Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)

Details Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt) Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)

Publishing date

2007-06-07

URL

arxiv.org/abs/0706.1072v3

Science

Mathematics

Field

Algebraic Geometry

Additional info

18 pages. Additional minor revisions and corrections

Type

Scientific paper

Abstract

We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $\alpha \in \Br(k)$, compute the index of the class $\alpha_{k(X)} \in \Br(X)$ obtained from $\alpha$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, giving a complete solution to the index reduction problem in this case. Using the twisted Fourier-Mukai transform, we show that a similarly simple formula describes homogeneous index reduction on torsors under higher-dimensional abelian varieties.

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