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The Stacks project

Lemma 101.41.3. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. If f is separated, then f satisfies the uniqueness part of the valuative criterion.

Proof. Since f is separated we see that all categories of dotted arrows are setoids by Lemma 101.39.2. Consider a diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} }

and a 2-morphism \gamma : y \circ j \to f \circ x as in Definition 101.39.1. Consider two objects (a, \alpha , \beta ) and (a', \beta ', \alpha ') of the category of dotted arrows. To finish the proof we have to show these objects are isomorphic. The isomorphism

f \circ a \xrightarrow {\beta ^{-1}} y \xrightarrow {\beta '} f \circ a'

means that (a, a', \beta ' \circ \beta ^{-1}) is a morphism \mathop{\mathrm{Spec}}(A) \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}. On the other hand, \alpha and \alpha ' define a 2-arrow

(a, a', \beta ' \circ \beta ^{-1}) \circ j = (a \circ j, a' \circ j, (\beta ' \star \text{id}_ j) \circ (\beta \star \text{id}_ j)^{-1}) \xrightarrow {(\alpha , \alpha ')} (x, x, \text{id}) = \Delta _ f \circ x

Here we use that both (a, \alpha , \beta ) and (a', \alpha ', \beta ') are dotted arrows with respect to \gamma . We obtain a commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[d]_ j \ar[rr]_ x & & \mathcal{X} \ar[d]^{\Delta _ f} \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a, a', \beta ' \circ \beta ^{-1})} & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} }

with 2-commutativity witnessed by (\alpha , \alpha '). Now \Delta _ f is representable by algebraic spaces (Lemma 101.3.3) and proper as f is separated. Hence by Lemma 101.39.13 and the valuative criterion for properness for algebraic spaces (Morphisms of Spaces, Lemma 67.44.1) we see that there exists a dotted arrow. Unwinding the construction, we see that this means (a, \alpha , \beta ) and (a', \alpha ', \beta ') are isomorphic in the category of dotted arrows as desired. \square


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