Extensions and homological algebra
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
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 .