# Twisted sheaves and the period-index problem

@article{Lieblich2008TwistedSA, title={Twisted sheaves and the period-index problem}, author={Max Lieblich}, journal={Compositio Mathematica}, year={2008}, volume={144}, pages={1 - 31} }

Abstract We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over…

## 71 Citations

### The period-index problem for twisted topological K-theory

- Mathematics
- 2014

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya…

### Period and index in the Brauer group of an arithmetic surface

- Mathematics
- 2007

Abstract In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector…

### Period -Index problem for hyperelliptic curves

- Mathematics
- 2022

Let C be a smooth projective curve of genus 2 over a number field k with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of X(Br(C)). In the appendix, an…

### A Gabriel Theorem for Coherent Twisted Sheaves and Picard Group and 2-factoriality of O'Grady's Examples of Irreducible Symplectic Varieties

- Mathematics
- 2008

This PhD thesis is divided in two parts: in the first one, we present a generalization of Gabriel's Theorem on coherent sheaves to twisted coherent sheaves. More precisely, we show that any…

### The topological period–index problem over 6‐complexes

- Mathematics
- 2014

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah–Hirzebruch spectral sequence, we deduce a lower bound on the…

### Cohomological obstruction theory for Brauer classes and the period-index problem

- Mathematics
- 2009

Let U be a connected scheme of finite cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that alpha is a class in H^2(U_et,Gm)_{tors}. For…

### Vector Bundles as Generators on Schemes and Stacks

- Mathematics
- 2010

We investigate the resolution property of quasicompact and quasiseparated schemes, or more generally of algebraic stacks with pointwise affine stabilizer groups. Such a space has the resolution…

### Experiments on the Brauer map in high codimension

- MathematicsAlgebra & Number Theory
- 2022

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that,…

### Fe b 20 20 EXPERIMENTS ON THE BRAUER MAP IN HIGH CODIMENSION

- Mathematics
- 2020

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that,…

## References

SHOWING 1-10 OF 61 REFERENCES

### Period and index in the Brauer group of an arithmetic surface

- Mathematics
- 2007

Abstract In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector…

### Moduli of twisted sheaves

- Mathematics
- 2004

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For…

### Moduli of vector bundles on projective surfaces: some basic results

- Mathematics
- 1994

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough.…

### Sheaves on Artin stacks

- Mathematics
- 2007

Abstract We develop a theory of quasi-coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological…

### Every rationally connected variety over the function field of a curve has a rational point

- Mathematics
- 2003

In a paper from 1992, Kollár, Miyaoka and Mori posed the following question: Given a proper flat morphism f : P → X with target a nonsingular curve and whose geometric generic fiber is…

### Compactified moduli of projective bundles

- Mathematics
- 2007

We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to…

### Some Elementary Examples of Unirational Varieties Which are Not Rational

- Mathematics
- 1972

An outstanding problem in the algebraic geometry of varieties of dimension n ^ 3 over an algebraically closed field k has been whether there exist unirational varieties which are not rational. Here V…

### Rational families of vector bundles on curves, II

- Mathematics
- 2003

Let C be a smooth complex projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k>1, we find two…