Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let A be a topological abelian group. For n≥ 0 set Cn(mathcal G;A) := Cc(mathcal Gn,A) and define partialnA=sumi=0n(-1)i(di)_*. This defines Hn(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. We reprove Matui type long exact sequences and identify the comparison maps at chain level. For discrete A we prove a natural universal coefficient short exact sequence 0→ Hn(mathcal G)otimesmathbb ZAxrightarrow iotanmathcal G Hn(mathcal G;A)xrightarrow kappanmathcal G operatornameTor1mathbb Zbigl(Hn-1(mathcal G),Abigr)→ 0. The key input is the chain level isomorphism Cc(mathcal Gn,mathbb Z)otimesmathbb ZAcong Cc(mathcal Gn,A), which reduces the groupoid statement to the classical algebraic UCT for the free complex Cc(mathcal G_bullet,mathbb Z). We also isolate the obstruction for non-discrete coefficients. For a locally compact totally disconnected Hausdorff space X with a basis of compact open sets, the image of PhiX:Cc(X,mathbb Z)otimesmathbb ZA→ Cc(X,A) is exactly the compactly supported functions with finite image. Thus PhiX is surjective if and only if every f∈ Cc(X,A) has finite image, and for suitable X one can produce compactly supported continuous maps X→ A with infinite image. Finally, for a clopen saturated cover mathcal G0=U1cup U2 we construct a short exact sequence of Moore complexes and derive a Mayer-Vietoris long exact sequence for H_bullet(mathcal G;A) for explicit computations.