Homological algebra MOC

Category of chain complexes

The category of chain complexes consists of chain complexes in as objects and chain maps as morphisms, with composition given by . homology For notational convenience, we will often use to refer to , and to refer to . Furthermore, for a ring we write .

Limits and colimits

• Both Initial and terminal objects are the trivial complexes.
• The coproduct is given by the sum of constituent modules

Proof of (co)limits

Consider the trivial chain complex of trivial modules with trivial homomorphisms between them. Clearly for any chain complex the following diagram commutes

Moreover no other vertical morphisms are definable, let alone commuting. Therefore is the initial and terminal object of .

Let and be chain complexes. The sequence is a well defined chain complex, since

The following diagram commutes with unique vertical morphisms due to the coproduct in .

Hence is the coproduct of and .

Homology functor

becomes a functor for each via induced homomorphisms (see chain map), where for we have

This functor preserves initial and terminal objects in a trivial fashion, as well as coproducts. prove

Proof of functor

Consider the identity chain map . Then already has the property that that , hence . Now consider chain complex , , and with chain maps and . Then

as required.

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