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Here is an exercise from Artin:

Let $V$ be an $F[t]$-module, and let $\mathbf B=(v_1,\dots,v_n)$ be a basis for $V$ as $F$-vector space. Let $B$ be the matrix of $T$ with respect to this basis. Prove that $A=tI-B$ is a presentation matrix for the module.

(Note that $T$denotes the linear operator $V\to V$ defined by $T(v)=tv$.)

Artin defines a presentation matrix as follows:

Left multiplication by an $m \times n$ matrix defines a homomorphism of $R$-modules $A: R^n \rightarrow R^m$. Its image consists of all linear combinations of the columns of $A$ with coefficients in the ring, and we may denote the image by $AR^n$. We say that the quotient module $V=R^m/AR^n$ is presented by the matrix $A$. More generally, we call any isomorphism $\sigma: R^m/AR^n \rightarrow V$ a presentation of a module $V$, and we say that the matrix $A$ is a presentation matrix for $V$ if there is such an isomorphism.

Back to the problem, I'm not even sure what I'm supposed to show. Presentation matrices are defined for quotient modules. Why is an arbitrary $F[t]$-module $V$ a quotient module? If $V$ were finitely generated, this would certainly be true since there would be a surjective $F[t]$-module homomorphism $(F[t])^n\to V$, and we could apply the First Isomorphism Theorem. So, why is a presentation matrix defined for $V$ in question, what exactly do I need to show, and at least how to get started?

Edit: alright, indeed, as @Mohan pointed out in the comments, since $V$ is generated by $\mathbf B$ as an $F$-vector space, then it is also generated by $\mathbf B$ as an $F[t]$-module. So there is a surjective module homomorphism $\phi: F[t]^n\to V$ given by $X\mapsto \mathbf B X$, where the RHS is the product of the row $\mathbf B$ by the column $X$. So $V$ is isomorphic to $F[t]^n/\ker \phi$. If we show that $\ker \phi= AF[t]^n$ where $AF[t]^n$ stands for the image of the map $A: F[t]^n\to F[t]^n$ given by $X\mapsto AX$ (here I denote the map and the matrix by the same letter $A$), then we will be done.

user557
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  • If $V$ is generated by the $v_i$s as an $F$-vector space, clearly, the same generators generate it as an $F[t]$-module. Thus, as you said, you have a surjection $F[t]^n\to V$. You should show that the kernel is also $F[t]^n$ and thus the map $F[t]^n\to F[t]^n$ is given by an $n\times n$ matrix over $F[t]$. You should show (for suitable basis for the $F[t]^n$), that this matrix is $tI-B$. – Mohan Apr 06 '18 at 22:49
  • @Mohan Thanks, I understood your first two sentences (but unfortunately not the succeeding ones). However, I think I got what I'm asked to do (I stated it in the edited version of my question). What I'm confused by now is perhaps some trivial fact: by definition, $T(v_i)=tv_i$, so it seems like the matrix $B$ is the scalar matrix $tI$. But a matrix of a linear operator of $F$-vector spaces is supposed to have entries in $F$... Also if $B=tI$, then $A=0$. – user557 Apr 07 '18 at 01:43
  • @Cary - Are you sure the exercise is stated as in your first block quote, and that $T$ is defined as you wrote? – Pierre-Yves Gaillard Apr 07 '18 at 15:32
  • @Pierre-YvesGaillard I checked the statement of the exercise, and I believe it is correct. However the original exercise doesn't talk about the definition of $T$; since throughout the corresponding chapter (14.8) $T$ stood for the operator I defined, I assumed the definition is the same in the exercise too. – user557 Apr 07 '18 at 16:06
  • There is a solution here (p. 68, exercise 14.8.4 https://www.csie.ntu.edu.tw/~b01902113/artin-sols.pdf), but I don't understand its solution mainly because I don't understand what exactly $B$ is and what the meaning of things like $Bv_i$ is ($B$ is a matrix, and it can only be multiplied by the coordinate vector of $v_i$, but it cannot be applied to $v_i$). – user557 Apr 07 '18 at 16:10
  • @TakumiMurayama Maybe you would be interested in and/or capable of answering this question (see my comments above). – user557 Apr 07 '18 at 16:43
  • @Cary - I think the intended exercise is: Let $V$ be an $F$-vector space with the indicated basis, and let $T\in\text{End}_F(V)$. Then $V$ admits a unique structure of $F[t]$-module which extends its $F$-vector space structure and satisfies $tv=Tv$ for all $v$. Show that $tI-B$ is a presentation matrix for this $F[t]$-module ($B$ being defined as in your question). (I think the statement of the exercise in the book doesn't make sense.) – Pierre-Yves Gaillard Apr 07 '18 at 16:49
  • @Pierre-YvesGaillard Why do you think the statement in the book doesn't make sense? – user557 Apr 07 '18 at 17:42
  • @Cary - Ok, I'm taking this claim back. I hadn't read the statement carefully enough. Now I think the statement in the book and the one I wrote are equivalent. I'll try to find a solution. – Pierre-Yves Gaillard Apr 07 '18 at 17:57

1 Answers1

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Set $R:=F[t]$ and identify $V$ to $F^n$ thanks to its $F$-basis. Consider the $R$-linear maps $$ R^n\xrightarrow{tI-B}R^n\xrightarrow\phi F^n\to0,\quad(\ast) $$ where $\phi$ is defined by $$ \phi\left(\sum t^ix_i\right)=\sum B^ix_i $$ for $x_i\in F^n$. It suffices to prove that $(\ast)$ is an exact sequence. The surjectivity of $\phi$ and the equality $\phi\circ(tI-B)=0$ are clear.

Let $$ \sum_{i=0}^dt^ix_i $$ be in $\ker\phi$. Set $$ y_{d-1}:=x_d, $$ and
$$ y_i:=By_{i+1}+x_{i+1}\quad\text{for}\quad i=d-2,d-3,\dots,0. $$ Then it is straightforward to check, using our assumption $$ \sum_{i=0}^dB^ix_i=0, $$ that we have $$ (tI-B)\ \sum_{i=0}^{d-1}t^iy_i=\sum_{i=0}^dt^ix_i, $$ as desired.

EDIT. Here is a variant: The equality $$ \sum_{i=0}^dB^ix_i=0, $$ implies $$ \sum_{i=0}^dt^ix_i=\sum_{i=0}^d\ (t^iI-B^i)\ x_i=(tI-B)\ \sum_{i=0}^d\ \sum_{j=0}^{i-1}\ t^jB^{i-1-j}x_i. $$

Pierre-Yves Gaillard
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  • Thanks! I'm not sure I understand what $\phi$ is (the formula is unclear since elements of the domain $R^n$ are column vectors with entries in $F[t]$, but I don't know what $x_i$s are). According to your verbal description, there is a map $F^n\to V$ given by $X\mapsto \mathbf BX$ with inverse $V\to F^n$ given by $v_1x_1+\dots v_nx_n\mapsto (x_1,\dots,x_n)^t$. There is also a map $R^n\to V$ defined in my question. So I guess your $\phi$ should be the composition of the latter map $R^n\to V$ with the former map $V\to F^n$. But I don't see why your formula defines this map. – user557 Apr 07 '18 at 19:40
  • And also I have another concern: if you show that $(*)$ is exact, then you will show that $A$ is a presentation matrix for $F^n$, not for $V$. Of course they are isomorphic as $F$-vector spaces, but why do the corresponding $F[t]$-modules have the same presentation matrix? – user557 Apr 07 '18 at 19:46
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    @Cary - First comment: $$R^n\ni u=(u_1,...,u_n),\ u_j\in R,\ u_j=\sum_iu_{ji}t^i,\ u_{ji}\in F,$$ $$x_i:=(u_{1i},...,u_{ni})\in F^n,\ t^ix_i:=(u_1t^i,...,u_nt^i)\in R^n,$$ $$u=\sum t^ix_i.$$ – Pierre-Yves Gaillard Apr 07 '18 at 19:57
  • @Cary - Comment 2: If $R^n\xrightarrow{tI-B}R^n\xrightarrow\phi F^n\to0$ is exact, and if $\psi:F^n\to V$ is an isomorphism, then $R^n\xrightarrow{tI-B}R^n\xrightarrow{\psi\circ\phi}V\to0$ is exact (because $\psi\circ\phi$ is surjective and has the same kernel as $\phi$). – Pierre-Yves Gaillard Apr 07 '18 at 20:13