Example 39.23.1. Let $k$ be a field. Let $U = \text{GL}_{2, k}$. Let $B \subset \text{GL}_2$ be the closed subgroup scheme of upper triangular matrices. Then the quotient sheaf $\text{GL}_{2, k}/B$ (in the Zariski, étale or fppf topology, see Definition 39.20.1) is representable by the projective line: $\mathbf{P}^1 = \text{GL}_{2, k}/B$. (Details omitted.) On the other hand, the ring of invariant functions in this case is just $k$. Note that in this case the morphisms $s, t : R = \text{GL}_{2, k} \times _ k B \to \text{GL}_{2, k} = U$ are smooth of relative dimension $3$.

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