Profinite completions of products

TL;DR

This paper introduces a new criterion ensuring that profinite completion preserves products in prospaces, particularly for étale homotopy types of schemes, addressing a key gap in existing proofs.

Contribution

It provides a checkable condition for profinite completion to preserve products, especially in étale homotopy types of schemes, enhancing the understanding of profinite homotopy theory.

Findings

Profinite completion preserves products under the new criterion.

Application to étale homotopy types of schemes.

Fills a gap in K"unneth formula proof for proper schemes.

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 \'{e}tale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the K\"{u}nneth formula for the \'{e}tale homotopy type of a product of proper schemes over a separably closed field.