Lagmental Vicfred

A Short Exact Sequence of Complexes Produces a Long Exact Sequence by Vicfred

tags: mathematics, homology, chain complexes, exact sequences

The connecting homomorphism measures the obstruction to lifting a homology class as a cycle. I will separate the object being defined from the consequence being claimed.

Definitions first

A chain complex \((C_\bullet,\partial)\) satisfies \(\partial_{n-1}\partial_n=0\). Homology \(H_n=\ker\partial_n/\operatorname{im}\partial_{n+1}\) measures cycles not explained as boundaries.

$$ 0\to A_\bullet\to B_\bullet\to C_\bullet\to0 $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \cdots\to H_n(A)\to H_n(B)\to H_n(C)\xrightarrow{\delta}H_{n-1}(A)\to\cdots $$

A small case in full

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

$$ \begin{array}{ccccccccc}0&\to&A_n&\to&B_n&\to&C_n&\to&0\\&&\downarrow&&\downarrow&&\downarrow\\0&\to&A_{n-1}&\to&B_{n-1}&\to&C_{n-1}&\to&0\end{array} $$

The reusable statement

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad 0\to A_\bullet\to B_\bullet\to C_\bullet\to0,\\[5pt] \mathsf{C}\;&:\quad \cdots\to H_n(A)\to H_n(B)\to H_n(C)\xrightarrow{\delta}H_{n-1}(A)\to\cdots. \end{aligned} $$

A nearby false statement

Homology depends on the chosen coefficient ring. Torsion may disappear over a field or change under reduction modulo a prime.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \cdots\to H_n(A)\to H_n(B)\to H_n(C)\xrightarrow{\delta}H_{n-1}(A)\to\cdots \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Wed 19 February 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.