---- > [!definition] Definition. ([[chain homotopy]]) > > Let $C_{\bullet}$, $D_{\bullet}$ be [[chain complex of modules|chain complexes]]. A **chain homotopy** $H$ from the [[chain map|chain map]] $f_{\bullet}:C_{\bullet} \to D_{\bullet}$ to the [[chain complex of modules|chain map]] $g_{\bullet}:C_{\bullet} \to D_{\bullet}$ is a sequence of [[linear map|homomorphisms]] $h_{n}:C_{n} \to D_{n+1}$ such that $f_{n}-g_{n} = d_{n+1}^{D} \circ h_{n} + h_{n-1} \circ d_{n}^{C} : C_{n} \to D_{n}. $ > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEB9YMAagEYAviEGl0mXPkIp+5KrUYs2XQqPHY8BIgCY51es1aIO3MAFohIsSAwapRMv3kGlxgCKmBwtTYmbpyLJO+opGIB6q1raSWii6wQqGbB48Ft7yMFAA5vBEoABmAE4QALZIZCA4EEhC1kWl5dRVSADM1ABGMGBQSGYtxD71ZYgVzYhtIJ3drQN1xcOyldWIACwdXT2IfRUuYfmmwtQMdJ0MAAp+9sZYYNiwVgXzNU3La5MbMyFJxlmchEcnGDnS6xEA3O6sQZPRC6JZIACs62mW36UIaMJeCKRm1Rc3RizGsN2bCgAD0VA8QENWpjVl9XCAySk+JYAacLnZQYUsFkABY4SnUulwxDwtELWkTBg3MJwCDSnpsoEcmLSMG3LD3elhXl-QXQ2FjN7HdkgtXgzWsbVsXWpVkgaVgWXyzUgai8mB0TZgJgMBhNOhYBhsSBOkQUQRAA > \begin{tikzcd} > C_{n+1} \arrow[r] \arrow[d, bend right] \arrow[d, bend left] & C_n \arrow[d, "f_{n}" description, bend right] \arrow[d, "g_n" description, bend left] \arrow[r, "d^C_n"] \arrow[ld, "h_n" description] & C_{n-1} \arrow[d, bend right] \arrow[d, bend left] \arrow[ld, "h_{n-1}" description] \\ > D_{n+1} \arrow[r, "d^D_{n+1}"'] & D_n \arrow[r] & D_{n-1} > \end{tikzcd} > \end{document} > ``` > ---- #### ---- #### 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 > ```