Recovering an affine group scheme from its representations

Question

I want to understand the proof of recovering an affine group scheme from its category of representations which is presented as proposition 2.8 in Milne's notes here.

1. Firstly I want to understand the group scheme $GL_X$ for a $k$-vector space $X$. I know it is isomorphic (non-canonically) to $GL_n$, by fixing a basis, which is represented by the $k$-algebra $k[X_{ij},Y]/(Y\cdot\det(X_{ij})-1)$. I want to what to know which $k$-algebra represents $GL_X$ for a $k$-vector space $X$. More specifically what is the $A$ which satisfies $$\operatorname{Hom}_{k-alg}(A,R)=\left \{f:X\otimes R\rightarrow X\otimes R \mid f \text{ is an } R-\text{linear isomorphism}\right \}.$$

Taking $C_X$ to be the full subcategory of $Rep_k(G)$ generated by objects isomorphic to subquotients of finite direct sums and tensor products of $X$ and $X^{\vee}$ and $Aut^\otimes(\omega| C_X)$ to be the functor sending $R$ to tensor automorphisms (endomorphisms due to rigidity) of the functor $\phi_R\circ\omega|_{C_X}$ ($\omega$ is the forgetful functor, $\phi_R$ is the functor from vacrtor spaces to R modules $V\mapsto V\otimes R$) we get $Aut^\otimes(\omega| C_X)(R)\subset GL_X(R)$.

2. Now, $G_X=Img(G\rightarrow GL_X)$ which is mentioned as a closed subgroup of $GL_X$. I want to know why image is representable and also closed inside $GL_X$ and how $G_X(R)$ lies inside $Aut^\otimes(\omega| C_X)(R)$.

Now Chevalleys theorem says that $G_X$ is a stabilizer of a 1- dimensional subspace of a finite dimensional $GL_X$ representation. $Aut^\otimes(\omega| C_X)$ also stabilizes that subspace and contains $G_X$. But how to conclude rigorously that they are equal.

Any help will be really appreciated. Thanks in advance

Answer 1

Question: "@hm2020, I was looking for A without using the basis of X which is finite dimensional. Does there exists a definition which also extends to infinite dimensional case. Jantzen's book does it for finitely generated projective. – bluebird yesterday"

Answer: @bluebird - If $A:=k[x_1,x_2,...]$ is a polynomial ring on a countably infinite set of variables $x_i$ and if $I\subseteq A$ is an ideal generated by a finite set of polynomials, consider $B:=A/I$. Let $\mathfrak{p} \subseteq B$ be a prime ideal and let $R:=B_{\mathfrak{p}}$ with maximal ideal $\mathfrak{m}$. Both entities $krdim(R)$ and $dim_k(\mathfrak{m}/\mathfrak{m}^2)$ are countably infinite. How do you make sense of the "inequality" $krdim(R) \leq dim_k(\mathfrak{m}/\mathfrak{m}^2)$ when both numbers are infinite? You need this type of inequality to define the notion "regular local ring" and it does not make any sense to speak of $\infty \leq \infty$.