I am trying to prove that a principal open set in $A^n$ is an affine variety in order to prove that $GL(n,K)$ is. Is there a way do this with a quite basic knowledge of algebraic geometry?

  • $\begingroup$ Do you define an affine variety as the max spec of a finitely generated $k$ algebra? Perhaps you could mention your definition. $\endgroup$ – R_D Nov 25 '16 at 15:26
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    $\begingroup$ What is "quite basic knowledge of algebraic geometry"? Here is a proof math.stackexchange.com/questions/375283/… $\endgroup$ – user302982 Nov 25 '16 at 15:27

So a principal open subset of $\mathbb A^n$ has the form $U_f = \{ x \in \mathbb A^n \mid f(x) \neq 0 \}$.

Then define $V \subset \mathbb A^{n+1}$ by $\{ (x,t) \in \mathbb A^{n+1} \mid f(x)t-1=0 \}$. $V$ is a closed subset of affine space, so is clearly an affine variety.

Also, $U_f$ and $V$ are isomorphic via the restriction of the projection map $\pi:\mathbb A^{n+1} \to \mathbb A^n$, namely send $(x,t) \in V$ to $x \in U_f$.


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