Let K be a field of transcendence degree d over an algebraically closed field k. For alpha ∈ operatornameBr(K), let operatornameper(alpha) be its period and let ind(alpha) be its index. Period-index conjecture. For every alpha ∈ operatornameBr(K), ind(alpha) | operatornameper(alpha)^d-1. This is a central open problem about the relationship between the period and index of Brauer classes. It is known for d≤ 2, while for d≥ 3 it is not known for any field K; the conjectural exponent is sharp.

The period-index conjecture for fields over algebraically closed fields

Let $K$ be a field of transcendence degree $d$ over an algebraically closed field $k$ . For $\alpha \in \operatorname{Br}(K)$ , let $\operatorname{per}(\alpha)$ be its period and let $\mathrm{ind}(\alpha)$ be its index.

Period-index conjecture. For every $\alpha \in \operatorname{Br}(K)$ ,

\mathrm{ind}(\alpha) \mid \operatorname{per}(\alpha)^{d-1}.

This is a central open problem about the relationship between the period and index of Brauer classes. It is known for $d\leq 2$ , while for $d\geq 3$ it is not known for any field $K$ ; the conjectural exponent is sharp.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The period-index conjecture for fields over algebraically closed fields
Let $k$ be an algebraically closed field, and let $K/k$ be an extension of transcendence degree $d$ . For any $\alpha \in \operatorname{Br}(K)$ , the period-index conjecture asserts:

Period-index conjecture.
$\operatorname{ind}(\alpha) \divides \operatorname{per}(\alpha)^{d-1}.$
The conjecture is trivial for $d=0$ , follows from Tsen's theorem for $d=1$ , and is known for $d=2$ by work of de Jong, Lieblich, and de Jong–Starr. For $d\geqslant 3$ , no field $K$ is known for which a uniform period-index bound is established.
source: James Hotchkiss, “Hodge theory of twisted derived categories and the period-index problem”, arXiv:2212.10638 (2022).

Sources & referencesView supporting material

Primary source

James Hotchkiss and Alexander Perry, “The period-index conjecture for abelian threefolds and Donaldson-Thomas theory”, arXiv:2405.03315 (2024).

1 week ago

The period-index conjecture for fields over algebraically closed fields

Equivalent formulations 1

The period-index conjecture for fields over algebraically closed fields

Primary source

Progress summary

Solutions 0