Extensions and homological algebra

10 January 2009 at 12:02 pm Leave a comment

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

0 \to B \to E \to A \to 0

where B is the kernel, and A is the quotient (cokernel). The group of extensions \mathrm{Ext}^1(A,B) 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 :

Baer sum

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

0 \to A \to B \to C \to 0

determines an exact sequence

0 \to Hom(M,A) \to Hom(M,B) \to Hom(M,C)
\to Ext(M,A) \to Ext(M,B) \to Ext(M,C)

and

0 \to Hom(C,M) \to Hom(B,M) \to Hom(A,M)
\to Ext(C,M) \to Ext(B,M) \to Ext(A,M)

Thus if A and B are projective (a thing A is projective if Hom(A,M) \to Hom(A,N) is surjective for any surjective morphism M \to N), Ext(A,M) = Ext(B,M) = 0, and it is not hard to compute Ext(C,M). More generally, it is possible to define groups \mathrm{Ext}^n(A,B) (see MacLane, Homology for an elementary definition), so that an exact sequence as above determines a long exact sequence

0 \to Hom(M,A) \to Hom(M,B) \to Hom(M,C)
\to Ext^1(M,A) \to Ext^1(M,B) \to Ext^1(M,C)
\to Ext^2(M,A) \to Ext^2(M,B) \cdots

We say that the Ext^n are the derived functors of Hom. A fundamental result is that if (P_\bullet) is a projective resolution of A i.e. a sequence of projective things with an exact sequence

\cdots \to P_n \to P_{n-1} \to \cdots \to P_1 \to P_0 \to A \to 0,

the cohomology groups of the complex \mathrm{Hom}(P_\bullet, M) are exactly the \mathrm{Ext}^n(A,M).

Entry filed under: commutative algebra, english, homological algebra. Tags: , , , .

Intégrales elliptiques et moyenne arithmético-géométrique Extensions of sheaves and the local-global spectral sequence

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