Lemma 101.40.1. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. If \Delta _ f is quasi-separated and if for every diagram (101.39.1.1) and choice of \gamma as in Definition 101.39.1 the category of dotted arrows is a setoid, then \Delta _ f is separated.
101.40 Valuative criterion for second diagonal
The converse statement has already been proved in Lemma 101.39.2. The criterion itself is the following.
Proof. We are going to write out a detailed proof, but we strongly urge the reader to find their own proof, inspired by reading the argument given in the proof of Lemma 101.39.2.
Assume \Delta _ f is quasi-separated and for every diagram (101.39.1.1) and choice of \gamma as in Definition 101.39.1 the category of dotted arrows is a setoid. By Lemma 101.6.1 it suffices to show that e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}} is a closed immersion. By Lemma 101.6.4 it in fact suffices to show that e = \Delta _{f, 2} is universally closed. Either of these lemmas tells us that e = \Delta _{f, 2} is quasi-compact by our assumption that \Delta _ f is quasi-separated.
In this paragraph we will show that e satisfies the existence part of the valuative criterion. Consider a 2-commutative solid diagram
and let \alpha : (a, \theta ) \circ j \to e \circ x be any 2-morphism witnessing the 2-commutativity of the diagram (we use \alpha instead of the letter \gamma used in Definition 101.39.1). Note that f \circ \theta = \text{id}; we will use this below. Observe that e \circ x = (x, \text{id}_ x) and (a, \theta ) \circ j = (a \circ j, \theta \star \text{id}_ j). Thus we see that \alpha is a 2-arrow \alpha : a \circ j \to x compatible with \theta \star \text{id}_ j and \text{id}_ x. Set y = f \circ x and \beta = \text{id}_{f \circ a}. Reading the arguments given in the proof of Lemma 101.39.2 backwards, we see that \theta is an automorphism of the dotted arrow (a, \alpha , \beta ) with
On the other hand, \text{id}_ a is an automorphism too, hence we conclude \theta = \text{id}_ a from the assumption on f. Then we can take as dotted arrow for the displayed diagram above the morphism a : \mathop{\mathrm{Spec}}(A) \to \mathcal{X} with 2-morphisms (a, \text{id}_ a) \circ j \to (x, \text{id}_ x) given by \alpha and (a, \theta ) \to e \circ a given by \text{id}_ a.
By Lemma 101.39.11 any base change of e satisfies the existence part of the valuative criterion. Since e is representable by algebraic spaces, it suffices to show that e is universally closed after a base change by a morphism I \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}} which is surjective and smooth and with I an algebraic space (see Properties of Stacks, Section 100.3). This base change e' : X' \to I' is a quasi-compact morphism of algebraic spaces which satisfies the existence part of the valuative criterion and hence is universally closed by Morphisms of Spaces, Lemma 67.42.1. \square
Comments (2)
Comment #2100 by Matthew Emerton on
Comment #2144 by Johan on