Period and index in the Brauer group of an arithmetic surface

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 bundles. By studying the structure of these spaces we prove new results on the standard period-index conjecture. The first yields new bounds on the periodindex relation for classes on curves over higher local fields, while the second can be used to relate the Hasse principle for forms of moduli spaces of stable vector bundles on pointed curves over global fields to the period-index problem for Brauer groups of arithmetic surfaces. We include an appendix by Daniel Krashen showing that the local period-index bounds are sharp.

On Brauer p-dimensions and index-exponent relations over finitely-generated field extensions

“…Indeed, if T ∈ F is a transcendental element over C ν , F 0 = C ν (T ), and f 0 is the restricted Gauss valuation of F 0 extending the natural Z ν -valued C-valuation of C ν (see [13], Example 4.3.2), then one may take as f ν any prolongation of f 0 on F . The equality trd( F /C) = 1 ensures that r p ( F ) = ∞, for all p ∈ P, which enables one to deduce from [31], Theorem 1, and [25], Corollary 1.4, that Brd p (F ) = abrd p (F ) = ν, p ∈ P and p = char(C) (see [25], page 37, for more details in case F/C ν is rational). At the same time, it follows from [8], Proposition 7.1, that if char(C) = 0, then Brd(C ν ) = abrd(C ν ) = [ν/2]; hence, by Proof.…”

Refinements to Patching and Applications to Field Invariants

“…Following [Lie11] and [HHK09], we define the Brauer dimension of a field k away from a prime p as follows: The value is 0 if the absolute Galois group of k is a pro-p group (e.g. if k is separably closed).…”

“…Previous results about the u-invariant and period-index bounds for related fields appeared in such papers as [PS10], [HHK09], [Lee13], [Sal08], [deJ04], and [Lie11]. To obtain our present results, we build on the patching framework that was used in our previous manuscripts [HH10], [HHK09], [HHK15], and [HHK13].…”

The bigger Brauer group and twisted sheaves

“…By composing with (U , u) → U , we obtain a functor G → (Aff/S). As Lieblich explains in [14,Proposition 2.4.3], this makes G into an S-stack, endowed with a 1-morphism of S-stacks F : G → X . Under fairly general assumptions, this S-stack is algebraic; the following criterion generalizes a result of de Jong [3] and Lieblich [ Proof.…”