General linear group is an affine variety

Question

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.

You mean $M_n(k)\cong\mathbb{A}^{n^2}$? – Krish Feb 23 '15 at 17:23 — Krish, Feb 23 '15 at 17:23

score 19 · Accepted Answer · answered Feb 23 '15 at 17:23

19

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

answered Feb 23 '15 at 17:23

MooS

32,317

Thank you for the response, great hint. I think I will be able to work out the details. – Marc Feb 23 '15 at 17:27

General linear group is an affine variety

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