Snake lemma

The Schlangenlemma, valid in all abelian categories statement from the mathematical subfield of homological algebra is a tool for the construction of the considered there long exact sequences. Important applications are found for example in algebraic topology. The constructed with the Schlangenlemma homomorphisms are commonly referred to as Verbindungshomomorphismen.

Statement

In an abelian category (eg the category of abelian groups or vector spaces over a given body ), is given the following commutative diagram:

Here, the rows are exact and denote the null object. Then there is an exact sequence of cokernels and the nuclei, is in Relationship:

Is also a monomorphism, so this is also the morphism. Is an epimorphism, so that also applies.

In the category of groups Schlangenlemma the other hand, applies only under additional assumptions on the homomorphisms, ( see below).

Origin of the name

If we extend the diagram to cores and cokernels, we see how the claimed exact sequence " snakes" through the diagram:

Evidence

For the proof, one first assumes that the chart relates to the category of modules over a ring. This allows to prove the assertion by diagram chase. The validity for the case of an arbitrary abelian category is then obtained from the embedding theorem of Mitchell.

Construction of the homomorphisms

The homomorphisms between the cores or Kokernen be induced in a natural way from the given horizontal homomorphisms of the universal properties of nuclear and cokernel. The main assertion of the lemma is the existence of Verbindungshomomorphismus that completes the sequence.

In the case of the category of abelian groups or modules over a ring can element by element construct by diagram chase: Be given, ie with a. Because of the surjectivity of there with a. Because there is one (because of the injectivity of clear ) with. Define as the image of in.

The choice of this case was not unique, because of the exactness, however, has any other choice in the form of suitable. As a result, replaced by what then, however, leads to the same value for. Thus, the map is well defined.

If you have elected to each and with and so you can obviously and choose: . It follows from this. Likewise follows when a ring member, and that is. Thus, the mapping is linear, ie a homomorphism.

Complex property

That the snake sequence forms a complex, ie that two "arrows" in succession always yield the zero mapping, rapidly follows:

  • The figure is induced by
  • And is for the picture. Then you can choose in the above construction of this very, resulting in, then and thus results.
  • Is for the picture. With the notation as in the construction above, the picture is from in. Since it is in, it follows 0
  • The figure is induced by

Accuracy

The accuracy of the homomorphisms between the cores, between the Kokernen as well as start and end points of the arrow d has again claimed by diagram chase:

  • Exactness: It has, after all, so for a. Paths and the injectivity of follows, so in fact as required for a.
  • Exactness: Be with. With the notation from above is then a. Then, is therefore a. This is
  • Exactness: An element of always comes from a. That it is mapped to, it means that the image of lies. Be with and sit. Then we have. Thus, and it is, as shown by the structure given.
  • Exactness: If the picture is of and is mapped to zero in, then for a. Because of the surjectivity of there with a. Then, ie one. The transition to the Kokernen falls away, so is the image of.

The last three points exploit the fact that the vertical sequences are exact.

Naturalness

For applications of Schlangenlemmas it is often necessary that the long exact sequences are "natural" (in the sense of a natural transformation). This then results from the naturalness of the sequence supplied by Schlangenlemma.

Is

A commutative diagram with exact rows, so you can Schlangenlemma once the " front " on the part and use the "back " once on the. The resulting two exact sequences with each other through a diagram of the form

In relationship.

It can be seen on the category of morphisms between objects of the original category also do this by applying the Schlangenlemmas.

Category of groups

Since a number of sets of homological algebra are valid not only for abelian categories, but also for the category of groups, it should be noted that this is not the case for the Schlangenlemma. Although it is also running a natural Verbindungshomomorphismus d, however, is the long sequence, only one chain of the complex, and not necessarily exact. Only if the vertical sequences are exact, ie the images under a, b and c are each normal subgroup in A ', B' or C ', does the proof of exactness for groups.

The simple alternating group contains an isomorphic to the symmetric group subgroup, in which again the cyclic group is a normal subgroup. This yields a commutative diagram

With exact rows.

Because simply, the cokernel of the right figure is trivial, while to be isomorphic. Long sequence therefore has the form

And, consequently, is not exact.

Find out more

  • In the movie It's my Turn ( 1980) Jill Clayburgh proves the Schlangenlemma.
  • Charles A. Weibel avoided in his book "An Introduction to Homological Algebra" ( Cambridge U. Press, 1994 ) on a proof by pointing to It's my turn.
  • At the very beginning of the film The Graduate (1967 ) we see the statement of the Schlangenlemmas on a blackboard behind Dustin Hoffman.
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