Profinite completions of products
Abstract
A source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. In this note, 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.
Fields of Study & Subjects
Computability
Logic
AI Algorithms
programming
and type systems
Publication details
- Journal / source
- Algebraic & Geometric Topology
- Volume
- 26
- Issue
- 1
- Pages
- 397–410
- Publication date
- 16 January 2026
- Language
- EN