83.8 Invariant functions
In some cases it is convenient to pin down the structure sheaf of a quotient by requiring any invariant function to be a local section of the structure sheaf of the quotient.
Definition 83.8.1. Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. Let $\phi : U \to X$ be an $R$-invariant morphism. Denote $\phi ' = \phi \circ s = \phi \circ t : R \to X$.
We denote $(\phi _*\mathcal{O}_ U)^ R$ the $\mathcal{O}_ X$-sub-algebra of $\phi _*\mathcal{O}_ U$ which is the equalizer of the two maps
\[ \xymatrix{ \phi _*\mathcal{O}_ U \ar@<1ex>[rr]^{\phi _*s^\sharp } \ar@<-1ex>[rr]_{\phi _*t^\sharp } & & \phi '_*\mathcal{O}_ R } \]
on $X_{\acute{e}tale}$. We sometimes call this the sheaf of $R$-invariant functions on $X$.
We say the functions on $X$ are the $R$-invariant functions on $U$ if the natural map $\mathcal{O}_ X \to (\phi _*\mathcal{O}_ U)^ R$ is an isomorphism.
Of course we can require this property holds after any (flat or any) base change, leading to a (uniform or) universal notion. This condition is often thrown in with other conditions in order to obtain a (more) unique quotient. And of course a good deal of motivation for the whole subject comes from the following special case: $U = \mathop{\mathrm{Spec}}(A)$ is an affine scheme over a field $S = B = \mathop{\mathrm{Spec}}(k)$ and where $R = G \times U$, with $G$ an affine group scheme over $k$. In this case you have the option of taking for the quotient:
\[ X = \mathop{\mathrm{Spec}}(A^ G) \]
so that at least the condition of the definition above is satisfied. Even though this is a nice thing you can do it is often not the right quotient; for example if $U = \text{GL}_{n, k}$ and $G$ is the group of upper triangular matrices, then the above gives $X = \mathop{\mathrm{Spec}}(k)$, whereas a much better quotient (namely the flag variety) exists.
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