Free, open-source online mathematics for students, teachers and workers

A complex $C_\ast$ is said to be an exact sequence if

$$\text{im }\partial_{k+1}= \text{ker }\partial_k$$

for all $k$

An exact sequence of the type

$$0\xrightarrow{} A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{} 0$$

is called short exact sequence


[Cohomological version]

A complex $C^\ast$ is said to be an exact sequence if

$$\text{im }\mathrm{d}^{k-1}= \text{ker }\mathrm{d}^k$$

for all $k$

An exact sequence of the type

$$0\xrightarrow{} A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{} 0$$

is called short exact sequence