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 10.95.5. $\square$

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 108.32.4. 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."


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05AY. Beware of the difference between the letter 'O' and the digit '0'.