---- > [!theorem] Theorem. ([[long exact sequence on homology induced by short exact sequence of chain complexes]]) > If $0 \to A_{\bullet} \xrightarrow{i_{\bullet}} B_{\bullet} \xrightarrow{j_{\bullet}} C_{\bullet} \to 0$ is a [[short exact sequence]] of [[chain complex of modules|chain complexes]], then there are natural "connecting morphisms" $\partial{_{*}}:H_{n}(C_{\bullet}) \to H_{n-1}(A_{\bullet})$ such that > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BjKCHBAL6l0mXPkIoAjOSq1GLNgAkA+mAAUAQWWcARkwYMYOAJQghI7HgJEATDOr1mrRCBXqAQtvZ6DR0+ZAMS3EiAGZ7OSclVTUAYS8fQxMzYUDRKwlkaRtZRwUXFWAwAFpJAU0E-ST-VKCxaxQ7HId5Z1dlItLyz10qvxSLeszw5sj89s6yuMrfZIC6jKIAFlJRvLbOHj5BWvSQxtJJXNa2TjQ6ACc8RmUAKjNZGCgAc3giUAAzC4gAWyQANmoOAgSAA7AEvr8kGQQMCkNIxht2OcrlgbvcId8-ogEXDEHZEWwsHcBiBIdiCXjwoSXAArEmYqGIal4gAc1B0MDAUCQoWIjOx7NhIMQKxAnO5SGKfIFSDFeIArC0oi5iRjUuSkErhQDleN6RiKAIgA > \begin{tikzcd} > \cdots \arrow[r, "\partial_*"] & H_n(A_\bullet) \arrow[r, "i_*"] & H_n(B_\bullet) \arrow[r, "j_*"] & H_n(C_\bullet) \arrow[ld, bend left] & \\ > & & \partial_* \arrow[ld, bend right] & & \\ > & H_{n-1}(A_\bullet) \arrow[r, "i_*"] & H_{n-1}(B_\bullet) \arrow[r, "j_*"] & H_{n-1}(C_\bullet) \arrow[r] & \cdots > \end{tikzcd} > \end{document} > ``` > is [[exact sequence|exact]]. Here, the $i_{*}$ and $j_{*}$ denote the [[homomorphism on homology induced by a chain map|morphisms on homology]] induced by the respective [[chain map|chain maps]]. ^theorem > [!proof]+ Proof. ([[long exact sequence on homology induced by short exact sequence of chain complexes]]) > The result should follow merely by inductively applying [[the snake lemma]]. (In fact, sometimes this theorem *called* the snake lemma — they essentially contain equivalent data.) - [ ] cohomology version ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```