Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

ArXi:2602.08998v4 Announce Type: replace-cross We study homology of ample groupoids via the compactly ed Moore complex of the nerve. Let $A$ be a topological abelian group. For $n\ge 0$ set $C_n(\mathcal G;A):= C_c(\mathcal G_n,A)$ and define $\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$. This defines $H_n(\mathcal G;A)$. The theory is functorial for continuous \'etale homomorphisms. It is compatible with standard reductions, including restriction to saturated clopen subsets. In the ample setting it is invariant under Kakutani equivalence.