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arXiv · 2406.00136

Profinite completions of products

Peter J. Haine
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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.

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Peter J. Haine. 2024-05-31. Profinite completions of products. https://doi.org/10.2140/agt.2026.26.397

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