Lemma
Let \(X\) be a DM stack with a torus action \(T\). Then the natural map \(\iota:X^{hT}\to X\) from the Romagny fixed stack to \(X\) is a closed immersion.
Proof
Proposition A.12 of Aranha et. al. in “Virtual Localization Revisted” says the following:
If \(G\) acts on a DM stack \(X\) which is locally of finite type over \(k\) and \(G\) has connected fibers over \(S\), then \(|X^G| \subset |X|\) is closed.
Here they’re using a different notion of fixed stack which guarantees that \(X^G\hookrightarrow X\). However,