Lagmental Vicfred

A Short Exact Sequence Is an Extension by Vicfred

tags: mathematics, modules, exact sequences, homological algebra

A short exact sequence identifies one module as a submodule and the other as the corresponding quotient. I will separate the object being defined from the consequence being claimed.

Objects and notation

An \(R\)-module generalises both vector spaces and abelian groups. A map \(f:M\to N\) is understood through its kernel, image, and cokernel, which exact sequences place on one line.

$$ 0\longrightarrow A\xrightarrow{\ i\ }B\xrightarrow{\ p\ }C\longrightarrow0 $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \ker p=\operatorname{im}i,\qquad C\cong B/i(A) $$

Push the symbols

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{array}{c@{\;}c@{\;}c@{\;}c@{\;}c}0&\to&2\mathbf Z&\hookrightarrow&\mathbf Z\\&&&&\downarrow\bmod2\\&&&&\mathbf Z/2\mathbf Z\to0\end{array} $$

Structural reading

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad 0\longrightarrow A\xrightarrow{\ i\ }B\xrightarrow{\ p\ }C\longrightarrow0,\\[5pt] \mathsf{C}\;&:\quad \ker p=\operatorname{im}i,\qquad C\cong B/i(A). \end{aligned} $$

A hypothesis worth keeping

Tensor product is right exact but not generally left exact, while \(\operatorname{Hom}_R(P,-)\) is exact precisely when \(P\) is projective.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \ker p=\operatorname{im}i,\qquad C\cong B/i(A) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sat 07 February 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.