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I am reading the first several pages of GIT by Mumford, and I need some clarification of one requirement in the definition of geometric quotient (c.f. Definition 0.4, GIT):

Suppose a group scheme $G/S$ acts on scheme $X/S$ by $\sigma$, where $G,X$ are schemes over $S$. If a pair $(Y, \phi)$ consisting of a scheme $Y$ over $S$ and an $S-$morphism $\phi: X \to Y$ is a geometric quotient, then one requirement is " the fundamental sheaf $\mathcal{O}_Y$ is the subsheaf of $\phi_*(\mathcal{O}_X)$ consisting of invariant functions" i.e.

If $f \in \Gamma(U,\phi_*(\mathcal{O}_X))= \Gamma(\phi^{-1}(U),\mathcal{O}_X)$, then $f \in \Gamma(U, \mathcal{O}_Y)$ if and only if:

$$\begin{matrix} G \times \phi^{-1}(U)&\stackrel{\sigma}{\longrightarrow}&\phi^{-1}(U)\\ \downarrow{p_2}&&\downarrow{F}\\ \phi^{-1}(U)&\stackrel{F}{\longrightarrow}&\mathbb{A}^1 \end{matrix} $$ commutes (where $F$ is the morphism defined by $f$, and $\mathbb{A}^1 = \operatorname{Spec}(\mathbb{Z}[x])$)

My questions is how to make sense of this $F$ ?

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1 Answer 1

up vote 2 down vote accepted

A regular function $f \in \mathcal O_X(U)$ defines a morphism $F: U \rightarrow \mathbb A^1$. In the classical case of a variety over an algebraically closed field, the affine line is identified with the base field $k$ this map is just evaluation.

More generally, if $f \in \mathcal O_X(U)$ is an element of the structure sheaf over an open affine subset $U$, then there is a natural ring homomorphism $\mathbb Z[X] \rightarrow \mathcal O_X(U)$ mapping $X$ to $f$. Taking Spec gives your morphism $F: U \rightarrow \mathbb A^1$. In your case your regular function is $f \circ \phi$ defined on $\phi^{-1}U$.

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