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Both sides are groups according to lemma 3.1. Define a map $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$ by $\\varphi([x]) = f(x)$ . This map is well defined since for $x' \\in [x]$ , there exists $h \\in \\ker f$ such that $x' = x + h$ and $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that $\\varphi$ is an isomorphism. First, $\\varphi$ is a homomorphism,", "citation": "according to lemma 3.1", "target_marker": "Lemma 3.1.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-581d033a4d3b64de8b484b60", "relation": "uses-author-statement", "basis": "explicit-author-reference", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-f5c8dc7bcb0740921fe8819b", "quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.", "citation": "From lemma 3.1", "target_marker": "Lemma 3.1.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-f504b265a21f6d412d13d8bb", "relation": "uses-author-statement", "basis": "explicit-author-reference", "from_statement": "statement-cf70f6e882eed0df163c05a1", "to_statement": "statement-dd7721e0c9adf5e7b891bfcd", "quote": "Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \\partial_{r+1}d$ for some $d \\in C_{r+1}(K)$ . Then we find $\\partial_r c = \\partial_r(\\partial_{r+1}d) = 0$ , that is, $c \\in Z_r(K)$ . This implies $Z_r(K) \\supset B_r(K)$ .", "citation": "from lemma 3.3", "target_marker": "Lemma 3.3.", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-d882a2d2a9998b9472515b5f", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-cf70f6e882eed0df163c05a1", "basis": "author-transition", "quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.", "citation": null, "target_marker": "Theorem 3.3. Let $Z_{r}(K)$ and $B_{r}(K)$ be the ", "reason": "作者的 Since 承接使用了定理3.3的包含关系，保证分母是分子中的子群；阿贝尔条件另由链群定义提供。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-5ab4c184178868d2f11eed23", "relation": "uses-author-statement", "from_statement": "statement-ae16e6e7da0a84f365502acf", "to_statement": "statement-45381cbdd77285069b293927", "basis": "construction-use", "quote": "Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined.", "citation": null, "target_marker": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r", "reason": "先确认 Zr、Br 的群资格，再结合定理3.3的包含关系组成商群。该前置结论单独不证明 Br 包含于 Zr。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-fd0436467a9a3a8bd6545786", "relation": "uses-author-statement", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-ac7699973242d9783a9bb1ca", "basis": "construction-use", "quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.", "citation": null, "target_marker": "Definition 3.3. If $c \\in C_r(K)$ satisfies", "reason": "引理3.1应用到核时，核正是定义3.3的循环群。", "evidence_status": "codex-source-reviewed-candidate", "review_status": "codex-source-reviewed; pending-independent-human-review" }, { "id": "dependency-514dec634fe89684d29346b0", "relation": "uses-author-statement", "from_statement": "statement-45381cbdd77285069b293927", "to_statement": "statement-e9cdbaabe0ae1a11cb9cc9b3", "basis": "construction-use", "quote": "From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.", "citation": null, "target_marker": "Definition 3.4. 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