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pl:Algebra homologicznade:Homologische Algebra
'''Homological algebra''' is that branch of
mathematics which studies the methods of
homology and
cohomology in a general setting. These concepts originated in
algebraic topology.
Cohomology theories have been described for
topological spaces,
sheaves, and
groups; also for
Lie algebras,
C-star algebras. The study of modern
algebraic geometry would be almost unthinkable without sheaf cohomology.
There are also other homological
functors that take their place in the theory, such as
Ext and
Tor. There have been attempts at 'non-commutative' theories, which extend first cohomology as ''
torsors'' (important in
Galois cohomology).
=== Foundational aspects ===
The methods of '''homological algebra''' start with use of the
exact sequence to perform actual calculations. With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts, before the subject settled down. An approximate history can be stated as follows:
* Cartan-Eilenberg: as in their eponymous book, used
projective and
injective module resolutions.
* 'Tohoku': the approach in a celebrated paper by
Alexander Grothendieck using the
abelian category concept (to include
sheaves of abelian groups).
* The
derived category of
Grothendieck and
Verdier, used in a number of modern theories.
These move from computability to generality. The computational sledgehammer ''par excellence'' is the
spectral sequence; in the derived category approach these don't appear at all, in an explicit way.
