# Is there an example of an algebraic stack whose closed points have affine stabilizers but whose diagonal is not affine?

Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence failing in general?

• Can you provide a link to a reference? What is the condition that guarantees that affine diagonal is equivalent to affine stabilizers? Commented Sep 29, 2009 at 17:53
• The paper is MR2108211. See also 5.4 of Jarod Alper's stack literature review on de Jong's web page. Commented Sep 29, 2009 at 18:28
• You should change your title to ask, "is there an example of an algebraic stack where all closed points have affine stabilizers by the diagonal is not affine?" As stated now, the answer is clearly "no." Commented Sep 29, 2009 at 19:04

The answer is yes since a fiberwise condition (such as affine stabilizers) does not imply a global condition (such as affine diagonal) without extra hypotheses (such as having the resolution property). Think of quasi-finite+proper <=> finite.

There are (non-separated) schemes with non-affine diagonal, for example, two copies of the affine plane glued together outside the origin.

Also see the related question.

• Excellent counterexample (though I think you meant to say "the answer is yes" rather than "the answer is no" since the question asked if a counterexample exists). You should pose your challenge as a separate question (phrased as a question rather than a challenge). Commented Oct 2, 2009 at 3:43
• Reposting a link mentioned in the post so that it appears in the "Linked" questions list: Stack with affine stabilizers but not quasi-affine diagonal Commented May 21, 2023 at 20:24

Let $X$ be an algebraic stack. Given a point $f:T\to X$, we define $Stab(f)=X\times_{X\times X}T$, where the map $X\to X\times X$ is the diagonal and $T\to X\times X$ is $(f,f)$. We say that the stabilizer if affine if $Stab(f)\to T$ is an affine morphism.

Since affine morphisms are stable under base extension, it is always true that if the diagonal is affine, then stabilizers are affine.

If I've got everything right so far, then I think I have an argument that shows that if stabilizers are affine, then the diagonal is affine. Let $h:U\to X$ be a smooth cover by an affine scheme, then $Stab(h)$ is affine over $U\times U$ by assumption. But affine morphisms are local on the base in the smooth topology, so the diagonal is affine.

Stab(h) --> U×U
|          |
|  cart    | smooth cover
v          v
X ------> X×X


But this uses that stabilizers of scheme-theoretic points are affine. Perhaps in the original question, you're only allowed to assume that stabilizers of geometric points are affine (or something like that).

Edit: Somehow I missed that "closed points" part of the question.