any face of σ is in ∆. 3. If σ ∈ ∆ and ~v ∈ σ, then R~ v ∩ σ = R≥0~ v ∩ σ (the cones in ∆ are strongly convex). 4. If N = Z is the set of lattice points in R and σ ∈ ∆, then there are ~v1, . . . , ~vs ∈ σ∩N with σ = R≥0~v1 + . . .+ R≥0~vs. We can make a fan ∆ a topological space by letting the open sets in the topology be the subsets of ∆ which are themselves fans (the open sets of ∆ are the subfans of ∆). The support |∆| of ∆ is ⋃ σ∈∆ σ ⊆ R . A support function h on ∆ is a real valued function on |∆| such that for each cone σ ∈ ∆, h|σ is linear on σ and integer valued on σ ∩ N . Let SF(∆) be the abelian group of support functions on ∆. Associate to ∆ the sheaf SF defined by SF(∆) = SF(∆) for each open set ∆ ⊆ ∆. The “restriction maps” in the sheaf SF are ordinary restriction. This article compares several cohomology groups and it will be helpful to the reader to lay out our notation with some care. If X is a irreducible scheme and F is a sheaf on the étale site, H (Xét,F) will denote the i-th étale cohomology group of X with values in F . If K is the function field of X, then H (K/Xét,F) is the subgroup of H (Xét,F) of cocycle classes split by K, i.e. are trivial when restricted to the generic point of X. The notation H i Z(Xét,F) denotes étale cohomology with supports. Similarly, if F is a Zariski sheaf on X, then H (XZar,F) will denote the i-th Zariski cohomology group of X with values in F . Finally, for any topological space D with open cover V, and F a sheaf on D, Ȟ (V/X,F) denotes the Čech cohomology of X with respect to the cover V with values in F [11]. The direct limit over all covers is denoted by Ȟ (X,F). The notation H (X,F) simply denotes the derived functor cohomology. The purpose of this paper is to compare these cohomology