69.7 Descending relative objects

The following lemma is typical of the type of results in this section.

Lemma 69.7.1. Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_ i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$. Assume

1. the morphisms $f_{ii'} : X_ i \to X_{i'}$ are affine,

2. the spaces $X_ i$ are quasi-compact and quasi-separated.

Let $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$. Then the category of algebraic spaces of finite presentation over $X$ is the colimit over $I$ of the categories of algebraic spaces of finite presentation over $X_ i$.

Proof. Pick $0 \in I$. Choose a surjective étale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma 65.6.3). Set $U_ i = X_ i \times _{X_0} U_0$. Set $R_0 = U_0 \times _{X_0} U_0$ and $R_ i = R_0 \times _{X_0} X_ i$. Denote $s_ i, t_ i : R_ i \to U_ i$ and $s, t : R \to U$ the two projections. In the proof of Lemma 69.4.1 we have seen that there exists a presentation $X = U/R$ with $U = \mathop{\mathrm{lim}}\nolimits U_ i$ and $R = \mathop{\mathrm{lim}}\nolimits R_ i$. Note that $U_ i$ and $U$ are affine and that $R_ i$ and $R$ are quasi-compact and separated (as $X_ i$ is quasi-separated). Let $Y$ be an algebraic space over $S$ and let $Y \to X$ be a morphism of finite presentation. Set $V = U \times _ X Y$. This is an algebraic space of finite presentation over $U$. Choose an affine scheme $W$ and a surjective étale morphism $W \to V$. Then $W \to Y$ is surjective étale as well. Set $R' = W \times _ Y W$ so that $Y = W/R'$ (see Spaces, Section 64.9). Note that $W$ is a scheme of finite presentation over $U$ and that $R'$ is a scheme of finite presentation over $R$ (details omitted). By Limits, Lemma 32.10.1 we can find an index $i$ and a morphism of schemes $W_ i \to U_ i$ of finite presentation whose base change to $U$ gives $W \to U$. Similarly we can find, after possibly increasing $i$, a scheme $R'_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $R'$. The projection morphisms $s', t' : R' \to W$ are morphisms over the projection morphisms $s, t : R \to U$. Hence we can view $s'$, resp. $t'$ as a morphism between schemes of finite presentation over $U$ (with structure morphism $R' \to U$ given by $R' \to R$ followed by $s$, resp. $t$). Hence we can apply Limits, Lemma 32.10.1 again to see that, after possibly increasing $i$, there exist morphisms $s'_ i, t'_ i : R'_ i \to W_ i$, whose base change to $U$ is $S', t'$. By Limits, Lemmas 32.8.10 and 32.8.14 we may assume that $s'_ i, t'_ i$ are étale and that $j'_ i : R'_ i \to W_ i \times _{X_ i} W_ i$ is a monomorphism (here we view $j'_ i$ as a morphism of schemes of finite presentation over $U_ i$ via one of the projections – it doesn't matter which one). Setting $Y_ i = W_ i/R'_ i$ (see Spaces, Theorem 64.10.5) we obtain an algebraic space of finite presentation over $X_ i$ whose base change to $X$ is isomorphic to $Y$.

This shows that every algebraic space of finite presentation over $X$ comes from an algebraic space of finite presentation over some $X_ i$, i.e., it shows that the functor of the lemma is essentially surjective. To show that it is fully faithful, consider an index $0 \in I$ and two algebraic spaces $Y_0, Z_0$ of finite presentation over $X_0$. Set $Y_ i = X_ i \times _{X_0} Y_0$, $Y = X \times _{X_0} Y_0$, $Z_ i = X_ i \times _{X_0} Z_0$, and $Z = X \times _{X_0} Z_0$. Let $\alpha : Y \to Z$ be a morphism of algebraic spaces over $X$. Choose a surjective étale morphism $V_0 \to Y_0$ where $V_0$ is an affine scheme. Set $V_ i = V_0 \times _{Y_0} Y_ i$ and $V = V_0 \times _{Y_0} Y$ which are affine schemes endowed with surjective étale morphisms to $Y_ i$ and $Y$. The composition $V \to Y \to Z \to Z_0$ comes from a (essentially unique) morphism $V_ i \to Z_0$ for some $i \geq 0$ by Proposition 69.3.10 (applied to $Z_0 \to X_0$ which is of finite presentation by assumption). After increasing $i$ the two compositions

$V_ i \times _{Y_ i} V_ i \to V_ i \to Z_0$

are equal as this is true in the limit. Hence we obtain a (essentially unique) morphism $Y_ i \to Z_0$. Since this is a morphism over $X_0$ it induces a morphism into $Z_ i = Z_0 \times _{X_0} X_ i$ as desired. $\square$

Lemma 69.7.2. With notation and assumptions as in Lemma 69.7.1. The category of $\mathcal{O}_ X$-modules of finite presentation is the colimit over $I$ of the categories $\mathcal{O}_{X_ i}$-modules of finite presentation.

Proof. Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X_0$. Set $U_ i = X_ i \times _{X_0} U_0$. Set $R_0 = U_0 \times _{X_0} U_0$ and $R_ i = R_0 \times _{X_0} X_ i$. Denote $s_ i, t_ i : R_ i \to U_ i$ and $s, t : R \to U$ the two projections. In the proof of Lemma 69.4.1 we have seen that there exists a presentation $X = U/R$ with $U = \mathop{\mathrm{lim}}\nolimits U_ i$ and $R = \mathop{\mathrm{lim}}\nolimits R_ i$. Note that $U_ i$ and $U$ are affine and that $R_ i$ and $R$ are quasi-compact and separated (as $X_ i$ is quasi-separated). Moreover, it is also true that $R \times _{s, U, t} R = \mathop{\mathrm{colim}}\nolimits R_ i \times _{s_ i, U_ i, t_ i} R_ i$. Thus we know that $\mathit{QCoh}(\mathcal{O}_ U) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{U_ i})$, $\mathit{QCoh}(\mathcal{O}_ R) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{R_ i})$, and $\mathit{QCoh}(\mathcal{O}_{R \times _{s, U, t} R}) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{R_ i \times _{s_ i, U_ i, t_ i} R_ i})$ by Limits, Lemma 32.10.2. We have $\mathit{QCoh}(\mathcal{O}_ X) = \mathit{QCoh}(U, R, s, t, c)$ and $\mathit{QCoh}(\mathcal{O}_{X_ i}) = \mathit{QCoh}(U_ i, R_ i, s_ i, t_ i, c_ i)$, see Properties of Spaces, Proposition 65.32.1. Thus the result follows formally. $\square$

Lemma 69.7.3. With notation and assumptions as in Lemma 69.7.1. Then

1. any finite locally free $\mathcal{O}_ X$-module is the pullback of a finite locally free $\mathcal{O}_{X_ i}$-module for some $i$,

2. any invertible $\mathcal{O}_ X$-module is the pullback of an invertible $\mathcal{O}_{X_ i}$-module for some $i$.

Proof. Proof of (2). Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Since invertible modules are of finite presentation we can find an $i$ and modules $\mathcal{L}_ i$ and $\mathcal{N}_ i$ of finite presentation over $X_ i$ such that $f_ i^*\mathcal{L}_ i \cong \mathcal{L}$ and $f_ i^*\mathcal{N}_ i \cong \mathcal{L}^{\otimes -1}$, see Lemma 69.7.2. Since pullback commutes with tensor product we see that $f_ i^*(\mathcal{L}_ i \otimes _{\mathcal{O}_{X_ i}} \mathcal{N}_ i)$ is isomorphic to $\mathcal{O}_ X$. Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that $f_{i'i}^*\mathcal{L}_ i \otimes _{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_ i$ is isomorphic to $\mathcal{O}_{X_{i'}}$ for some $i' \geq i$. It follows that $f_{i'i}^*\mathcal{L}_ i$ is invertible (Modules on Sites, Lemma 18.32.2) and the proof is complete.

Proof of (1). Omitted. Hint: argue as in the proof of (2) using that a module (on a locally ringed site) is finite locally free if and only if it has a dual, see Modules on Sites, Section 18.29. Alternatively, argue as in the proof for schemes, see Limits, Lemma 32.10.3. $\square$

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