vector space as a k-module

Motivation

Let $k$ be a field. A $k$-module is a vector space over $k$. This can be formalized using categories, which gives us a way to study vector spaces using module theory or to understand certain modules given linear algebra.

Equivalence of categories of vector spaces

The categories $k-{\text{mod}}^{\text{f.g.}}$ and ${\text{vect}}_k^{\text{f.d.}}$ are equivalent.

Proof

We define a functor

$$F:k-{\text{mod}}^{\text{f.g.}}⟶{\text{vect}}_k^{\text{f.d.}}.$$

Given a $k$-module $M$, we choose that $M$ is the data of a vector space over $k$. Therefore, $F(M)=M$ under reinterpretation (noting that $M$) satisfies all of the axioms for a vector space.

Let $f:M\to N$ be a $k$-module morphism. Thus, $f$ is an abelian group which satisfies

$$f(K\cdot m)=K\cdot f(m)\forall K\in k,m\in M$$

Therefore, $f$ is $k$-linear so it is a morphism of $k$-vector spaces, and we may set $F(f)=f$.

Somewhat trivially, $F$ is fully faithful and essentially surjective.

Interpretation

With the above equivalence, we can interpret $k$-module morphisms as matrices with entries in $k$. These matrices will differ with choices of bases, but different bases will lead to similar matrices. We can say this more formally using the language from above.

Choosing a basis is the same as choosing a $k$-linear map

$$\alpha :k^{\oplus d}⟶Md={\dim }_{k}M.$$

(We know this morphism exists and is an isomorphism since $k$-modules don’t have any torsion (module theory), and are thus free.) Under this morphism, the ordered basis of $M$ is $\{\alpha (e_1),\dots ,\alpha (e_d)\}$.

Given an ordered basis $\beta :k^{\oplus e}\to N$ for a $k$-module $N$, we have the following commutative diagram.

where $f_{\alpha ,\beta }\in {\text{Mat}}_{d\times e}(k)$.

If ${\alpha }^{\prime }$ and ${\beta }^{\prime }$ are a second choice of ordered basis, then we can extend the diagram to

Note that ${\alpha }^{-1}\circ {\alpha }^{\prime }$ and ${\beta }^{-1}\circ {\beta }^{\prime }$ are both invertible matrices, we can call them

$$A\in {\text{GL}}_d(k)\text{and}B\in {\text{GL}}_e(k).$$

Therefore, by commutative of the diagram above, we get

$$f_{\alpha ,\beta }=B{f}_{{\alpha }^{\prime },{\beta }^{\prime }}{A}^{-1}.$$

Equivalence of categories with endomorphisms

Definition

Let $\mathcal{C}$ be category with

• Objects: pairs $(V,\varphi )$ where $V$ is a finite dimensional vector space over $k$ and $\varphi :V\to V$ is a $k$-linear map.
• Morphisms: $f:(V,\varphi )\to (W,\psi )$ are $k$-linear maps $f:V\to W$ such that the following diagram commute

Morphisms are composed by composing $k$-linear maps.

Lemma

The categories $\mathcal{C}$ and $k[x]-{\text{mod}}^{\text{f.g.tor}}$ are equivalent.

Proof

We define a restriction functor using any ring homomorphism $h:R\to S$. Given an $S$-module $M$,

$$S\to {\text{End}}_{\text{Ab}}(M)$$

we define $h^{\ast }M$ to the composition

$$R\stackrel{h}{\to }S\to {\text{End}}_{\text{Ab}}(M).$$

Given a morphism $\varphi :M\to N$ of $S$-modules define

$$\begin{array}{rl}{h}^{\ast }\varphi :h^{\ast }M& ⟶N\\ m& ⟼\varphi (m)\end{array}$$

We can check this is $R$-linear:

$$\begin{array}{rl}(h^{\ast }\varphi )(r\cdot m)& =(h^{\ast }\varphi )(h(r)\cdot m)\\ & =\varphi (h(r)\cdot m)\\ & =h(r)\varphi (m)\\ & =r\cdot (h^{\ast }\varphi )(m)\end{array}$$

We can use this and apply it to the inclusion $h:k\hookrightarrow k[x]$ gives

$$h:k[x]\text{-mod}⟶k\text{-mod}.$$

Therefore, we define a functor $F(M)=(h^{\ast }M,X:M\stackrel{x}{\to }M)$ where $X$ is the action of $x\in k[x]$ on the vector space $h^{\ast }M$. On morphisms

$$F(f:M\to N)=h^{\ast }f$$

$F$ is fully faithful and surjective.

Interpretation

Consider the subcategory ${\mathcal{C}}_0$ of objects of the form $k^{\oplus n}$ for $n\ge 0$. This is a skeleton of $\mathcal{C}$, so we may interpret the $k$-linear map $\varphi ,\psi$ as matrices in $\mathcal{C}$. Therefore, to classify the morphisms of finite generated torsion $k[x]$ modules is the same as classifying isomorphism classes of objects in ${\mathcal{C}}_0$. This is the same as classifying square matrices up to conjugation.