----- > [!proposition] Proposition. ([[exact sequence wrangling]]) > If we have an [[exact sequence]] where we know $A,B$, $f$, $D,E$, $g$: $A \xrightarrow{f} B \to C \to D \xrightarrow{g} E$ > then the sequence $0 \to \text{coker }f \to C \to \text{ker }g \to 0$is [[short exact sequence|short exact]]. This can help us determine $C$. Note that (at least in the [[category|categories]] we care about here) $\text{coker }f \cong B / \text{im }f$ (see [[cokernel of a module homomorphism]], [[cokernel of an abelian group homomorphism]], etc.) > [!proof]+ Proof. ([[exact sequence wrangling]]) > The map $B / \text{im }f \to C$ is [[injection|injective]] by the [[first isomorphism theorem]], which is applicable by the assumption that $\text{im }f \subset \text{ker }(B \to C)$. The map $C \to \text{ker }g$ is a [[surjection]] because exactness means $\text{im } (C \to D)=\text{ker }g$. Since the induced map from FIT $\text{im }(B / \text{im }f \to C)$ has the same image as the map $B \to C$, the middle is exact and we are done. ----- #### ---- #### 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 > ```