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Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ) and abstract algebra (theory of modules and syzygies ) at the end of the 19th century, chiefly by ...
In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X . The chain complex for X is defined as follows: first find a free module F 1 {\displaystyle F_{1}} and a surjective homomorphism p 1 : F 1 → X . {\displaystyle p_{1}:F_{1 ...
Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X ...
This is a list of homological algebra topics, by Wikipedia page. Basic techniques. Cokernel; Exact sequence; Chain complex; Differential module; Five lemma; Short ...
In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic.
The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides.
From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra .
The method of proof we shall use is commonly referred to as diagram chasing. [1] We shall prove the five lemma by individually proving each of the two four lemmas. To perform diagram chasing, we assume that we are in a category of modules over some ring , so that we may speak of elements of the objects in the diagram and think of the morphisms ...
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