Homological algebra MOC

Chain complex

A chain complex¹ in an Abelian category is a sequence of objects of -chains² with homomorphisms called boundary operators between them

such that is the trivial homomorphism for all .³ homology Each has two important subobjects, the object of -cycles and the object of -boundaries . Hence, for all , i.e. all -boundaries are -cycles. The -chain homology is defined as

with -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 trivial for all .
• 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 and . 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 ↩