We study homology of ample groupoids via the compactly supported 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 étale 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 $$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ ι_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ κ_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$ The key input is the chain level isomorphism $C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$, which reduces the groupoid statement to the classical algebraic UCT for the free complex $C_c(\mathcal G_\bullet,\mathbb Z)$. 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 $Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$ is exactly the compactly supported functions with finite image. Thus $Φ_X$ is surjective if and only if every $f\in C_c(X,A)$ has finite image, and for suitable $X$ one can produce compactly supported continuous maps $X\to A$ with infinite image. Finally, for a clopen saturated cover $\mathcal G_0=U_1\cup U_2$ 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.

翻译：我们通过神经的紧支撑Moore复形研究充足广群的同调。设$A$为拓扑阿贝尔群。对于$n\ge 0$，定义$C_n(\mathcal G;A) := C_c(\mathcal G_n,A)$及$\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$。由此定义$H_n(\mathcal G;A)$。该理论对连续étale同态具有函子性，并与标准约化（包括限制于饱和闭开子集）相容。在充足框架下，该理论在角谷等价下保持不变。我们重新证明了Matui型长正合序列，并在链层次上识别了比较映射。对于离散$A$，我们证明了自然统一系数短正合序列$$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ ι_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ κ_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$关键输入是链层次同构$C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$，它将广群陈述约化为自由复形$C_c(\mathcal G_\bullet,\mathbb Z)$的经典代数UCT。我们还孤立了非离散系数的障碍。对于具有紧开集基的局部紧全不连通豪斯多夫空间$X$，映射$Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$的像恰为具有有限像的紧支撑函数。因此$Φ_X$是满射当且仅当每个$f\in C_c(X,A)$具有有限像，且对于适当的$X$，可构造具有无限像的紧支撑连续映射$X\to A$。最后，对于闭开饱和覆盖$\mathcal G_0=U_1\cup U_2$，我们构造了Moore复形的短正合序列，并推导出$H_\bullet(\mathcal G;A)$的Mayer-Vietoris长正合序列以用于显式计算。