1
$\begingroup$

We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for every $k$-algebra $R$, $N(R):=\{g\in G(R): gY(R)\subseteq Y(R)\}.$

Is this functor representable? I know the proof for varieties over algebraically closed fields (looking at the ideal of $Y$ gives equations for $N(k)$), but I was wondering if I could produce it as a fiber product of some sort in a more functorial way

$\endgroup$

1 Answer 1

2
$\begingroup$

When $Y$ is closed, this is a standard result (possibly with some finiteness conditions). See Demazure Gabriel, Groupes Algebriques, I, section 2, no. 7 or Milne, Algebraic Groups, 1.79.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.