Holmstrom Homotopy invariance

In algebraic geometry, this can mean either that a functor sends $X \times {A}^1 \to X$ to an isomorphism, or the same statement for $A_X^m \to X$ (affine m-space over $X$).

The latter statement holds for etale cohomology with torsion coeffs prime to the characteristic.

Grayson (K-th handbook p 47) mentions that there is a standard way of converting a functor into a homotopy invariant functor. Check his references to Gersten and Karoubi-Villamayor. Very briefly following Grayson: for a contravariant functor $F$ from a category of varieties containg the affine spaces to the category of spaces, we consider the functor $G: X \mapsto | n \mapsto F(X \times A^n ) |$ . The map $F \to G$ is, in a certain up-to-homotopy sense, the universal map from $F$ to a homotopy invariant functor.

nLab page on Homotopy invariance