为群胚同调建立通用系数公式与梅耶-维托里斯序列，统一处理拓扑阿贝尔系数。
Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology
- 基于紧支集摩尔复形构建群胚同调理论，支持连续埃坦同态与饱和开子集限制。
- 对离散系数证明了标准的万有系数短正合列，关键在链级同构 $C_c(ullet,bZ) en A o C_c(ullet,A)$。
- 构造了开饱和覆盖下的梅耶-维托里斯长正合列，适用于具体同调计算，尤其对局部紧零维空间有效。
我们通过群胚的神经复形的紧支集摩尔复形研究了稠密群胚的同调。设 $A$ 为拓扑阿贝尔群，对 $n/ge 0$ 定义 $C_n( rak G;A) := C_c( rak G_n,A)$，并令 $oundary_n^A = extstyleackslashsum_{i=0}^n (-1)^i (d_i)_*$，从而定义同调 $H_n( rak G;A)$。该理论对连续埃坦同态是函子性的，并兼容标准约化，包括对饱和紧开子集的限制。在稠密情形下，它关于卡库塔尼等价不变。我们重新证明了马秋类型长正合列，并在链级别识别了比较映射。当 $A$ 离散时，我们证明了自然的万有系数短正合列：$$0\to H_n(\frak G)\otimes_{\mathbb Z}A \xrightarrow{\ ι_n^{\frak G}} H_n(\frak G;A) \xrightarrow{\ κ_n^{\frak G}} \operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\frak G),A\bigr)\to 0.$$ 关键在于链级别同构 $C_c(\frak G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\frak G_n,A)$，将群胚命题归约为自由复形 $C_c(\frak G_\bullet,\mathbb Z)$ 的经典代数UCT。我们还刻画了非离散系数的障碍：对局部紧、零维、豪斯多夫空间 $X$ 且具有紧开基，映射 $Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$ 的像恰好是有限像的紧支集函数。因此 $Φ_X$ 满射当且仅当每个 $f\in C_c(X,A)$ 具有限像；对合适 $X$ 可构造具有无限像的紧支集连续映射 $X\to A$。最后，对饱和开覆盖 $\frak G_0=U_1\cup U_2$，我们构造了摩尔复形的短正合列，并导出 $H_\bullet(\frak G;A)$ 的梅耶-维托里斯长正合列，用于显式计算。
原文摘要 · Abstract (English)
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.
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