Exact sequence

The term of the exact sequence plays a central role in the mathematical subfield of homological algebra. Particularly important are the short exact sequences.

Definition

A sequence

Objects and morphisms, in a suitable category is exactly at the point where

Applies, that is, when the image of an arrow is equal to the core of the next. A longer sequence

Is called exact if it is exact at the points, and (analog for shorter or longer sequences ).

Appropriate in this sense, a category is obviously only if it can be useful spoken of core and image. This is the case for all abelian categories, but also, for example, the category Grp of groups and group homomorphisms. In fact Grp is usually the only non- abelian category will be considered in the exact sequences.

Examples

• A sequence

• A sequence

• For each homomorphism of vector spaces ( abelian groups, modules, each morphism of an abelian category) exists an exact sequence as follows:

• Are a group   the center,
• The group of automorphisms,
• The group of inner automorphisms and
• The group of outer automorphisms

Short exact sequences

Definition

Exact sequence of the form

Is called short exact sequence.

Crumbling short exact sequences

A short exact sequence breaks down when one section. Chance is instead disintegrates, the term splits used, which is due to a not quite correct translation of the English term split.

In an additive category it follows also that a retraction is that the resulting sequence

Is also exact and that these sequences is isomorphic to

Or

Are.

Decomposes a short exact sequence in the category of groups, the result is merely an operation of on, and that semi- direct product of and with respect to this operation. For example, the cyclic group subgroup of the symmetric group, hence the short exact sequence

Results; by the non- neutral element of an element of order 2 in maps, you get a split.

Division of a long exact sequence

Each long exact sequence can be divided into short exact sequences decompose by inserting cores and cokernels: Is

An exact sequence, it should be

Then there are short exact sequences

Is a chain complex, so the accuracy is all this short sequence is equivalent to the exactness of the long sequence.

Extensions

In the context of a short exact sequence

We also say that is an extension of by.

For example, if a normal subgroup in the group and the factor group, we obtain a short exact sequence

Where the second arrow is the embedding in the third and the quotient map. This is an extension of and and you can ask for a classification of all possible extensions of and. Relevant questions to get about in the category of rings or modules over a fixed ring. This leads to mathematical terms such as Ext ( mathematics) or Gruppenkohomologie.