Homology and K-theory of dynamical systems. III. Beyond totally disconnected case

We study homological invariants of \'etale groupoids continuing on our previous work, but going beyond the ample case by incorporating resolutions in the space direction. We prove analogues of the K\"unneth formula and the Poincar\'e duality in this framework. For non-wandering Smale spaces, we show that Putnam's homology is isomorphic to the groupoid homology with integer coefficients, and that the K-groups of C*-algebras of stable and unstable equivalence groupoids have finite rank.