The concept of the exact sequence or exact sequence plays a central role in the mathematical subfield of homological algebra . The short, exact sequences are particularly important .
of objects and morphisms in a suitable category
means exactly at the point if
holds, d. H. when the image of one arrow is equal to the core of the next. A longer sequence
means exact , if it is exactly at the places , and (analogous for shorter or longer sequences).
In this sense, a category is apparently only suitable if one can meaningfully speak of the core and the image. This is the case for all Abelian categories , but also, for example, for the category Grp of groups and group homomorphisms .
Examples
If there is a homomorphism between Abelian groups , then is and . The sequence is therefore exactly at the point if is.
A sequence is exact if and only if a monomorphism , i. H. is injective . Using a hook arrow, this can also be written with 2 terms:
One sequence
is exact if and only if an epimorphism , i. H. is surjective . Using a double-headed arrow, this can also be written with 2 terms:
For every homomorphism of vector spaces ( Abelian groups , modules , every morphism of an Abelian category) there is an exact sequence as follows:
In Grp , however, the sequence is only exact if the image is a normal divider in . Even in additive but not Abelian categories, the exactness is not necessarily given. This denotes the coke core of .
A short exact sequence disintegrates when has
a cut . Occasionally, instead of disintegrating , the term splits is also used, which is due to an incorrect translation of the English term split .
In an additive category it also follows from this that a retraction has the resulting sequence
is also exact and that these sequences are isomorphic to
or.
are.
If a short exact sequence falls apart in the category of groups, this only results in an operation of on , and that is a semi-direct product of and
with regard to this operation . For example, the cyclic group is a subgroup of the symmetric group , which is the short exact sequence
results; by mapping the non-neutral element of to an element of order 2 in , a split is obtained.
Splitting a long exact sequence
Every long exact sequence can be broken down into short exact sequences by inserting cores and coke cores : is
an exact sequence, so be
Then there are short exact sequences
If there is a chain complex , the exactness of all this short sequence is equivalent to the exactness of the long sequence.
where the second arrow is the embedding of in and the third is the quotient map . This is an extension of and and one can ask the question of a classification of all possible extensions of and . Corresponding questions can be found in the category of rings or modules over a solid ring. This leads to mathematical terms like Ext or group cohomology .