Timothy J. Ford

On the Brauer group of a localization

Let X be a smooth variety over an algebraically closed field k and Y a closed subvariety of codimension 1. We associate to Y a graph Γ and use the homology of Γ to study the Brauer group of X — Y.

Computing the Brauer-Long group of Z/2 dimodule algebras

The Brauer-Long group of Z/2 dimodule algebras over a connected commutative ring R in which 2 is invertible is computed in terms of the cohomology of R. Consequences of this calculation are given.

On the Brauer group of toric varieties

We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to 2 everywhere

The Brauer group of an affine cone

Abstract Let V be a reduced projective hypersurface in P v and C the affine cone over V in X = A v + 1 . We employ cohomological methods to study the Brauer groups of the varieties C , C - P , X - C and X ¯ - C ¯ , where P is the vertex of the cone, X ¯ = P v + 1 and C is the completion of C in X .

Topological invariants of a fan associated to a toric variety

Associated to a toric variety X of dimension r over a field k is a fan Δ on R1. The fan Δ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on X. The fan Δ inherits the Zariski topology from X. In this article some cohomological invariants of X are studied in terms of whether or not they depend only on Δ and not k. Secondly some numerical invariants of X are studied in terms of whether or not they are topological invariants of the fan Δ. That is, whether or not they depend only on the finite topological space defined on Δ. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan ...

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z 2 = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z 2 − f), and if R = k[x, y][f −1] and S = R[z]/(z 2 − f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H1(G, Cl(T)) → H1(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H1(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.

On the brauer group of a designularization of a normal surface 1

If ‰ : ś S → is a desingularization of the norm3 surface S, then it is shown that the induced map H2 et:(S, Gm) → H2: et(ś, Gm) is surjective. It follows that if all of the singularities of S are rational, the Brauer group map B(S) → B(ś) is surjective. An example is given to show that this property fails if the dimension of S ≥ 3.

Homomorphisms of progenerator modules

Abstract Homomorphisms defined on progenerator modules over a commutative ring are studied through the introduction of a relation called homotopy, which is coarser than the usual notion of equivalence. The set of homotopy classes of homomorphisms forms a commutative monoid with operation induced by tensor product. For a given commutative ring, classification of homomorphisms is carried out by determining the algebraic structure of the monoid of homotopy classes and then giving a representing homomorphism for each class. This classification is carried out explicitly for Dedekind domains.

The Brauer group of an affine double plane associated to a hyperelliptic curve

ABSTRACT We study the Brauer group of an affine double plane π:X→𝔸2 defined by an equation of the form z2 = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ℙ1×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y2−p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.

THE PICARD GROUP OF A GENERAL TORIC VARIETY OF DIMENSION THREE

ABSTRACT Let be a complete fan in such that every three-dimensional cone in is non-simplicial. In any non-empty open neighborhood of there is a fan such that every -linear support function is linear and the Picard group of the associated toric variety is zero.