13
$\begingroup$

I am trying to prove that the general linear group $GL(n)$ is an $\underline{\text{affine}}$ variety. Unfortunately, I am having trouble with showing that $GL(n)$ is indeed affine.

Before I show my progress I present the definitions as given to me during the lecture:

  • A quasiprojective variety is a locally closed set of affine or projective space.
  • An affine variety is a quasiprojective variety isomorphic to a closed subset of affine space.

Thank you in advance!

Let $k$ be the underlying field and let $\mathbb{A}^n$ represent affine $n$-space. The first step is that $GL(n)\subset M_n(k)\cong\mathbb{A}^{n^2}$ is given by all matrices with determinant unequal to zero. Since the determinant is a polynomial in the coefficients of a matrix we conclude that $GL(n)$ is open in $\mathbb{A}^{n^2}$. Therefore, it is locally closed in $\mathbb{A}^{n^2}$ and thus a (quasiprojective) variety. However, to show it is affine I have to show that it is isomorphic to a closed subset of some affine space, which I don't know how to tackle.

$\endgroup$
1
  • 1
    $\begingroup$ You mean $M_n(k)\cong\mathbb{A}^{n^2}$? $\endgroup$
    – Krish
    Commented Feb 23, 2015 at 17:23

1 Answer 1

Reset to default
19
$\begingroup$

Add one variable $T$ and look at the equation $\det(X) \cdot T=1$.

$\endgroup$
1
  • $\begingroup$ Thank you for the response, great hint. I think I will be able to work out the details. $\endgroup$
    – Marc
    Commented Feb 23, 2015 at 17:27

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .