## Extensions and homological algebra

*10 January 2009 at 12:02 pm* *
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If you ever followed a course in algebra, you may have heard about *extensions* of “things”. Most “things” fit in what is called a *category*, that is, an abstract structure remembering how to compose morphisms between these “things”, and sometimes what is the sum of such “things”, the kernel of a morphism: we are interested in the structure of abelian category, which is the framework of vector spaces, modules over a ring, sheaves of modules… An extension of A by B is an exact sequence

where is the kernel, and is the quotient (cokernel). The *group of extensions * is the set of isomorphism classes of such exact sequences. It is a group by means of the *Baer sum*, which takes two extensions *E* and *E’*, and considers the diagram :

Here *S* is the fiber product of *E* and *E’* over B. Then the Baer sum of *E* and *E’* is the quotient of *S* where the two copies of A are identified. But this approach doesn’t really help knowing what the Ext group looks like. The machinery of homological algebra tells us that Ext is a *derived functor* for Hom. This essentially means that an exact sequence

determines an exact sequence

and

Thus if *A* and *B* are projective (a thing *A* is *projective* if is surjective for any surjective morphism ), , and it is not hard to compute . More generally, it is possible to define groups (see MacLane, *Homology* for an elementary definition), so that an exact sequence as above determines a long exact sequence

We say that the are the derived functors of . A fundamental result is that if is a projective resolution of i.e. a sequence of projective things with an exact sequence

the cohomology groups of the complex are exactly the .

Entry filed under: commutative algebra, english, homological algebra. Tags: abelian category, baer sum, cohomology, extension.

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