Chain complex

A chain complex¹ $(𝐴 ∙, 𝜕 ∙)$ in an Abelian category $𝖠$ is a sequence of objects $𝐴 𝑘$ of $𝑘$ -chains² with homomorphisms $𝜕 𝑘 : 𝐴 𝑘 \to 𝐴 𝑘 - 1$ called boundary operators between them

\dots 𝜕 𝑘 - 1 \leftarrow \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow 𝐴 𝑘 - 1 𝜕 𝑘 ⟵ 𝐴 𝑘 𝜕 𝑘 + 1 \leftarrow \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow 𝐴 𝑘 + 1 𝜕 𝑘 + 2 \leftarrow \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \dots

such that $𝜕 𝑘 𝜕 𝑘 + 1 = 0$ is the trivial homomorphism for all $𝑘 \in ℤ$ .³ #m/def/homology Each $𝐴 𝑘$ has two important subobjects, the object of $𝑘$ -cycles $𝑍 𝑘 (𝐴, 𝜕) = k e r 𝜕 𝑘$ and the object of $𝑘$ -boundaries $𝐵 𝑘 (𝐴, 𝜕) = 𝜕 𝑘 + 1 (𝐴 𝑘 + 1)$ . Hence, $𝐵 𝑘 (𝐴, 𝜕) \leq \subseteq 𝑍 𝑘 (𝐴, 𝜕)$ for all $𝑘 \in ℤ$ , i.e. all $𝑘$ -boundaries are $𝑘$ -cycles. The $𝑘$ -chain homology is defined as

𝐻 𝑘 (𝐴, 𝜕) = 𝑍 𝑘 (𝐴, 𝜕) / 𝐵 𝑘 (𝐴, 𝜕)

with $𝑘$ -homology classes of chains as its elements, and two cycles in the same homology class are called homologous.

Additional terminology

A non-negative chain complex has $𝐴 𝑘 = 0$ trivial for all $𝑘 < 0$ .
A structure-preserving map of chain complexes is a Chain map, which form the morphisms in $𝖢 𝗁 𝖠$ .
One can form the Direct sum of chain complexes.

Properties

A chain complex with only trivial homologies is an Exact sequence.

Dual

A cochain complex is the exact same construction but with $𝑑 𝑘 : 𝐴 𝑘 \to 𝐴 𝑘 + 1$ and $𝑑 𝑘 + 1 𝑑 𝑘 = 0$ . #m/def/homology All other constructions above follow directly, yielding cochains, cocycles, coboundaries, and cohomologies.

#state/develop | #lang/en | #SemBr

German Kettenkomplex, Randoperator. ↩
In this abstract setting chains, cycles, and boundaries refer simply to the elements of each of these groups/modules as they are defined. ↩
2010, Algebraische Topologie, §3.1, p. 127 ↩

Chain complex

Additional terminology

Properties

Dual

Footnotes