I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting is as follows. Let $S$ be a scheme and $U \longrightarrow X$ be a morphism of $S$-algebraic spaces endowed with $\mathbb{G}_{m,S}$-actions. Lemma 1.10 in loc. cit. says that if $u \colon U \longrightarrow X$ is étale then $U^0 = U \times_X X^0$ where $(-)^0$ denotes the functor of fixed points (of the given $\mathbb{G}_{m,S}$-actions).
In case $U \longrightarrow X$ is an open immersion, the lemma seems to be obvious but I could not understand the proof of the étale case. Here the proof: let $T/S$ be a scheme with trivial $\mathbb{G}_{m,S}$-action and an equivariant morphism $T \longrightarrow X$. Given a morphism $\widetilde{f} \colon T \longrightarrow U$ such that $u \circ \widetilde{f} = f$, then the problem is to prove that $\widetilde{f}$ is equivariant.
- The author claims that it suffices to prove that $\widetilde{f}$ is locally étale equivariant.
- If $T$ is the spectrum of a strictly henselian local ring then $\widetilde{f}$ is equivariant because $\mathbb{G}_{m,S}$ is connected and $u$ is étale.
- Finally, he finishes the proof by some limit argument.
Since I am not really familiar with neither algebraic spaces nor algebraic groups, I hope you can help me to understand the proof or give me some references for each of my confusions. Thank you in advance.
