{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:5ed77be4-d851-51e9-b457-f87d7d1b9deb", "labels": { "zh-CN": "屈内特公式", "en": "Künneth formula" }, "alternative_labels": [], "scope_note": "Nakahara §6.4.3给出实de Rham上同调的乘积公式；保留系数及空间假设。", "identity_status": "active", "identity_revision": "fe50c71a625cdcf6e5b7ecc980b8c28cf01e13f82d3c585f7e56c5dc0cf6973d", "occurrences": [ { "concept_id": "urn:uuid:5ed77be4-d851-51e9-b457-f87d7d1b9deb", "book_id": "nakahara-geometry-topology-physics", "edition_id": "second-edition-local", "lang": "zh-CN", "route": "books/nakahara-geometry-topology-physics/second-edition-local/zh/chapter-06/04-section-6-4", "anchor": "bound-definition-c8669578e9146d7b4241a2fe", "chapter": "6", "section": "6.4", "offset": 21258, "range": [ 39181, 39673 ], "source_revision": "6aab657c41d9d1ac76549c604724105327bc219fe0fc97cf77e858dec3b440ae", "source_category": "machine_draft", "review_status": "unreviewed", "role": "definition", "role_basis": { "kind": "explicit-source-uuid-marker", "source_bound_role": "definition", "first_mention": true, "source_node": "a-0e27122179728509c056758b", "source_sidecar": { "path": "library/books/nakahara-geometry-topology-physics/second-edition-local/provenance/chapter-06/concept-annotations.json", "sha256": "8e24dc92eba8bc34f780867e45c26d2a5c8c45b757e7b0baaa2fb1e5a283fd69", "page": "6.4", "annotation_count": 46, "policy": "explicit-source-bound-UUIDs; no string-based identity allocation", "rendered_annotation_count": 27, "plain_mention_count": 19, "display_policy": "clickable-only" } }, "label": "屈内特公式(Künneth formula)", "context": { "range": [ 38731, 39677 ], "quote": "比较左端与右端，可得 $\\alpha^{p - 1} = \\delta^{r - p - 1} = 0$，从而 $\\omega_{i}^{p}\\wedge \\eta_{j}^{r - p} = 0$，这与我们的假设矛盾。因此，$\\omega_{i}^{p}\\wedge \\eta_{j}^{r - p}$ 是 $H^{r}(M)$ 中的一个非平凡元素。反之，$H^{r}(M)$ 中的任一元素都可分解为 $H^{p}(M_{1})$ 的元素与 $H^{r - p}(M_2)$ 的元素之积的和，其中 $0\\leq p\\leq r$。至此我们得到了屈内特公式(Künneth formula)", "quote_sha256": "3f94af051cdaf8ab865088d5a0a1f3c71a11096261def28245d2dba34470b2ee", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/nakahara-geometry-topology-physics/second-edition-local/en/chapter-06/04-section-6-4", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-9276869c2a882d908143ba83" } ], "registered_positions": [], "statements": [], "resolved_references": [], "reading_order": [ "occ-9276869c2a882d908143ba83" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 0, "located_occurrences": 1, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "reviewed_definitions": [ { "concept_id": "urn:uuid:5ed77be4-d851-51e9-b457-f87d7d1b9deb", "definition_id": "urn:uuid:095e5c60-f20b-566a-9c8b-f6e56f219d60", "definition_group": null, "path": "library/concepts/source-evidence/mcp/4f1eb261589629f2f3aeb64cbb98236d3f280b084c39c867c4f0ab1c31a8524d.txt", "range": [ 0, 469 ], "quote": "By comparing the LHS with the RHS, we find $\\alpha^{p - 1} = \\delta^{r - p - 1} = 0$ , hence $\\omega_{i}^{p}\\wedge \\eta_{j}^{r - p} = 0$ in contradiction to our assumption. Thus, $\\omega_{i}^{p}\\wedge \\eta_{j}^{r - p}$ is a nontrivial element of $H^{r}(M)$ . Conversely, any element of $H^{r}(M)$ can be decomposed into a sum of a product of the elements of $H^{p}(M_{1})$ and $H^{r - p}(M_2)$ for $0\\leq p\\leq r$ . 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