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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