Artin-Verdier duality is a duality theorem for constructible abelian sheaves defined over the spectrum of a ring of algebraic numbers, introduced by Michael Artin and Jean-Louis Verdier in 1964. It generalizes an earlier duality theorem due to John Tate. The theorem shows that, as far as etale or flat cohomology is concerned, the ring of integers in a number field behaves like a three-dimensional mathematical object. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor the spectrum of the ring of integers in a totally imaginary number field, and a constructible etale abelian sheaf on it, a Yoneda pairing between an etale cohomology group and an extension group into the sheaf of units is a non-degenerate pairing of finite abelian groups, in every relevant degree, playing the role Poincare duality plays for a compact three dimensional manifold. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Artin-Verdier duality (Wikipedia)
Sources
1. Artin-Verdier duality (Wikipedia)
Statement section, the duality pairing
Let X be the spectrum of the ring of integers in a totally imaginary number field K, and F a constructible etale abelian sheaf on X.
Lead section, attribution and year
In mathematics, Artin-Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Michael Artin and Jean-Louis Verdier (1964), that generalizes Tate duality.
Proved By: Emil Artin, Lead paragraph
In mathematics, Artin-Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers,
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