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.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2406.00136
 arXiv:
 arXiv:2406.00136
 Bibcode:
 2024arXiv240600136H
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory
 EPrint:
 Comments very welcome. 13 pages. A version of this note originally appeared on the author's website in November 2022