---- Let $R$ be a (say, [[commutative ring|commutative]]) [[ring]]. > [!definition] Definition. ([[relative homology of an embedding of chain complexes]]) > Let $(M_{\bullet}, d_{\bullet})$ be a [[chain complex of modules|chain complex]] of $R$-[[module|modules]]. Then any left-[[exact sequence]] $0 \to N_{\bullet} \to M_{\bullet}$ of [[chain complex of modules|chain complexes]] of $R$-[[module|modules]] extends to a [[short exact sequence]] $0 \to N_{\bullet} \xrightarrow{\iota_{\bullet}} M_{\bullet} \xrightarrow{q_{\bullet}} L_{\bullet} \to 0 ,$ where $L_{\bullet}=M_{\bullet} / N_{\bullet}$. Specifically, each [[chain complex of modules|differential]] $d_{n}:M_{n} \to M_{n-1}$ restricts to a map $N_{n} \to N_{n-1}$, and thus [[characterization of quotienting a module|descends]] to a [[well-defined]] differential $\overline{d_{n}}:L_{n} \to L_{n-1}$ sending $[c] \mapsto [d_{n} c]$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkB9QgX1PUy58hFAEZyVWoxZsuwMAFpRPEHwHY8BIgCYJ1es1aIQAGU7ylPALwAdGwDMATnQDGwOYuU9gAOXOeeFTUQDA1hIjJRSQMZYzsnV3d-QN9k6zNeSRgoAHN4IlAnCABbJDIQHAgkcSlDNihuVX4QItLEGsqkXVrYkABHRuoACxg6KDYcAHcIEbGEYNay6k7EAGZh0fHjKZnN+ebFteWqxG6YoxA7GAAPLDgcOAACAEJHuwgaGEcGLDAYYAaYBU1AYdAARjAGAAFQSaEQgRxYHJDHCqCg8IA > \begin{tikzcd} > M_n \arrow[r, "d_n"] \arrow[d, two heads] & M_{n-1} \arrow[r, "q_n", two heads] & L_{n-1}=\frac{M_{n-1}}{N_{n-1}} \\ > \frac{M_{n}}{N_{n}}=L_n \arrow[rru, "\exists ! \overline{d_n}"'] & & > \end{tikzcd} > \end{document} > ``` > > The $i$th [[(co)homology of a complex|homology]] $H^{i}\big( L_{\bullet}, \overline{d_{\bullet}} \big)$ of $L_{\bullet}=M_{\bullet}/N_{\bullet}$ is called the **$i$th relative homology of the inclusion/embedding $N_{\bullet} \hookrightarrow M_{\bullet}$**. > > > Let $\mathsf{ChainPairs}$ be the [[category]] of embeddings of [[chain complex of modules|chain complexes]] of $R$-[[module|modules]], a morphism $(N_{\bullet} \hookrightarrow M_{\bullet}) \xrightarrow{f} (N_{\bullet}' \hookrightarrow M_{\bullet}')$ in which looks like a [[diagram]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADkB9AHW4CMmDBjBwgAvqXSZc+QigCM5KrUYs2AWR79Bw0RKnY8BImXnL6zVog5aBQkQHJxkkBkOyiis9Qtrrm3jtdJzFlGCgAc3giUAAzACcIAFskMhAcCCQAJh9VKxBYgD12EGoGOj4YBgAFaSM5EHisCIALPRcE5NTqDKRFdLosBjYWiAgAa2c4xJTEHPTMxABmHsHh61GJqYKZvp7FlZVLNiL1UpByypq6j2thWL0KMSA > \begin{tikzcd} > N_\bullet \arrow[d, "f^N"'] \arrow[r, hook] & M_\bullet \arrow[d, "f^M"] \\ > N_\bullet' \arrow[r, hook] & M_\bullet' > \end{tikzcd} > \end{document} > ``` > > > The $i$th relative homology can be viewed as a [[covariant functor]] $H_{i}(- \hookrightarrow -):\mathsf{ChainPairs} \to R\text{-}\mathsf{Mod}$. $H_{i}(- \hookrightarrow -)$ behaves on a morphism $f$ as follows. With $L_{\bullet}=M_{\bullet} / N_{\bullet}$ and $L_{\bullet}'=M_{\bullet}' / N_{\bullet}'$ as usual, there is a natural map $f^{L}:L_{\bullet} \to L_{\bullet}'$ (well-)defined so that the following diagram commutes: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRADkB9AHW4CMmDBjBwgAvqXSZc+QigCM5KrUYs2AWR79Bw0RKnY8BImXnL6zVog5aBQkQHJxkkBkOyiis9Qtrrm3jtdJ31XaSM5ZAAmJR9VKxAAGVsdEWcDGWMUGO8VSzZkwNScEOUYKABzeCJQADMAJwgAWyQyEBwIJBi8vxBagD12EGoGOj4YBgAFcI9reqwKgAs9Fwbm1uoOpEV2uiwGNkWICABrdL7GlsRurcQAZk29g+sj0-O1q53bh56EgfVhiBRuMpjMskCYLUVnVLttNp1EAAWaiLGB0KBsHAAdwgqPRCFCHyQP1uAFYUWiMdZsbjKQTVrCkfCkOTfmwBolxBQxEA > \begin{tikzcd} > N_\bullet \arrow[d, "f^N"'] \arrow[r, hook] & M_\bullet \arrow[d, "f^M"] \arrow[r, two heads] & L_\bullet \arrow[d, "f^L"] \\ > N_\bullet' \arrow[r, hook] & M_\bullet' \arrow[r, two heads] & L_\bullet' > \end{tikzcd} > \end{document} > ``` > that is, $f^{L}([x]):=[f^{M}(x)]$. Now, $f^{L}$ is an ordinary [[chain map]], and the ordinary [[homomorphism on homology induced by a chain map|homology functor]] [[(co)homology of a complex|induces]] a map $f_{*}:H_{i}(L_{\bullet}) \to H_{i}(L_{\bullet}')$ on relative homology. Explicitly, $f_{*}$ maps $\big[[x]\big] \mapsto \big[ [f_{i}^{M}(x)] \big]$. ---- #### ---- #### 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 > ```