Homological algebra MOC

Chain complex

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

⋯𝜕𝑘−1←←←←←←←𝐴𝑘−1𝜕𝑘⟵𝐴𝑘𝜕𝑘+1←←←←←←←𝐴𝑘+1𝜕𝑘+2←←←←←←←⋯

such that 𝜕𝑘𝜕𝑘+1 =0 is the trivial homomorphism for all 𝑘 ∈ℤ.³ homology Each 𝐴𝑘 has two important subobjects, the object of 𝑘-cycles 𝑍𝑘(𝐴,𝜕) =ker⁡𝜕𝑘 and the object of 𝑘-boundaries 𝐵𝑘(𝐴,𝜕) =𝜕𝑘+1(𝐴𝑘+1). Hence, 𝐵𝑘(𝐴,𝜕) ≤⊆𝑍𝑘(𝐴,𝜕) for all 𝑘 ∈ℤ, i.e. all 𝑘-boundaries are 𝑘-cycles. The 𝑘-chain homology is defined as

𝐻𝑘(𝐴,𝜕)=𝑍𝑘(𝐴,𝜕)/𝐵𝑘(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 𝐴𝑘 =0 trivial for all 𝑘 <0.
• A structure-preserving map of chain complexes is a Chain map, which form the morphisms in Category of chain complexes.
• 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 𝑑𝑘 :𝐴𝑘 →𝐴𝑘+1 and 𝑑𝑘+1𝑑𝑘 =0. homology All other constructions above follow directly, yielding cochains, cocycles, coboundaries, and cohomologies.

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Footnotes

1. German Kettenkomplex, Randoperator. ↩

2. In this abstract setting chains, cycles, and boundaries refer simply to the elements of each of these groups/modules as they are defined. ↩

3. 2010, Algebraische Topologie, §3.1, p. 127 ↩