Let k be a field of dimension d, and let b1∈operatornameBr(k). The index ind(alpha) is the square root of the rank of the unique division algebra representing alpha. Period-index conjecture. ind(alpha)| (per(alpha))^d-1 for all alpha∈operatornameBr(k). This is the algebraic period-index conjecture attributed in the source to Colliot-Thélène. The surrounding text presents a topological period-index theorem as evidence, but gives no resolution of this conjecture.

Colliot-Thélène's period-index conjecture

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Let $k$ be a field of dimension $d$ , and let $b1\in\operatorname{Br}(k)$ . The index $ind(\alpha)$ is the square root of the rank of the unique division algebra representing $\alpha$ .

Period-index conjecture.

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

for all $\alpha\in\operatorname{Br}(k)$ .

This is the algebraic period-index conjecture attributed in the source to Colliot-Thélène. The surrounding text presents a topological period-index theorem as evidence, but gives no resolution of this conjecture.

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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.

Colliot-Thélène's period-index conjecture
Let $X$ be a smooth projective variety over an algebraically closed field, and let $\alpha\in\operatorname{Br}(X)$ be a Brauer class. Write $\operatorname{per}(\alpha)$ for its period and $\operatorname{ind}(\alpha)$ for its index. Colliot-Thélène's conjecture. Every such class satisfies
$\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^{\dim(X)-1}.$
This is the general period-index conjecture. The conjecture is wide open in general, although several cases and partial bounds are known.
source: Alessio Bottini and Daniel Huybrechts, “The period-index problem for hyperkähler varieties: Lower and upper bounds”, arXiv:2512.15131 (2025).

Sources & referencesView supporting material

Primary source

Benjamin Antieau, “Cohomological obstruction theory for Brauer classes and the period-index problem”, arXiv:0909.2352 (2010).

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