# Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms

I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though.

## First definition: isovariant morphisms

Throughout, I fix a finite group $$G$$. For a $$G$$-scheme $$X$$ we have the stabilizer scheme $$Stab(X) \subset G \times X$$, which may be informally described as the scheme of pairs $$(g, x)$$ with $$gx=x$$. In other words the fibers of $$Stab(X) \to X$$ are given by the scheme-theoretic stabilizers of the points. If $$f: X \to Y$$ is a morphism of $$G$$-schemes, there is a natural map $$Stab(X) \to Stab(Y) \times_Y X =: f^* Stab(Y)$$. We call $$f$$ isovariant (a.k.a. stabilizer preserving, fixed point reflecting) if $$Stab(X) \to f^* Stab(Y)$$ is an isomorphism.

## Preliminary question

If $$f: X \to Y$$ is isovariant, then for every $$x \in X$$ the stabilizer of $$G$$ at $$x$$ is equal to the stabilizer of $$G$$ at $$f(x)$$.

Question: are there conditions for the converse to hold?

If I interpret the stacks project correctly, the converse does hold if $$f$$ is unramified. Can we do better?

## More definitions: strong morphisms

From now on, I fix a $$G$$-scheme $$S$$ and assume that the quotient $$S/G$$ exists (in algebraic spaces, if we want to be fancy). For safety, assume that the quotient is universal (e.g. $$|G|$$ invertible). Let $$P$$ be some property of morphisms of schemes, such as flat, smooth, etc. A morphism $$f: X \to S$$ is called strongly $$P$$ if $$X/G$$ exists, $$f/G$$ is $$P$$, and $$X = X/G \times_{S/G} S$$.

Note that if $$f': X' \to S/G$$ is any morphism, then $$f = f' \times_{S/G} S$$ is isovariant. Morphisms that arise in this way are called strongly isovariant.

## Main questions

Basically I want to know: if $$f: X \to S$$ is $$P$$ and isovariant, then under what additional hypotheses is $$f$$ strongly $$P$$?

I'm particularly interested in the case $$P = \emptyset$$ (which isovariant morphisms are strongly isovariant), $$P$$ means "flat" and $$P$$ means "smooth".

It is easy to see that this is true (possibly under mild assumptions) if $$P$$ means "étale". Also if I'm not mistaken, smooth + strongly flat implies strongly smooth.
• With considerable pain, I seem to have convinced myself if the following. Perhaps let everything be affine over a noetherian base with trivial action, to be safe, but surely this cannot matter too much. Consider $\pi: X \to S$. (1) suppose $\pi$ is smooth and preserves stabilizers in the naive sense (pointwise). Then $\pi$ is isovariant. (2) suppose $\pi$ is flat, isovariant and finite type. Then $\pi$ is strongly flat. (3) suppose $\pi$ is smooth and isovariant. Then $\pi$ is strongly smooth. I'd still be very interested in references for statements like this. – Tom Bachmann Jan 29 at 22:16