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Moore复形研究富足广群的同调。设$A$为拓扑阿贝尔群。对$n\ge 0$，令$C_n(\mathcal G;A) := C_c(\mathcal G_n,A)$并定义$\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$。由此定义$H_n(\mathcal G;A)$。该理论对连续étale同态具有函子性，且与标准约化（包括限制到饱和既开又闭子集）相容。在富足设定下，该理论在Kakutani等价下保持不变。我们重新证明了Matui型长正合序列，并在链水平上确定了比较映射。对于离散的$A$，我们证明了一个自然的万有系数短正合序列$$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ ι_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ κ_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$关键输入是链水平的同构$C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$，它将广群陈述约化为自由复形$C_c(\mathcal G_\bullet,\mathbb Z)$的经典代数万有系数定理。我们还分离了非离散系数的障碍。对于具有紧开集基的局部紧致完全不连通Hausdorff空间$X$，映射$Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$的像恰好是具有限像的紧支函数。因此$Φ_X$是满射当且仅当每个$f\in C_c(X,A)$具有有限像，且对于合适的$X$可以构造出具有无限像的紧支连续映射$X\to A$。最后，对于既开又闭的饱和覆盖$\mathcal G_0=U_1\cup U_2$，我们构造了Moore复形的短正合序列，并为$H_\bullet(\mathcal G;A)$导出了一个用于显式计算的Mayer-Vietoris长正合序列。