The Stacks project

35.7 Descent of finiteness properties of modules

In this section we prove that one can check quasi-coherent module has a certain finiteness conditions by checking on the members of a covering.

Lemma 35.7.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

Proof. Omitted. For the affine case, see Algebra, Lemma 10.83.2. $\square$

Lemma 35.7.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of locally ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. If

  1. $f$ is open as a map of topological spaces,

  2. $f$ is surjective and flat, and

  3. $f^*\mathcal{F}$ is of finite type,

then $\mathcal{F}$ is of finite type.

Proof. Let $y \in Y$ be a point. Choose a point $x \in X$ mapping to $y$. Choose an open $x \in U \subset X$ and elements $s_1, \ldots , s_ n$ of $f^*\mathcal{F}(U)$ which generate $f^*\mathcal{F}$ over $U$. Since $f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{O}_ X$ we can after shrinking $U$ assume $s_ i = \sum t_{ij} \otimes a_{ij}$ with $t_{ij} \in f^{-1}\mathcal{F}(U)$ and $a_{ij} \in \mathcal{O}_ X(U)$. After shrinking $U$ further we may assume that $t_{ij}$ comes from a section $s_{ij} \in \mathcal{F}(V)$ for some $V \subset Y$ open with $f(U) \subset V$. Let $N$ be the number of sections $s_{ij}$ and consider the map

\[ \sigma = (s_{ij}) : \mathcal{O}_ V^{\oplus N} \to \mathcal{F}|_ V \]

By our choice of the sections we see that $f^*\sigma |_ U$ is surjective. Hence for every $u \in U$ the map

\[ \sigma _{f(u)} \otimes _{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} : \mathcal{O}_{X, u}^{\oplus N} \longrightarrow \mathcal{F}_{f(u)} \otimes _{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} \]

is surjective. As $f$ is flat, the local ring map $\mathcal{O}_{Y, f(u)} \to \mathcal{O}_{X, u}$ is flat, hence faithfully flat (Algebra, Lemma 10.39.17). Hence $\sigma _{f(u)}$ is surjective. Since $f$ is open, $f(U)$ is an open neighbourhood of $y$ and the proof is done. $\square$

Lemma 35.7.3. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is an $\mathcal{O}_{X_ i}$-module of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Proof. Omitted. For the affine case, see Algebra, Lemma 10.83.2. $\square$

Lemma 35.7.4. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_ X$-module.

Proof. By Lemma 35.7.1 we see that $\mathcal{F}$ is of finite type. Hence Nakayama's lemma (Algebra, Lemma 10.20.1) implies that $\mathcal{F}$ is generated by $r$ sections in the neighbourhood of a point $x \in X$ if and only if $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) \leq r$. Choose an $i$ and a point $x_ i \in X_ i$ mapping to $x$. Then $\dim _{\kappa (x)} \mathcal{F}_ x \otimes \kappa (x) = \dim _{\kappa (x_ i)} (f_ i^*\mathcal{F})_{x_ i} \otimes \kappa (x_ i)$ which is $\leq r$ as $f_ i^*\mathcal{F}$ is locally generated by $r$ sections. $\square$

Lemma 35.7.5. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a flat $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module.

Proof. Omitted. For the affine case, see Algebra, Lemma 10.83.2. $\square$

Lemma 35.7.6. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module.

Proof. This follows from the fact that a quasi-coherent sheaf is finite locally free if and only if it is of finite presentation and flat, see Algebra, Lemma 10.78.2. Namely, if each $f_ i^*\mathcal{F}$ is flat and of finite presentation, then so is $\mathcal{F}$ by Lemmas 35.7.5 and 35.7.3. $\square$

The definition of a locally projective quasi-coherent sheaf can be found in Properties, Section 28.21.

Lemma 35.7.7. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a locally projective $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a locally projective $\mathcal{O}_ X$-module.

Proof. Omitted. For Zariski coverings this is Properties, Lemma 28.21.2. For the affine case this is Algebra, Theorem 10.95.6. $\square$

Remark 35.7.8. Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that $R$ is a ring and that $M$ is a projective $R$-module which is a countable direct sum $M = \bigoplus L_ n$ of rank 1 locally free modules, but not locally free, see Examples, Lemma 110.34.4. Then $M$ becomes free on making the faithfully flat base change

\[ R \longrightarrow \bigoplus \nolimits _{m \geq 1} \bigoplus \nolimits _{(i_1, \ldots , i_ m) \in \mathbf{Z}^{\oplus m}} L_1^{\otimes i_1} \otimes _ R \ldots \otimes _ R L_ m^{\otimes i_ m} \]

But we don't know what happens for fppf coverings. In other words, we don't know the answer to the following question: Suppose $A \to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \otimes _ A B$ is free. Is $M$ a locally free $A$-module? It turns out that if $A$ is Noetherian, then the answer is yes. This follows from the results of [Bass]. But in general we don't know the answer. If you know the answer, or have a reference, please email stacks.project@gmail.com.

We also add here two results which are related to the results above, but are of a slightly different nature.

Lemma 35.7.9. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is a finite morphism. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_ Y$-module of finite type.

Proof. As $f$ is finite it is affine. This reduces us to the case where $f$ is the morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ given by a finite ring map $A \to B$. Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules associated to the $B$-module $M$. Note that $M$ is finite as a $B$-module if and only if $M$ is finite as an $A$-module, see Algebra, Lemma 10.7.2. Combined with Properties, Lemma 28.16.1 this proves the lemma. $\square$

Lemma 35.7.10. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is finite and of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_ Y$-module of finite presentation.

Proof. As $f$ is finite it is affine. This reduces us to the case where $f$ is the morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ given by a finite and finitely presented ring map $A \to B$. Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules associated to the $B$-module $M$. Note that $M$ is finitely presented as a $B$-module if and only if $M$ is finitely presented as an $A$-module, see Algebra, Lemma 10.36.23. Combined with Properties, Lemma 28.16.2 this proves the lemma. $\square$

Comments (2)

Comment #256 by Keenan Kidwell on

In 082U, "generate" should be "generated."

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