Going-Down functors and the K\"unneth formula for crossed products by \'etale groupoids

We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the K\"unneth formula for the 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 into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the K\"unneth formula. We also provide a stability result for the K\"unneth formula using controlled K-theory, and apply it to give an example of a space that does not admit a coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies the K\"unneth formula. As a by-product of our methods, we also prove a permanence property for the Baum-Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.