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Let $$A=\begin{pmatrix} 3&2&0 \\0&1&-1\\1&1&1\end{pmatrix}. $$The $\mathbb C$-vector space $\mathbb C^3$ becomes a $\mathbb C[T]$-module via

$$\left(\sum_{j=0}^{m}a_jT^j\right)v:=\sum_{j=0}^{m}a_j\left(A^jv\right).$$What are the associated prime ideals of this $\mathbb C[T]$-module?

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Compute the minimal polynomial $f$ of $A$. Then you have the module $\mathbb{C}[T]/(f)$. – Martin Brandenburg Feb 13 '13 at 3:16
@MartinBrandenburg two questions, why to compute the minimal polynomial, and what to do with $\mathbb C[T]/(f)$ – i.a.m Feb 13 '13 at 3:18
Repeat the definitions, and think a while about it, then you can answer this for yourself. And since this is homework, you should learn by doing. – Martin Brandenburg Feb 13 '13 at 3:32
@MartinBrandenburg, the minimal polynomial is $(x-2)^2(x-1)$ and this is not a homework – i.a.m Feb 13 '13 at 3:33
@i.a.m If that is the minimal polynomial polynomial, and I agree it is, what is the annihilator of $V_{{\mathbb C}[T]}$ in ${{\mathbb C}[T]}$, what is the Noether Lasker decomposition of $V_{{\mathbb C}[T]}$, and finally what are the associated primes. Look up any terms there you do not know. – Barbara Osofsky Feb 13 '13 at 4:05
up vote 0 down vote accepted

The associated primes in this case are the ideals of ${\mathbb C}[x]$ generated by the primes which divide the characteristic polynomial, namely $<x-2>$ and $<x-1>$ where $<\cdot>$ indicates the ideal of ${\mathbb C}[x]$ generated by $\cdot$. This set of primes are the same as the set of primes which divide the minimal polynomial. This is based on a standard definition of associated primes used in the tagged area of commutative algebra.

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Let $V$ be a finite dimensional vectorspace over a field $K$ and $A$ a $K$-linear endomorphism of $V$. Determine the associated prime ideals of $V$ as a $K[T]$-module (via $A$), that is, $\operatorname{Ass}_{K[T]}(V)$.

Note that $V$ is a torsion finitely generated $K[T]$-module. From the structure theorem there is a sequence of monic polynomials $d_1,\dots,d_r\in K[T]$ with $d_1\mid \cdots\mid d_r$ and such that, as $K[T]$-modules, $$V\simeq K[T]/(d_1)\oplus\cdots\oplus K[T]/(d_r).$$ Furthermore, one can write $d_i=\prod_{j=1}^sf_j^{a_{ij}}$, where $f_j\in K[T]$ are monic irreducible polynomials, and $a_{ij}$ are nonnegative integers. Obviously, $V\simeq\bigoplus_{i,j} K[T]/(f_j^{a_{ij}})$. Now we get $$\operatorname{Ass}_{K[T]}(V)=\bigcup_{i,j}\operatorname{Ass}_{K[T]}(K[T]/(f_j^{a_{ij}}))=\{(f_1),\dots,(f_s)\}.$$ Since the characteristic polynomial of $A$ is $d_1\cdots d_r=\prod_{i,j}f_j^{a_{ij}}$, we found that $\operatorname{Ass}_{K[T]}(V)$ is the set of principal ideals generated by the irreducible polynomials that appear in the decomposition of the characteristic polynomial of $A$.

In this concrete example the characteristic polynomial of $A$ is $(T-1)(T-2)^2$ and therefore $\operatorname{Ass}_{\mathbb C[T]}(\mathbb C^3)=\{(T-1),(T-2)\}$.

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