The Stacks project

Proof. Denote $\omega _{Y/X}$ the coherent $\mathcal{O}_ Y$-module such that there is an isomorphism

of $\pi _*\mathcal{O}_ Y$-modules, see Morphisms of Spaces, Lemma 67.20.10 and Descent on Spaces, Lemma 74.6.6. The canonical map $\mathcal{O}_ X \to \pi _*\mathcal{O}_ Y$ produces a canonical map

Since $V$ is Noetherian affine we may choose sections

generating the coherent module $\pi ^*\mathcal{F} \otimes _{\mathcal{O}_ X} \omega _{Y/X}$ over $V$. By Cohomology of Spaces, Lemma 69.13.4 we can choose integers $n_ i \geq 0$ such that $s_ i$ extends to a map $s_ i' : \mathcal{I}^{n_ i} \to \pi ^*\mathcal{F} \otimes _{\mathcal{O}_ Y} \omega _{Y/X}$. Pushing to $X$ we obtain maps

where the equality sign is Cohomology of Spaces, Lemma 69.4.3. To finish the proof we will show that the sum of these maps is surjective.

Let $x \in |X|$ be a point of $X$. Let $v \in |V|$ be a point mapping to $x$. We may choose an étale neighbourhood $(U, u) \to (X, x)$ such that

(disjoint union of algebraic spaces) such that $W \to U$ is an isomorphism and such that the unique point $w \in W$ lying over $u$ maps to $v$ in $V \subset Y$. To see this is true use Lemma 76.33.2 and Étale Morphisms, Lemma 41.18.1. After shrinking $U$ further if necessary we may assume $W$ maps into $V \subset Y$ by the projection. Since the formation of $\omega _{Y/X}$ commutes with étale localization we see that

We have $(\pi |_ W)_*\omega _{W/U} = \mathcal{O}_ U$ and this isomorphism is given by the trace map $\text{Tr}_\pi |_ U$ restricted to the first summand in the decomposition above. Since $W$ maps into $V$ we see that $\mathcal{I}^{n_ i}|_ W = \mathcal{O}_ W$. Hence

Chasing diagrams the reader sees (details omitted) that $\sigma _ i|_ U$ on the summand $\mathcal{O}_ U$ is the map $\mathcal{O}_ U \to \mathcal{F}$ corresponding to the section

Since the sections $s_ i$ generate the module $\pi ^*\mathcal{F} \otimes _{\mathcal{O}_ Y} \omega _{Y/X}$ over $V$ and since $W$ maps into $V$ we conclude that the restriction of $\bigoplus \sigma _ i$ to $U$ is surjective. Since $x$ was an arbitrary point the proof is complete. $\square$

Proof. The module $\pi _*\mathcal{I}$ is coherent by Cohomology of Spaces, Lemma 69.12.9. Hence if $X$ has the resolution property then $\pi _*\mathcal{I}$ is the quotient of a finite locally free $\mathcal{O}_ X$-module. Conversely, assume given a surjection $\mathcal{E} \to \pi _*\mathcal{I}$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. Observe that for all $n \geq 1$ there is a surjection

Hence $\mathcal{E}^{\otimes n}$ surjects onto $\pi _*\mathcal{I}^ n$ for all $n \geq 1$. We conclude that $X$ has the resolution property if we combine this with the result of Lemma 76.56.2. $\square$

Proof. The pushforward $\pi _*\mathcal{G}$ of a finite type quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{G}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module by Descent on Spaces, Lemma 74.6.6. In particular, if $X$ has the resolution property, then $\pi _*\mathcal{I}$ is the quotient of a finite locally free $\mathcal{O}_ X$-module by Derived Categories of Spaces, Definition 75.28.1.

Assume that we have a surjection $\mathcal{E} \to \pi _*\mathcal{I}$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. In the rest of the proof we show that $X$ has the resolution property by reducing to the Noetherian case handled in Lemma 76.56.3. We suggest the reader skip the rest of the proof.

A first reduction is that we may view $X$ as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 65.16.2. (This doesn't affect the conditions nor the conclusion of the lemma.)

By Limits of Spaces, Lemma 70.11.3 we can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ with $Y_ i$ finite and of finite presentation over $X$ and where the transition maps are closed immersions. Consider the closed subspace $Z = V(\mathcal{I})$ of $Y$. Since $\mathcal{I}$ is of finite type, the morphism $Z \to Y$ is of finite presentation. Hence we can find an $i$ and a morphism $Z_ i \to Y_ i$ of finite presentation whose base change to $Y$ is $Z \to Y$, see Limits of Spaces, Lemma 70.7.1. For $i' \geq i$ denote $Z_{i'} = Z_ i \times _{Y_ i} Y_{i'}$. After increasing $i$ we may assume $Z_ i \to Y_ i$ is a closed immersion (of finite presentation), see Limits of Spaces, Lemma 70.6.8. Denote $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$ the ideal sheaf of $Z_ i$ and denote $\pi _ i : Y_ i \to X$ the structure morphism. Similarly for $i' \geq i$. Since $Z = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} Z_{i'}$ we have

The transition maps in the system are all surjective as follows from the surjectivity of the maps $\pi _{i, *}\mathcal{O}_{Y_ i} \to \pi _{i', *}\mathcal{O}_{Y_{i'}}$ and the fact that $Z_{i'} = Z_ i \times _{Y_ i} Y_{i'}$. By Cohomology of Spaces, Lemma 69.5.3 for some $i' \geq i$ the map $\mathcal{E} \to \pi _*\mathcal{I}$ lifts to a map $\mathcal{E} \to \pi _{i', *}\mathcal{I}_{i'}$. After increasing $i'$ this map $\mathcal{E} \to \pi _{i', *}\mathcal{I}_{i'}$ becomes surjective (since if not the colimit of the cokernels, having surjective transition maps, is nonzero). This reduces us to the case discussed in the next paragraph.

Assume $X$ is an algebraic space over $\mathbf{Z}$ and that $Y \to X$ is of finite presentation. By absolute Noetherian approximation we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit, where each $X_ i$ is a quasi-separated algebraic space of finite type over $\mathbf{Z}$ and the transition morphisms are affine, see Limits of Spaces, Proposition 70.8.1. Since $\pi : Y \to X$ is of finite presentation we can find an $i$ and a morphism $\pi _ i : Y_ i \to X_ i$ of finite presentation whose base change to $X$ is $\pi $, see Limits of Spaces, Lemma 70.7.1. After increasing $i$ we may assume $\pi _ i$ is finite, see Limits of Spaces, Lemma 70.6.7. Next, we may assume there exists a finite locally free $\mathcal{O}_{X_ i}$-module $\mathcal{E}_ i$ whose pullback to $X$ is $\mathcal{E}$, see Limits of Spaces, Lemma 70.7.3. We may also assume there is a map $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ whose pullback to $X$ is the composition $\mathcal{E} \to \pi _*\mathcal{I} \to \pi _*\mathcal{O}_ Y$, see Limits of Spaces, Lemma 70.7.2. The cokernel

is a coherent $\mathcal{O}_{Y_ i}$-module whose pullback to $X$ is the (finitely presented) cokernel $\mathcal{Q}$ of the map $\mathcal{E} \to \pi _*\mathcal{O}_ Y$. In other words, we have $\mathcal{Q} = \pi _*(\mathcal{O}_ Y/\mathcal{I})$. Consider the map

where the second arrow is given by the algebra structure on $\pi _{i, *}\mathcal{O}_{Y_ i}$. The pullback of this map to $Y$ is zero because the image of $\mathcal{E} \to \pi _*\mathcal{O}_ Y$ is the ideal $\pi _*\mathcal{I}$. Hence by Limits of Spaces, Lemma 70.7.2 after increasing $i$ we may assume the displayed composition is zero. This exactly means that the imag of $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ is of the form $\pi _{i, *}\mathcal{I}_ i$ for some coherent ideal sheaf $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$. Since $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ pulls back to $\mathcal{E} \to \pi _*\mathcal{O}_ Y$ we see that the pullback of $\mathcal{I}_ i$ to $Y$ generates $\mathcal{I}$. Denote $V_ i \subset Y_ i$ the open subspace whose complement is $V(\mathcal{I}_ i) \subset Y_ i$. Then $V$ is the inverse image of $V_ i$ by the comments above. After increasing $i$ we may assume that $V_ i$ is affine and that $\pi _ i|_{V_ i} : V_ i \to X_ i$ is étale, see Limits of Spaces, Lemmas 70.5.10 and 70.6.2. Having said all of this, we may apply Lemma 76.56.3 to conclude that $X_ i$ has the resolution property. Since $X \to X_ i$ is affine we conclude that $X$ has the resolution property too by Derived Categories of Spaces, Lemma 75.28.3. $\square$

Proof. If $X_ i$ has the resolution property, then $X$ does by Derived Categories of Spaces, Lemma 75.28.3. Assume $X$ has the resolution property. Choose $i \in I$. We may choose an affine scheme $V_ i$ and a surjective étale morphism $V_ i \to X_ i$ (Properties of Spaces, Lemma 66.6.3). We may choose an embedding $j : V_ i \to Y_ i$ with $Y_ i$ finite and finitely presented over $X_ i$ (Lemma 76.34.4). We may choose a finite type quasi-coherent ideal $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$ such that $V_ i = Y_ i \setminus V(\mathcal{I}_ i)$ (Limits of Spaces, Lemma 70.14.1). Denote $V \to Y \to X$ the base changes of $V_ i \to Y_ i \to X_ i$ to $X$. Denote $\mathcal{I} \subset \mathcal{O}_ Y$ the pullback of the ideal $\mathcal{I}_ i$. By the easy direction of Lemma 76.56.4 there exists a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and a surjection $\mathcal{E} \to \pi _*\mathcal{I}$. Note that since $\pi _ i : Y_ i \to X_ i$ is finite and of finite presentation we also have that $\pi : Y \to X$ is finite and of finite presentation and that the $\mathcal{O}_{X_ i}$-modules $\pi _{i, *}\mathcal{O}_{Y_ i}$ and $\pi _{i, *}(\mathcal{O}_{Y_ i}/\mathcal{I}_ i)$ are of finite presentation and pullback to $X$ to give $\pi _*\mathcal{O}_ Y$ and $\pi _*(\mathcal{O}_ Y/\mathcal{I})$. Thus by Limits of Spaces, Lemma 70.7.2 after increasing $i$ we can find a finite locally free $\mathcal{O}_{X_ i}$-module $\mathcal{E}_ i$ and a map $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ whose base change to $X$ recovers the composition $\mathcal{E} \to \pi _*\mathcal{I} \to \pi _*\mathcal{O}_ Y$. The pullbacks of the finitely presented $\mathcal{O}_{X_ i}$-modules $\mathop{\mathrm{Coker}}(\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i})$ and $\pi _{i, *}(\mathcal{O}_{Y_ i}/\mathcal{I}_ i)$ to $X$ agree as quotients of $\pi _*\mathcal{O}_ Y$. Hence by Limits of Spaces, Lemma 70.7.2 we may assume that these agree, in other words that the image of $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{X_ i}$ is equal to $\pi _{i, *}\mathcal{I}_ i$. Then we conclude that $X_ i$ has the resolution property by Lemma 76.56.4. $\square$

Proof. We could prove this as in the case of schemes, but instead we will deduce the lemma from the case of schemes. First, we may and do assume $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Next, we choose a scheme $Y$ and a surjective integral morphism $f : Y \to X$, see Decent Spaces, Lemma 68.9.2. Then $f$ is affine, hence $Y$ has the resolution property by Derived Categories of Spaces, Lemma 75.28.3. Hence by the case of schemes, the scheme $Y$ has affine diagonal, see Derived Categories of Schemes, Lemma 36.36.10. Next, we consider the commutative diagram

Observe that the right vertical arrow is integral, in particular affine. Let $W \to X \times _{\mathbf{Z}} X$ be a morphism with $W$ affine. Then we see that

is affine. On the other hand, $Y \to X$ is integral and surjective hence

is integral surjective as the base change of $Y \to X$ to $W$. We conclude that the target of this arrow is affine by Limits of Spaces, Proposition 70.15.2. It follows that $\Delta _ X$ is affine as desired. $\square$