| Homological Algebra |
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Central to homological algebra is the notion of Exact Sequence ; these can be used to perform actual calculations. A classical tool of homological algebra is that of Derived Functor ; the most basic examples are Ext and Tor . FOUNDATIONAL ASPECTS 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:
These move from computability to generality. The computational sledgehammer ''par excellence'' is the Spectral Sequence ; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary. There have been attempts at 'non-commutative' theories which extend first cohomology as '' Torsor s'' (important in Galois Cohomology ). |
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