Following the definition of $\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ found here: functor from $\mathbf{Alg}$ to $\mathbf{Set}$

I would like to show that it is a representable functor.

EDIT: wrong!

For ease of notation, $F:=\text{GL}_n$ and $G:= \text{Hom}(k,-)$

Where $\text{Hom}(k,-):\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{Hom}(k,R):f\mapsto (\sigma\mapsto f\circ \sigma)$

I'll define the natural transformation as:

$$\alpha_R:\text{GL}_n(R)\rightarrow \text{Hom}(k,R):(a_{i,j})\mapsto (k\mapsto \sum a_{i,j}k)$$

this defines a natural transformation. But how natural is it? I.e. are there any more intuitive ways of doing this?

EDIT: answer (from answer below and adapted to my notations)

For ease of notation, $F:=\text{GL}_n$ and $G:= \text{Hom}(A_n,-)$

where $$A_n=k[X_{11},\ldots,X_{nn},Y]/\left<1-Y\det(X_{ij})\right>.$$

Where $\text{Hom}(A_n,-):\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{Hom}(A_n,R):f\mapsto (\sigma\mapsto f\circ \sigma)$

I'll define the natural transformation as:

$$\alpha_R:\text{GL}_n(R)\rightarrow \text{Hom}(A_n,R)\\(a_{i,j})\mapsto (P(X_{11},\ldots,X_{nn},Y)+ (1-Y\det(X_{ij}))\mapsto P(a_{1,1},\ldots,a_{n,n},\text{det}(a_{i,j})))$$

  • $\begingroup$ It will certaibly not be represented by $k$. Do you think $GL_n(k) \simeq Hom(k,k)$ ? $\endgroup$ – Max Dec 30 '17 at 11:52
  • $\begingroup$ Oups, I forgot to check that it is a bijection. What I gave is a natural transformation but not necessarily a natural isomorphism. $\endgroup$ – tomak Dec 30 '17 at 12:00
  • $\begingroup$ Only other idea is to represent it by $k^n$ but not sure that works $\endgroup$ – tomak Dec 30 '17 at 12:01
  • $\begingroup$ There's the same problem with $Hom(k^n, k) \simeq GL_n(k)$ $\endgroup$ – Max Dec 30 '17 at 12:40

Yes, $\text{GL}_n$ is an affine group scheme. It is represented by the $k$-algebra $$A_n=k[X_{11},X_{12},\ldots,X_{21},\ldots,X_{nn},Y]/\left<1-Y\det(X_{ij})\right>.$$ An algebra homomorphism $\phi:A_n\to R$ is given by choosing the images of $X_{ij}$ and $Y$, which must go to the inverse of $\det(\phi(X_{ij}))$ which ensures that $(\phi(X_{ij}))\in\text{GL}_n(R)$.

Waterhouse's book is a good introduction to affine group schemes.

  • $\begingroup$ I modified my answer. Can you check that it makes sense? $\endgroup$ – tomak Dec 30 '17 at 14:43

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