The Stacks project

Proof. Write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ as a limit of algebraic spaces over a directed set $I$ with affine transition morphisms and with $Y_ i$ Noetherian, see Limits of Spaces, Proposition 70.8.1. We will use the material from Limits of Spaces, Section 70.23.

Choose a presentation $\mathcal{X} = [U/R]$. Denote $(U, R, s, t, c, e, i)$ the corresponding groupoid in algebraic spaces over $Y$. We may and do assume $U$ is affine. Then $U$, $R$, $R \times _{s, U, t} R$ are quasi-separated algebraic spaces of finite type over $Y$. We have two morpisms $s, t : R \to U$, three morphisms $c : R \times _{s, U, t} R \to R$, $\text{pr}_1 : R \times _{s, U, t} R \to R$, $\text{pr}_2 : R \times _{s, U, t} R \to R$, a morphism $e : U \to R$, and finally a morphism $i : R \to R$. These morphisms satisfy a list of axioms which are detailed in Groupoids, Section 39.13.

According to Limits of Spaces, Remark 70.23.5 we can find an $i_0 \in I$ and inverse systems

1. $(U_ i)_{i \geq i_0}$,

2. $(R_ i)_{i \geq i_0}$,

3. $(T_ i)_{i \geq i_0}$

over $(Y_ i)_{i \geq i_0}$ such that $U = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} U_ i$, $R = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} R_ i$, and $R \times _{s, U, t} R = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} T_ i$ and such that there exist morphisms of systems

1. $(s_ i)_{i \geq i_0} : (R_ i)_{i \geq i_0} \to (U_ i)_{i \geq i_0}$,

2. $(t_ i)_{i \geq i_0} : (R_ i)_{i \geq i_0} \to (U_ i)_{i \geq i_0}$,

3. $(c_ i)_{i \geq i_0} : (T_ i)_{i \geq i_0} \to (R_ i)_{i \geq i_0}$,

4. $(p_ i)_{i \geq i_0} : (T_ i)_{i \geq i_0} \to (R_ i)_{i \geq i_0}$,

5. $(q_ i)_{i \geq i_0} : (T_ i)_{i \geq i_0} \to (R_ i)_{i \geq i_0}$,

6. $(e_ i)_{i \geq i_0} : (U_ i)_{i \geq i_0} \to (R_ i)_{i \geq i_0}$,

7. $(i_ i)_{i \geq i_0} : (R_ i)_{i \geq i_0} \to (R_ i)_{i \geq i_0}$

with $s = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} s_ i$, $t = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} t_ i$, $c = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} c_ i$, $\text{pr}_1 = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} p_ i$, $\text{pr}_2 = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} q_ i$, $e = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} e_ i$, and $i = \mathop{\mathrm{lim}}\nolimits _{i \geq i_0} i_ i$. By Limits of Spaces, Lemma 70.23.7 we see that we may assume that $s_ i$ and $t_ i$ are smooth (this may require increasing $i_0$). By Limits of Spaces, Lemma 70.23.6 we may assume that the maps $R \to U \times _{U_ i, s_ i} R_ i$ given by $s$ and $R \to R_ i$ and $R \to U \times _{U_ i, t_ i} R_ i$ given by $t$ and $R \to R_ i$ are isomorphisms for all $i \geq i_0$. By Limits of Spaces, Lemma 70.23.9 we see that we may assume that the diagrams

are cartesian. The uniqueness of Limits of Spaces, Lemma 70.23.4 then guarantees that for a sufficiently large $i$ the relations between the morphisms $s, t, c, e, i$ mentioned above are satisfied by $s_ i, t_ i, c_ i, e_ i, i_ i$. Fix such an $i$.

It follows that $(U_ i, R_ i, s_ i, t_ i, c_ i, e_ i, i_ i)$ is a smooth groupoid in algebraic spaces over $Y_ i$. Hence $\mathcal{X}_ i = [U_ i/R_ i]$ is an algebraic stack (Algebraic Stacks, Theorem 94.17.3). The morphism of groupoids

over $Y \to Y_ i$ determines a commutative diagram

(Groupoids in Spaces, Lemma 78.21.1). We claim that the morphism $\mathcal{X} \to Y \times _{Y_ i} \mathcal{X}_ i$ is a closed immersion. The claim finishes the proof because the algebraic stack $\mathcal{X}_ i \to Y_ i$ is of finite presentation by construction. To prove the claim, note that the left diagram

is cartesian by Groupoids in Spaces, Lemma 78.25.3 and the results mentioned above. Hence the right commutative diagram is cartesian too. Then the desired result follows from the fact that $U \to Y \times _{Y_ i} U_ i$ is a closed immersion by construction of the inverse system $(U_ i)$ in Limits of Spaces, Lemma 70.23.3, the fact that $Y \times _{Y_ i} U_ i \to Y \times _{Y_ i} \mathcal{X}_ i$ is smooth and surjective, and Properties of Stacks, Lemma 100.9.4. $\square$