Profinite completions of products

Haine, Peter J.

doi:10.2140/agt.2026.26.397

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Profinite completions of products

Peter J. Haine

Algebraic & Geometric Topology 26 (2026) 397–410

DOI: 10.2140/agt.2026.26.397

Abstract

A source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. We provide a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the profinite completion of $X \times Y$ agrees with the product of the profinite completions of $X$ and $Y$ . Using this criterion, we show that profinite completion preserves products of étale homotopy types of qcqs schemes. This fills a gap in Chough’s proof of the Künneth formula for the étale homotopy type of a product of proper schemes over a separably closed field.

Keywords

profinite completion, profinite space, étale homotopy type

Mathematical Subject Classification

Primary: 55P99

References

Publication

Received: 3 July 2024

Revised: 27 January 2025

Accepted: 14 February 2025

Published: 16 January 2026

Authors

Peter J. Haine
Department of Mathematics University of Southern California Los Angeles, CA United States

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Volume 26, issue 1 (2026)