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ag.algebraic geometry – How do the invariants of a bunch strategy motion examine to the invariants of the group motion by world sections

If $G$ is a bunch strategy over $S$ appearing on an $S$-scheme $X$, I’d love to grasp the algebra of invariants $(mathcal{O}_X)^G$. Specifically, I’d love to grasp its relation to invariants $(mathcal{O}_X)^{G(S)}$.

To simplify notation, say every little thing is affine: $G = operatorname{Spec}R$, $X = operatorname{Spec}A$, and $S = operatorname{Spec}ok$, the place $ok$ is an capricious ring (not essentially a bailiwick). If it helps we are able to occupy $G$ is flush. We labor within the class of $ok$-schemes.

The motion is given by a map $sigma : Gtimes Xrightarrow X$. Let $p : Gtimes Xrightarrow X$ breathe the projection map. Then there’s a unaffected bijection $A = operatorname{Hom}(X,mathbb{A}^1)$, and by definition the subalgebra of invariants $A^G$ is the clique of $fin A$ whose corresponding map $F : Xrightarrowmathbb{A}^1$ satisfies

$$Fcircsigma = Fcirc p$$

Via $sigma$, the group $G(ok)$ acts on $X(ok)$, and for any $ok$-scheme $T$, $G(ok)$ maps to $G(T)$ and therefore too acts on $X(T)$, so $G(ok)$ acts on $X$. Thus, we could too deem the ring of invariants $A^{G(ok)}$. Certainly we’ve

$$A^Gsubset A^{G(ok)}$$

My important query is: What is the clearest route to specific this relationship? I’m in search of an announcement of the figure $fin A$ is $G$-invariant if and solely whether it is fastened by $G(ok)$ and another situations.

I cerebrate one can say that

$$A^G = {fin A| fotimes_k 1in Aotimes_k B textual content{ is fastened by $G(B)$ for each $ok$-algebra $B$}}$$

Is this rectify? Is it workable to additional limit the category of $B$‘s that you need to deem? Are there different methods of occupied with this?

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