Commutative Algebra/Normal and composition series

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Normal series[edit]

Definition 12.1:

Let be an -module. A finite sequence of submodules

is called normal series of .

Note that a normal series of a module is a normal series of the underlying group ; indeed, each subgroup of an abelian group is normal, hence each normal series in modules gives rise to a normal series of groups. The other direction is not true, since additive subgroups need not be closed under multiplication by elements of .

Definition 12.2:

Let be an -module, and let a normal series

be given. This normal series is said to be without repetitions if each inclusion of modules is, in fact, strict.

Refinements and composition series[edit]

Definition 12.3:

Let be an -module, and let a normal series

be given. A refinement of this normal series is another normal series

such that .

Note that this implies . Refinements arise from a normal series

by inserting submodules between two modules of the composition series and ; that is, we start with two modules of a composition series and , find a submodule of such that , and then just insert this into the normal series.

Definition 12.4:

Let be an -module. We say that is simple if and only if it has no proper submodules (i.e. no submodules of which are neither equal to nor equal to ).

Definition 12.5:

Let be an -module. A composition series of is a normal series of , say

,

such that

  1. there are no repetitions, and
  2. each so-called composition factor of that series, namely for , is simple (where we set and ).

Equivalently, a composition series is a normal series without repetitions, such that any proper refinement of it has repetitions.

To any module, we may associate a so-called length. This concept is justified by the following theorem:

Theorem 12.6 (Jordan):

Let be an -module which has a composition series

.

We say that this composition series has length , and then it follows that

  1. each normal series in without repetitions has length ,
  2. every other composition series of also has length , and
  3. every normal series in can be refined to a composition series of .

Proof:

First, we note that 1. implies 3., since whenever a normal series has a refinement that has no repetitions, we may apply that refinement, and due to 1., we must eventually reach a composition series.

Then we prove 1. and 2. by induction on . Indeed, for , this theorem follows since then is simple, and therefore any normal series of length must have repetitions, which is why the trivial normal series is the only one without repetitions, and there is only one composition series.

Assume now the case to be valid. Let there be a composition series

of length , and assume that there is any other normal series

without repetition of length . Now hence has a composition series of length . By induction, we have:

  1. If , then is a normal series in and hence has length at most , whence the complete normal series has length at most .
  2. If , then has a normal series without repetitions of length , which is a contradiction.
  3. If not , we have , for otherwise the composition series would have a proper refinement. Then we have two normal series
and
.
Now has a composition series of length , whence has a composition series of length . Furthermore, , which is why any such composition series then extends to a composition series of of length . Therefore, the partial series
has length at most .

This proves 1. by induction. Furthermore, by induction, can not have a composition series of length , since then also the composition series above would have length , whence 2. is proven by 1. and induction.

Definition 12.7:

Assume has a composition series. Then the length of the module is defined to be the length of such a composition series.

If doesn't have a composition series, we set the length of to be .

Furthermore, composition series are essentially unique, as given by the following theorem:

Theorem 12.8 (Hölder):

If

and

are two composition series, then there exists a permutation such that for all

(again and , and analogous for ).

We say that the two series are equivalent.

Proof:

We proceed by induction on . For , we have only the trivial composition series as composition series. Now assume the theorem for . Let two composition series

and

be given. If , we have equivalence by induction. If not, we have once again (since neither can be properly contained in the other, for else we would obtain a contradiction to the previous theorem of Jordan). Now must have a composition series, since by the previous theorem we may refine the series

to a composition series of . Further, we again have

and ;

both modules on the right of the isomorphisms are simple, whence we get two composition series of given by

and

.

Now the two above isomorphisms also imply that these two are equivalent, and by induction, the first one is equivalent to the first composition series, and the second one equivalent to the second composition series.

Proposition 12.9:

Let be an -module, let be a submodule.

has a composition series if and only if both and have composition series.

Proof:

If has a composition series, then intersecting this series or projecting this series gives normal series of or respectively. When the repetitions are crossed out, no refinements are possible (else they induce a refinement of the original composition series, in the latter case by the correspondence theorem).

If and both have composition series, we take a composition series

of and another one of given by

.

By the correspondence theorem, we write for suitable . Then

is a composition series of .

Normal series between modules[edit]

Definition 12.10:

Let be a module and a submodule. A series

is called a normal series between and , and if each inclusion is strict and there does not exist a refinement which leaves each inclusion strict, it is called a composition series between and .

By the correspondence theorem, we get a bijection between normal (or composition) series

between and on the one hand, and of normal (or composition) series

of . Then by the above and the third isomorphism theorem, composition series between and are essentially unique. Further, if there is a composition series, normal series can be refined to composition series of same length.