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The answer to your question is yes in positive characteristic. Check Kollár's paper "Quotient Spaces Modulo Algebraic Groups" (Ann. of Math. 1997), Lem 8.4. The reason is that universal homeomorphisms …

As noted in the comments, the question has a positive answer for tame Artin stacks (see e.g. [Ols12, Prop 6.1]) and also for non-tame Deligne–Mumford stacks (see [KV04, Lem. 2]). It is however also tr …

Various people have imposed finite presentation instead of finite type: most notably Grothendieck for unramified and Demazure–Gabriel for proper maps. Both these cases seem to have been motivated by t …

Question: Let $X$ be a noetherian integral scheme. Is there a dense open subscheme $U\subset X$ such that $U$ is Jacobson? I am happy to allow $X$ to be excellent and then the question of course imme …

To complement Laurent Moret-Bailly and Karl Schwede's answers (10 years ago!), the pushout of a closed immersion $Z\to X$ along an affine morphism $Z\to Y$ always exists in the category of algebraic s …

The answer is no, even for the sheaf-case. First of all, you would have to make some assumptions such as assume that Xλ→ Spec(Bλ) is locally of finite presentation. For simplicity also assume that th …

If $A$ is noetherian, then every integral epimorphism $f \colon A \to B$ is surjective. This has been proven by Ferrand: Prop. 3.8 in Monomorphismes de schémas noethérien", Exp. 7 in Séminaire Samuel, …

This is answered in general by Theorem 13.1 in our paper The étale local structure of algebraic stacks (arXiv:1912.06162). Let $\mathcal{X}$ be a stack with a good moduli space $X$ such that $\mathca …

No, if X is any algebraic space such that X_red is an affine scheme, then X is an affine scheme. This follows from Chevalley's theorem. For X noetherian scheme/alg. space this theorem is in EGA/Knutso …

I think that it is highly unlikely that there exists a separification functor. What does exist is the following: Theorem (Raynaud-Gruson): Let $S$ be a base scheme and work relative to $S$. Given a no …

No. Whereas $A/I\otimes_A B=B/IB$ by right-exactness $I\otimes_A B\to IB$ is not an isomorphism in general. The exact sequence $$0\to I_{i-1}\to I_i\to I_i/I_{i-1} \to 0$$ only gives rise to the right …

The recent effectivity result of Brown-Geraschenko (arXiv:1208.2882) for coherent sheaves on stacks with the resolution property and admitting good moduli spaces does not reduce to the usual ones. The …

Theorem (EGA IV 13.1.3): Let $f \colon X \to Y$ be a morphism of schemes, locally of finite type. Then $$x \mapsto \dim_x(X_{f(x)})$$ is upper semi-continuous. Corollary (Chevalley's upper semi-conti …

The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. …

No, not every DM-stack has a coarse moduli space. The following is a counter-example (see my paper on geometric quotients): Let X be two copies of the affine plane glued outside the y-axis (a non-sep …