The Stacks project

This is a special case of [Lemma 2.3.15, Emerton-Gee]

Proposition 99.3.8. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $f$ is limit preserving,

  2. $f$ is limit preserving on objects, and

  3. $f$ is locally of finite presentation.

Proof. Assume (3). Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a directed limit of affine schemes. Consider the functor

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{T_ i} \longrightarrow \mathcal{X}_ T \times _{\mathcal{Y}_ T} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{T_ i} \]

Let $(x, y_ i, \beta )$ be an object on the right hand side, i.e., $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ T)$, $y_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_{T_ i})$, and $\beta : f(x) \to y_ i|_ T$ in $\mathcal{Y}_ T$. Then we can consider $(x, y_ i, \beta )$ as an object of the algebraic stack $\mathcal{X}_{y_ i} = \mathcal{X} \times _{\mathcal{Y}, y_ i} T_ i$ over $T$. Since $\mathcal{X}_{y_ i} \to T_ i$ is locally of finite presentation (as a base change of $f$) we see that it is limit preserving by Lemma 99.3.7. This means that $(x, y_ i, \beta )$ comes from an object over $T_{i'}$ for some $i' \geq i$ and unwinding the definitions we find that $(x, y_ i, \beta )$ is in the essential image of the displayed functor. In other words, the displayed functor is essentially surjective. Another formulation is that this means $f$ is limit preserving on objects. Now we apply this to the diagonal $\Delta $ of $f$. Namely, by Morphisms of Stacks, Lemma 98.26.7 the morphism $\Delta $ is locally of finite presentation. Thus the argument above shows that $\Delta $ is limit preserving on objects. By Lemma 99.3.5 this implies that $\Delta $ is limit preserving. By Lemma 99.3.6 we conclude that the displayed functor above is fully faithful. Thus it is an equivalence (as we already proved essential surjectivity) and we conclude that (1) holds.

The implication (1) $\Rightarrow $ (2) is trivial. Assume (2). Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. By Criteria for Representability, Lemma 94.5.1 the base change $\mathcal{X} \times _\mathcal {Y} V \to V$ is limit preserving on objects. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. Since a smooth morphism is locally of finite presentation, we see that $U \to \mathcal{X} \times _\mathcal {Y} V$ is limit preserving (first part of the proof). By Criteria for Representability, Lemma 94.5.2 we find that the composition $U \to V$ is limit preserving on objects. We conclude that $U \to V$ is locally of finite presentation, see Criteria for Representability, Lemma 94.5.3. This is exactly the condition that $f$ is locally of finite presentation, see Morphisms of Stacks, Definition 98.26.1. $\square$


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