The Stacks project

36.36 The resolution property

This notion is discussed in the paper [totaro_resolution]. It is currently not known if a proper scheme over a field always has the resolution property or if this is false. If you know the answer to this question, please email stacks.project@gmail.com

We can make the following definition although it scarcely makes sense to consider it for general schemes.

Definition 36.36.1. Let $X$ be a scheme. We say $X$ has the resolution property if every quasi-coherent $\mathcal{O}_ X$-module of finite type is the quotient of a finite locally free $\mathcal{O}_ X$-module.

If $X$ is a quasi-compact and quasi-separated scheme, then it suffices to check every $\mathcal{O}_ X$-module module of finite presentation (automatically quasi-coherent) is the quotient of a finite locally free $\mathcal{O}_ X$-module, see Properties, Lemma 28.22.8. If $X$ is a Noetherian scheme, then finite type quasi-coherent modules are exactly the coherent $\mathcal{O}_ X$-modules, see Cohomology of Schemes, Lemma 30.9.1.

Lemma 36.36.2. Let $X$ be a scheme. If $X$ has an ample invertible $\mathcal{O}_ X$-module, then $X$ has the resolution property.

Proof. Immediate consquence of Properties, Proposition 28.26.13. $\square$

Lemma 36.36.3. Let $f : X \to Y$ be a morphism of schemes. Assume

  1. $Y$ is quasi-compact and quasi-separated and has the resolution property,

  2. there exists an $f$-ample invertible module on $X$.

Then $X$ has the resolution property.

Proof. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an $f$-ample invertible module. Choose an affine open covering $Y = V_1 \cup \ldots \cup V_ m$. Set $U_ j = f^{-1}(V_ j)$. By Properties, Proposition 28.26.13 for each $j$ we know there exists finitely many maps $s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_ j} \to \mathcal{F}|_{U_ j}$ which are jointly surjective. Consider the quasi-coherent $\mathcal{O}_ Y$-modules

\[ \mathcal{H}_ n = f_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \]

We may think of $s_{j, i}$ as a section over $V_ j$ of the sheaf $\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally free $\mathcal{O}_ Y$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}}$ such that $s_{j, i}$ is in the image. Then the corresponding maps

\[ f^*\mathcal{E}_{i, j} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F} \]

are going to be jointly surjective and the lemma is proved. By Properties, Lemma 28.22.3 for each $i, j$ we can find a finite type quasi-coherent submodule $\mathcal{H}'_{i, j} \subset \mathcal{H}_{-n_{j, i}}$ which contains the section $s_{i, j}$ over $V_ j$. Thus using the resolution property of $Y$ to get surjections $\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$ and we conclude. $\square$

Lemma 36.36.4. Let $f : X \to Y$ be an affine morphism of schemes with $Y$ quasi-compact and quasi-separated. If $Y$ has the resolution property, so does $X$.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. The pushforward $f_*\mathcal{F}$ is quasi-coherent, see Schemes, Lemma 26.24.1. The adjunction map $f^*f_*\mathcal{F} \to \mathcal{F}$ is surjective; this follows from Schemes, Lemma 26.7.3 after restricting to $f^{-1}(V)$ for $V \subset Y$ affine open. Write $f_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit with $\mathcal{G}_ i$ quasi-coherent $\mathcal{O}_ Y$-modules of finite type, see Properties, Lemma 28.22.3. Then we see that $\mathop{\mathrm{colim}}\nolimits f^*\mathcal{G}_ i \to \mathcal{F}$ is surjective. Since $\mathcal{F}$ is of finite type and $X$ is quasi-compact, we conclude that for some $i$ the map $f^*\mathcal{G}_ i \to \mathcal{F}$ is surjective (details omitted; look at generators on affines). Hence if $\mathcal{G}_ i$ is a quotient of a finite locally free $\mathcal{O}_ Y$-module, then $\mathcal{F}$ is a quotient of the pullback which is a finite locally free $\mathcal{O}_ X$-module. $\square$

Lemma 36.36.5. Let $X$ be a scheme. Suppose given

  1. a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$

  2. finite type quasi-coherent ideals $\mathcal{I}_ j$ with $V(\mathcal{I}_ j) = X \setminus U_ j$

Then $X$ has the resolution property if and only if $\mathcal{I}_ j$ is the quotient of a finite locally free $\mathcal{O}_ X$-module for $j = 1, \ldots , m$.

Proof. One direction of the lemma is trivial. For the other, say $\mathcal{E}_ j \to \mathcal{I}_ j$ is a surjection with $\mathcal{E}_ j$ finite locally free. In the next paragraph, we reduce to the Noetherian case; we suggest the reader skip it.

The first observation is that $U_ j \cap U_{j'}$ is quasi-compact as the complement of the zero scheme of the quasi-coherent finite type ideal $\mathcal{I}_{j'}|{U_ j}$ on the affine scheme $U_ j$, see Properties, Lemma 28.24.1. Hence $X$ is quasi-compact and quasi-separated, see Schemes, Lemma 26.21.6. By Limits, Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as the limit of a direct system of Noetherian schemes with affine transition morphisms. For each $j$ we can find an $i$ and a finite type quasi-coherent ideal sheaf $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_ j}$ pulling back to $\mathcal{I}_ j$, see Limits, Lemma 32.10.3. Denoting $U_{i, j} \subset X_ i$ the complementary opens, we may assume these are affine for all $i, j$, see Limits, Lemma 32.4.13. Similarly, we may assume the maps $\mathcal{E}_ j \to \mathcal{I}_ j$ are the pullbacks of surjections $\mathcal{E}_{i, j} \to \mathcal{I}_{i, j}$ with $\mathcal{E}_{i, j}$ finite locally free on $X_ i$, see Limits, Lemmas 32.10.3 and 32.10.2. Using this and Lemma 36.36.4 we reduce to the case of a Noetherian scheme.

Assume $X$ is Noetherian. For every coherent module $\mathcal{F}$ we can choose a finite list of sections $s_{jk} \in \mathcal{F}(U_ j)$, $k = 1, \ldots , e_ j$ which generate the restriction of $\mathcal{F}$ to $U_ j$. By Cohomology of Schemes, Lemma 30.10.4 we can extend $s_{jk}$ to a map $s'_{jk} : \mathcal{I}_ i^{n_{jk}} \to \mathcal{F}$ for some $n_{jk} \geq 1$. Then we can consider the compositions

\[ \mathcal{E}_ j^{\otimes n_{jk}} \to \mathcal{I}_ j^{n_{jk}} \to \mathcal{F} \]

to conclude. $\square$

Lemma 36.36.6. Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has the resolution property.

Proof. Observe that $X$ is a finite disjoint union of integral schemes (Properties, Lemmas 28.9.4 and 28.7.6). Thus we may assume that $X$ is integral as well as Noetherian, regular, and having affine diagonal. Choose an affine open covering $X = U_1 \cup \ldots \cup U_ m$. We may and do assume $U_ j$ nonempty for all $j$. By More on Algebra, Lemma 15.118.2 the local rings of $X$ are UFDs and hence by Divisors, Lemma 31.16.7 we can find an effective Cartier divisors $D_ j \subset X$ whose complement is $U_ j$. Then the ideal sheaf of $D_ j$ is invertible, hence a finite locally free module and we conclude that $X$ has the resolution property by Lemma 36.36.5. $\square$

Lemma 36.36.7. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $X$ has the resolution property if and only if $X_ i$ has the resolution properties for some $i$.

Proof. If $X_ i$ has the resolution property, then $X$ does by Lemma 36.36.4. Assume $X$ has the resolution property. Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$. For each $j$ choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}_ j \subset \mathcal{O}_ X$ such that $X \setminus V(\mathcal{I}_ j) = U_ j$, see Properties, Lemma 28.24.1. Choose finite locally free $\mathcal{O}_ X$-modules and surjections $\mathcal{E}_ j \to \mathcal{I}_ j$. By Limits, Lemmas 32.10.3 and 32.10.2 we can find an $i$ and finite locally free $\mathcal{O}_{X_ i}$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i}$ whose base changes to $X$ recover the maps $\mathcal{E}_ j \to \mathcal{I}_ j$, $j = 1, \ldots , m$. Denote $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_ i}$ the image of these maps. Set $U_{i, j} = X_ i \setminus V(\mathcal{I}_{i, j})$. After increasing $i$ we may assume $U_{i, j}$ is affine, see Limits, Lemma 32.4.13. Then we conclude that $X_ i$ has the resolution property by Lemma 36.36.5. $\square$

reference

Lemma 36.36.8. Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then $X$ has affine diagonal.

Proof. Combining Limits, Proposition 32.5.4 and Lemma 36.36.7 this reduces to the case where $X$ is Noetherian (small detail omitted). Assume $X$ is Noetherian. Recall that $X \times X$ is covered by the affine opens $U \times V$ for affine opens $U$, $V$ of $X$, see Schemes, Section 26.17. Hence to show that the diagonal $\Delta : X \to X \times X$ is affine, it suffices to show that $U \cap V = \Delta ^{-1}(U \times V)$ is affine for all affine opens $U$, $V$ of $X$, see Morphisms, Lemma 29.11.3. In particular, it suffices to show that the inclusion morphism $j : U \to X$ is affine if $U$ is an affine open of $X$. By Cohomology of Schemes, Lemma 30.3.4 it suffices to show that $R^1j_*\mathcal{G} = 0$ for any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{G}$. By Proposition 36.8.3 (this is where we use that we've reduced to the Noetherian case) we can represent $Rj_*\mathcal{G}$ by a complex $\mathcal{H}^\bullet $ of quasi-coherent $\mathcal{O}_ X$-modules. Assume

\[ H^1(\mathcal{H}^\bullet ) = \mathop{\mathrm{Ker}}(\mathcal{H}^1 \to \mathcal{H}^2)/\mathop{\mathrm{Im}}(\mathcal{H}^0 \to \mathcal{H}^1) \]

is nonzero in order to get a contradiction. Then we can find a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and a map

\[ \mathcal{F} \longrightarrow \mathop{\mathrm{Ker}}(\mathcal{H}^1 \to \mathcal{H}^2) \]

such that the composition with the projection onto $H^1(\mathcal{H}^\bullet )$ is nonzero. Namely, we can write $\mathop{\mathrm{Ker}}(\mathcal{H}^1 \to \mathcal{H}^2)$ as the filtered union of its coherent submodules by Properties, Lemma 28.22.3 and then one of these will do the job. Next, we choose a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and a surjection $\mathcal{E} \to \mathcal{F}$ using the resolution property of $X$. This produces a map in the derived category

\[ \mathcal{E}[-1] \longrightarrow Rj_*\mathcal{G} \]

which is nonzero on cohomology sheaves and hence nonzero in $D(\mathcal{O}_ X)$. By adjunction, this is the same thing as a map

\[ j^*\mathcal{E}[-1] \to \mathcal{G} \]

nonzero in $D(\mathcal{O}_ U)$. Since $\mathcal{E}$ is finite locally free this is the same thing as a nonzero element of

\[ H^1(U, j^*\mathcal{E}^\vee \otimes _{\mathcal{O}_ U} \mathcal{G}) \]

where $\mathcal{E}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X)$ is the dual finite locally free module. However, this group is zero by Cohomology of Schemes, Lemma 30.2.2 which is the desired contradiction. (If in doubt about the step using duals, please see the more general Cohomology, Lemma 20.47.5.) $\square$


Comments (2)

Comment #5478 by Nick Addington on

Typo in the second paragraph: scarecely -> scarcely.


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