Function field (scheme theory)

The sheaf of rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, KX(U) is the set of fractions of regular functions on U. Despite its name, KX does not always give a field for a general scheme X. In the simplest cases, the definition of KX is straightforward. If X is an (irreducible) affine algebraic variety, and if U is an open subset of X, then KX(U) will be the field of fractions of the ring of regular functions on U. Because X is affine, the ring of regular functions on U will be a localization of the global sections of X, and consequently KX will be the constant sheaf whose value is the fraction field of the global sections of X. If X is integral but not affine, then any non-empty affine open set will be dense in X. This means there is not enough room for a regular function to do anything interesting outside of U, and consequently the behavior of the rational functions on U should determine the behavior of the rational functions on X. In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any U, KX(U) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point. The trouble starts when X is no longer integral. Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.

Diagonal Quartic Surfaces And Transcendental Elements Of The Brauer Group

Evis Ieronymou

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic sur ...

Cambridge University Press2010

Rigidity of diamond reinforced metals featuring high particle contents

Ludger Weber, Reza Tavangar Hamzehkolaei

Diamond/metal composites with diamond contents between 57 and 72 vol% have been produced by gas pressure assisted liquid metal infiltration using Ag-3 wt% Si and Al-2 wt% Cu as matrix. The experimental data cover a range of Young's moduli from 300 to 425 G ...

2009