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I'm reading my class notes of Commutative Algebra. I'm stuck on an exercise given in my notes:

Definition: Let $M$ be a $A$-module. If $M$ has a composition series then the length of the composition series is called the length of the module otherwise we say $M$ has infinite Length.

Exercise: Let $M=\frac {m^n}{ m^{n+1}}$ where $m=(x_1,x_2,...,x_n) \subset k[[x_1,x_2,...,x_n]]=A$. Compute the length of $M$.

I'm unable in finding a composition series for this module $M$? How do we "guess" composition series in general? Any hints/ideas?

user26857
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Arpit Kansal
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1 Answers1

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Let $(R,m,k)$ be a local ring, and $M$ a finitely generated $R$-module such that $mM=0$. Then $l(M)=\dim_kM$.

In fact, $M$ is an $R/m=k$-vector space and its submodules (as $R$-module) are the same as its subspaces (as a $k$-vector space). Since it is finitely generated it follows that $\dim_kM<\infty$, so $M$ is a module of finite length, and $l(M)=\dim_kM$ (since the composition factors have all dimension one). Furthermore, $\dim_kM$ equals the minimal number of generators of $M$. Can you find this in your case?

user26857
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  • Sorry..I'm confused.Is there any relation between $l(M)$ as an $R$-module and $l(M)$ as an $ \frac{R}{m}$-module? – Arpit Kansal Oct 26 '15 at 09:25
  • They are equal (because the submodules are the same and this follows from $mM=0$). For more details you can also have a look here: http://math.stackexchange.com/questions/325825/if-m-is-an-r-module-and-i-subseteq-mathrmannm-then-m-has-the-struct – user26857 Oct 26 '15 at 10:07
  • +1 I see,Thanks.But how do you count generators in this case? – Arpit Kansal Oct 26 '15 at 14:08
  • @ArpitKansal They should be all the products of $n$ (not necessarily distinct) variables, that is, all the degree $n$ monomials in $x_1,\dots,x_n$. – user26857 Oct 26 '15 at 14:24
  • correct,but how do we count them? – Arpit Kansal Oct 26 '15 at 14:28
  • @ArpitKansal See http://math.stackexchange.com/questions/551214/the-number-of-monomials-of-a-given-degree – user26857 Oct 26 '15 at 14:32