The homological algebra is a branch of mathematics that has its origins in algebraic topology. The methods used there can be considerably generalized and used in other mathematical areas. The appearance of the now classical work Homological Algebra by Henri Cartan and Samuel Eilenberg in 1956 can be considered as the beginning of homological algebra. The following year, Alexander Grothendieck generalized these ideas for abelian categories.
Origins in algebraic topology
In algebraic topology certain topological spaces called chain complexes or Kokettenkomplexe and then formed therefrom homology and cohomology groups are assigned in functorial way first. Chain complexes are consequences
So that is always true of groups, modules, vector spaces or other structures and morphisms between them, that is, that the image of lies in the core of. Therefore, one can form the factor group, called the -th homology group. A typical example is simplicial, the derived homology groups then called simplicial homology groups. If you turn in all the above considerations arrows around, we obtain an analogous manner, the cohomology groups. The general procedure can therefore be summarized as follows:
In a first step we abstract from the topological spaces and goes directly from chain complexes. So you can build homology theories for other mathematical structures ( Ko). This results, for example, the Hochschild homology of a chain complex which is associated with an algebra over a field. This approach leads to the informal investigation of exact sequences and their behavior under functors. Large parts of the theory can be executed in arbitrary abelian categories. But is sufficient for many applications, the category of modules over a ring, in which the basic ideas can be developed. In this context it also refers to the embedding theorem of Mitchell.
Hom functor and tensor functor
Of special significance is the application of the Hom - functor on sequences. Be
A short exact sequence, for example in the category of modules over a ring. This means precisely that at any point in the core and the image morphisms involved are the same, in particular, the accuracy in the injectivity of the accuracy is in the equivalent of surjectivity. Short represents the length of the sequence 3, the terminal null objects are not counted. Note that even shorter sequences are trivial: An exact sequence of length 2 only states that are isomorphic and an exact sequence of length 1 is only possible. If we apply it now to the Hom - functor on, with a further module is, or another object of the category under consideration, we obtain an exact sequence
Being and analogy given by. In general, this sequence does not extend exactly to the zero object, that is, is not surjective in general. One hand this leads to the notion of projective module, because that for projective modules can be all those sequences continue exactly with the zero object, on the other hand, the concept of the Ext- functor, the in the general case when an exact continuation of the above sequence in place of the zero object on right side of the sequence occurs.
If one replaces the Hom - functor by the tensor product with a module, it is found before similar conditions. Applying the functor to the above short exact sequence, so we obtain the exact sequence
Which is now defined as, and analog. This sequence can not be extended by exactly 0 on the left in general, that is, is not injective in general. One hand this leads to the concept of the flat module, because that for flat modules can be all those sequences continue exactly with the zero object, on the other hand, the concept of the Tor functor, the case of an exact continuation of the above sequence in place of the zero object on the left side sequence occurs.
Considering one obtains leave the term of the derived functor, Ext and Tor to understand the just means of functors and featured designs, as derivatives of these two functors have in common.
Sequences of homology groups
Another important topic of homological algebra are certain exact sequences of ( co ) homology groups to assist their calculation, which should be touched upon briefly here. Under a homomorphism between two chain complexes and is defined as a sequence of homomorphisms such that
A commutative diagram. Kernels and images of such homomorphisms are the chain complexes from the kernels and images of. Thus one can speak of exact sequences of chain complexes and moving in a category that does not consist of modules over a ring. The homomorphism between the chain complexes induces homomorphisms by
Sets and is convinced of the well- definedness. A typical and fundamental result of homological algebra states:
Is a short exact sequence of chain complexes, the Schlangenlemma provides homomorphisms such that
An exact sequence.
Are some of the homology groups occurring 0, then one can construct isomorphisms between other and so arrive at statements about homology groups. Above theorem is sometimes called the main theorem on chain complexes and speaks of so-called long exact sequences. Similar sequences can be constructed for discharges of additive functors. Further generalizations lead to the so-called spectral sequences.