The \'etale symmetric K\"unneth theorem

TL;DR

This paper proves an étale homotopy analogue of the classical Künneth theorem for symmetric powers of schemes, relating their étale homotopy types to symmetric powers of the original scheme's étale homotopy type.

Contribution

It establishes an étale symmetric Künneth theorem connecting symmetric powers of schemes to symmetric powers of their étale homotopy types, extending classical topological results.

Findings

Étale homotopy type of symmetric powers is homologically equivalent to symmetric powers of the original type.

The $bZ/l$-local étale homotopy type of motivic Eilenberg-Mac Lane spaces is an ordinary Eilenberg-Mac Lane space.

Provides tools for understanding étale homotopy types in algebraic geometry.

Abstract

Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the \'etale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the \'etale homotopy type of $X$. We deduce that the $\mathbb Z/l$-local \'etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.