For the group $G=\mathrm{GL}_n(\mathbb{K})$ of invertible matrices over any field, there is the well-known determinant character $\det:G\to\mathbb{K}^\times$ which, well, maps any matrix to its determinant. It has the property that
\[ [G,G]=\mathrm{SL}_n(\mathbb{K})=\{ g\in G \mid \det(g)=1 \}, \]
where $[G,G]$ denotes the derived subgroup of $G$. In more geometric terms, $[G,G]$ is the vanishing of the regular function $\det-1$ on $G$. Because I find it rather cute, I will show that you can find a similar function for all linear, reductive groups.
(more…)

**Theorem.**Let $X$ be a normal, separated, Noetherian scheme and $U\subseteq X$ a nonempty affine open subset. Then, $X\setminus U$ is pure of codimension one.

**Proof.**We need to show that for each generic point $y\in Y=X\setminus U$, the local ring $\mathcal{O}_{X,y}$ has dimension one. Let $y$ be the generic point of a component of $Y$. Denote by $\mathfrak{m}$ the maximal ideal of the local ring $A:=\mathcal{O}_{X,y}$. Let $S:=\mathrm{Spec}(A)$. The inclusion morphism $f:S\to X$ is affine because $X$ is separated (Lemma 1). Thus, $V:=f^{-1}(U)=S\setminus\{ \mathfrak{m} \}$ is an affine, open subscheme of $S$. The morphism on global sections $A=\mathcal{O}_S(S)\to\mathcal{O}_{V}(V)$, corresponding to the inclusion $V\to S$, is therefore injective. It can not be surjective because $V\ne S$. Pick some $a\in\mathcal{O}_{V}(V)\setminus A$. If we now had $\dim(A)>1$, then the prime ideals of height one $\mathfrak{p}\subseteq A$ would satisfy $\mathfrak{p}\ne\mathfrak{m}$, i.e. they would correspond to points contained in $V$. Consequently, we would have $a\in \mathcal{O}_{V,\mathfrak{p}} = A_{\mathfrak{p}}$. Because $A$ is a normal Noetherian domain, it is the intersection of all localizations at prime ideals of height one - this is Corollary 11.4 in Eisenbud's book. This yields the contradition that $a\in A$.

**Lemma 1.**If $f:X\to Y$ is a morphism of schemes with $Y$ separated and $X$ affine, then $f$ is an affine morphism.

**Proof.**This is exercise 3.3.6 in Qing Liu's book. Let $V\subseteq Y$ be an open affine subset. Proposition 3.9(f) in the same book tells you that there is a closed immersion $f^{-1}(V)\cong X\times_Y V\to X\times V$, so $f^{-1}(V)$ is isomorphic to a closed subscheme of an affine scheme, hence affine.