Finiteness of Brauer and X
نویسنده
چکیده
The relation ∼ is transitive because End(k) ⊗ End(kn) ' End(kn+n). The operation ⊗ on algebras gives Br(k) a group structure, since A ⊗ A ' End(A). The latter is an equivalent definition of a (finite-dimensional) central simple algebra. For any extension k′/k, there is a map Br(k)→ Br(k′) given by base change. The Artin-Wedderburn Theorem classifies all finite-dimensional simple k-algebras as End(D) for a unique finite-dimensional central division algebra D over k. If
منابع مشابه
A Finiteness Theorem for the Brauer Group of Abelian Varieties and K3 Surfaces
Let k be a field finitely generated over the field of rational numbers, and Br (k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer–Grothendieck group Br (X). We prove that the quotient of Br (X) by the image of Br (k) is finite if X is a K3 surface. When X is an abelian variety over k, and X is the variety over an algebraic closure k of k obtained f...
متن کاملA Finiteness Theorem for the Brauer Group of K3 Surfaces in Odd Characteristic
Let p be an odd prime and let k be a field finitely generated over the finite field with p elements. For any K3 surface X over k we prove that the the cokernel of the natural map Br(k) → Br(X) is finite modulo the p-primary torsion subgroup.
متن کاملArithmetic of Del Pezzo Surfaces of Degree 4 and Vertical Brauer Groups
We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane g : X 99K P, i.e., that every Brauer class is obtained by pullback from an element of Brk(P). As a consequence, we prove that a Brauer class obstructs the existence of a k-rational point if and only if all k-fibers of g fail to be locally solvable, or in other words,...
متن کاملKato Homology of Arithmetic Schemes and Higher Class Field Theory over Local Fields
For arithmetical schemes X, K. Kato introduced certain complexes C(X) of Gersten-Bloch-Ogus type whose components involve Galois cohomology groups of all the residue fields of X. For specific (r, s), he stated some conjectures on their homology generalizing the fundamental isomorphisms and exact sequences for Brauer groups of local and global fields. We prove some of these conjectures in small ...
متن کاملOn the Brauer-Manin obstruction for zero-cycles on curves
We wish to give a short elementary proof of S. Saito’s result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group X1(A) of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in ...
متن کامل