##
Atiyah-Macdonald: Modules and Exact Sequences *June 23, 2010*

*Posted by Sarah in Uncategorized.*

Tags: algebra

trackback

Tags: algebra

trackback

A module is a generalization of the notion of ideals. Rings, ideals, quotient rings, and vector spaces are all modules. If A is a ring, then an A-module is an abelian group M on which A acts linearly:

$1x = x$

Submodules are subgroups of M closed under multiplication by elements of A; the sum and intersection of submodules is again a submodule.

The annihilator of M is the set of all a in A such that aM = 0. A module is called faithful if its annihilator is zero.

Direct sum of two modules is just defined as an ordered pair (x, y), with componentwise addition and scalar multiplication. An A-module is called free if it’s isomorphic to the direct sum of modules isomorphic to A. (A finitely generated A-module is isomorphic to a quotient of for some n.)

Proposition.

Let M be a finitely generated A-module. Let a be an ideal of A and let be an A-module automorphism of M st . Then satisfies an equation of the form

.

Proof.

Let be a set of generators of M. Then for some . That is,

Left-multiplying by the adjoint of the matrix , we see that the determinant annihilates each , hence is the zero endomorphism of M. Expanding out the determinant gives us an equation of the desired form.

Exact sequences. Suppose we have a sequence of homomorphisms between modules,

This is exact at if $Im(f_i) = Ker(f_{i+1})$. The sequence is exact if it is exact at every module.

is exact iff f is injective;

is exact iff g is surjective.

A short exact sequence:

.

Every long exact sequence can be split into these.

If the above is an exact sequence, then

is also exact; note that the order reverses when we go to the dual space.

Next time I’ll introduce the boundary map and the Snake Lemma.

## Comments»

No comments yet — be the first.