Tag Info

A sequence of morphisms where the image of one is the kernel of the next. It is a useful thing to examine in the study of abstract algebra and homological algebra. Use this tag if your question is about the general theory of exact sequences, not just because an exact sequence appears in it. For sequences of numbers use the tag (sequences-and-series) instead.

An exact sequence in a category a sequence of of morphisms in that category

$$\dotsb \xrightarrow{\;\;\varphi_{i-1}\;\;} X_i\xrightarrow{\;\;\varphi_{i}\;\;} X_{i+1}\xrightarrow{\;\;\varphi_{i+1}\;\;} \dotsb$$

such that the image of $$\varphi_j$$ is equal to the kernel of $$\varphi_{j+1}$$ for any $$j$$. Any (long) exact sequence can be decomposed in a reasonable way into short exact sequences, so these are more often the objects that we examine. A short exact sequence is a sequence

$$0 \to B \xrightarrow{\;\;\varphi\;\;} C \xrightarrow{\;\;\psi\;\;} A \to 0$$

Such that $$\mathrm{Im}(\varphi) = \mathrm{Ker}(\psi)$$, $$\varphi$$ is an monomorphism, and $$\psi$$ is an epimorphism. The object $$C$$ is referred to as an extension of $$A$$ by $$B$$. Exact sequences are major objects of study in the broader areas of abstract algebra and homological algebra.