Short exact sequence

Short exact sequence

Short exact sequences are exact sequences of the form $${\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0.}$$ As established above, for any such short exact sequence, ${\displaystyle f}$ is a monomorphism and ${\displaystyle g}$ is an epimorphism. Furthermore, the image of ${\displaystyle f}$ is equal to the kernel of ${\displaystyle g}$. It is helpful to think of ${\displaystyle A}$ as a subobject of ${\displaystyle B}$ with ${\displaystyle f}$ embedding ${\displaystyle A}$ into ${\displaystyle B}$, and of ${\displaystyle C}$ as the corresponding factor (or quotient) object, ${\displaystyle B/A}$, with ${\displaystyle g}$ inducing an isomorphism $${\displaystyle C\cong B/\operatorname {im} (f)=B/\operatorname {ker} (g)}$$