Going-down functors and the Künneth formula for crossed products by étale groupoids

We study the connection between the Baum-Connes conjecture for an ample groupoid $ G$ with coefficient $ A$ and the Künneth formula for the $ {\mathrm K}$-theory of tensor products by the crossed product $ A\rtimes _r G$. To do so, we develop the machinery of going-down functors for ample groupoids. As an application, we prove that both the uniform Roe algebra of a coarse space which uniformly embeds in a Hilbert space and the maximal Roe algebra of a space admitting a fibered coarse embedding in a Hilbert space satisfy the Künneth formula. Additionally, we give an example of a space that does not admit a coarse embedding in a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula and provides a stability result for the Künneth formula using controlled $ {\mathrm K}$-theory. As a byproduct of our methods, we also prove a permanence property for the Baum-Connes conjecture with respect to equivariant inductive limits of the coefficient algebra