The Stacks project

Proof. Let $U \subset X$ be affine open. Looking at Topologies, Definitions 34.8.4 and 34.10.7 it suffices to show that the base change $\{ X_ i \times _ X U \to U\} $ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Thus we may assume $X$ is affine and we have to show $\{ X_ i \to X\} $ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Since a standard ph covering is a standard V covering, see Topologies, Lemma 34.10.3 it suffices to prove the other implication.

Assume $X$ is affine and assume $\{ f_ i : X_ i \to X\} _{i \in I}$ can be refined by a standard V covering $\{ g_ j : Y_ j \to X\} _{j = 1, \ldots , m}$. For each $j$ choose an $i_ j$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ such that $g_ j = f_{i_ j} \circ h_ j$. Since $Y_ j$ is affine hence quasi-compact, for each $j$ we can find finitely many affine opens $U_{j, k} \subset X_{i_ j}$ such that $\mathop{\mathrm{Im}}(h_ j) \subset \bigcup U_{j, k}$. Then $\{ U_{j, k} \to X\} _{j, k}$ refines $\{ X_ i \to X\} $ and is a standard V covering (as it is a finite family of morphisms of affines and it inherits the lifting property for valuation rings from the corresponding property of $\{ Y_ j \to X\} $). Thus we reduce to the case discussed in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation. We have to show that $\{ X_ i \to X\} $ can be refined by a standard ph covering. Choose a generic flatness stratification

as in More on Morphisms, Lemma 37.54.2 for the finitely presented morphism

of affines. We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_ i$ to $U_ k = S_ k \setminus S_{k + 1}$ is flat. Denote $Y_ k$ the scheme theoretic closure of $U_ k$ in $S_ k$. Since $U_ k \to S_ k$ is a quasi-compact open immersion (see Properties, Lemma 28.24.1), we see that $U_ k \subset Y_ k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma 29.6.3. The morphism $\coprod _{k = 0, \ldots , t - 1} Y_ k \to X$ is finite surjective, hence $\{ Y_ k \to X\} $ is a standard ph covering and hence a standard V covering (see above). By the transitivity property of standard V coverings (Topologies, Lemma 34.10.5) it suffices to show that the pullback of the covering $\{ X_ i \to X\} $ to each $Y_ k$ can be refined by a standard V covering. This reduces us to the case described in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation and there is a dense quasi-compact open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat. By Theorem 38.30.7 there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_ i : X'_ i \to X'$ of $f_ i$ is flat. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is scheme theoretically dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a valuation ring $V$ and a morphism $g : \mathop{\mathrm{Spec}}(V) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(V)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(V)$ maps to $x$, see Morphisms, Lemma 29.6.5. Since $\{ X_ i \to X\} $ is a standard V covering, we can choose an extension of valuation rings $V \subset W$, an index $i$, and a morphism $\mathop{\mathrm{Spec}}(W) \to X_ i$ such that the diagram

is commutative. Since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme containing the open $U \times _ X X_ i$, since $\mathop{\mathrm{Spec}}(W)$ is an integral scheme, and since the induced morphism $h : \mathop{\mathrm{Spec}}(W) \to X' \times _ X X_ i$ maps the generic point of $\mathop{\mathrm{Spec}}(W)$ into $U \times _ X X_ i$, we conclude that $h$ factors through the closed subscheme $X'_ i \subset X' \times _ X X_ i$. We conclude that $\{ f'_ i : X'_ i \to X'\} $ is a V covering. In particular, $\coprod f'_ i$ is surjective. In particular $\{ X'_ i \to X'\} $ is an fppf covering. Since an fppf covering is a ph covering (More on Morphisms, Lemma 37.48.7), we can find a standard ph covering $\{ Y_ j \to X'\} $ refining $\{ X'_ i \to X\} $. Say this covering is given by a proper surjective morphism $Y \to X'$ and a finite affine open covering $Y = \bigcup Y_ j$. Then the composition $Y \to X$ is proper surjective and we conclude that $\{ Y_ j \to X\} $ is a standard ph covering. This finishes the proof. $\square$

Proof. The implication (2) $\Rightarrow $ (1) follows from the more general Topologies, Lemma 34.10.14 and our definition of h covers. Assume $\coprod X_ i \to X$ is universally submersive. We will show that $\{ X_ i \to X\} $ can be refined by a ph covering; this will suffice by Topologies, Lemma 34.8.7 and our definition of h coverings. The argument will be the same as the one used in the proof of Lemma 38.34.1.

Choose a generic flatness stratification

as in More on Morphisms, Lemma 37.54.2 for the finitely presented morphism

We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_ i$ to $U_ k = S_ k \setminus S_{k + 1}$ is flat. Denote $Y_ k$ the scheme theoretic closure of $U_ k$ in $S_ k$. Since $U_ k \to S_ k$ is a quasi-compact open immersion (all schemes in this paragraph are Noetherian), we see that $U_ k \subset Y_ k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma 29.6.3. The morphism $\coprod _{k = 0, \ldots , t - 1} Y_ k \to X$ is finite surjective, hence $\{ Y_ k \to X\} $ is a ph covering. By the transitivity property of ph coverings (Topologies, Lemma 34.8.8) it suffices to show that the pullback of the covering $\{ X_ i \to X\} $ to each $Y_ k$ can be refined by a ph covering. This reduces us to the case described in the next paragraph.

Assume $\coprod X_ i \to X$ is universally submersive and there is a dense open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat for all $i$. By Theorem 38.30.7 there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_ i : X'_ i \to X'$ of $f_ i$ is flat for all $i$. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a discrete valuation ring $A$ and a morphism $g : \mathop{\mathrm{Spec}}(A) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$, see Limits, Lemma 32.15.1. Set

Since $\coprod X_ i \to X$ is universally submersive, there is a specialization $w' \leadsto w$ in $W$ such that $w'$ maps to the generic point of $\mathop{\mathrm{Spec}}(A)$ and $w$ maps to the closed point of $\mathop{\mathrm{Spec}}(A)$. (If not, then the closed fibre of $W \to \mathop{\mathrm{Spec}}(A)$ is stable under generalizations, hence open, which contradicts the fact that $W \to \mathop{\mathrm{Spec}}(A)$ is submersive.) Say $w' \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ so of course $w \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ as well. Let $x'_ i \leadsto x_ i$ be the image of $w' \leadsto w$ in $X' \times _ X X_ i$. Since $x'_ i \in X'_ i$ and since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme we see that $x_ i \in X'_ i$. Since $x_ i$ maps to $x \in X'$ we conclude that $\coprod X'_ i \to X'$ is surjective! In particular $\{ X'_ i \to X'\} $ is an fppf covering. But an fppf covering is a ph covering (More on Morphisms, Lemma 37.48.7). Since $X' \to X$ is proper surjective, we conclude that $\{ X'_ i \to X\} $ is a ph covering and the proof is complete. $\square$

Proof. By Definition 38.34.2 a h covering is a ph covering. Conversely, if $\{ f_ i : X_ i \to X\} $ is a ph covering, then the morphisms $f_ i$ are locally of finite type (Topologies, Definition 34.8.4). Since $X$ is locally Noetherian, each $f_ i$ is locally of finite presentation and we see that we have a h covering by definition. $\square$

Proof. By assumption there exists a proper surjective morphism $Y \to X$ and a finite affine open covering $Y = \bigcup _{j = 1, \ldots , m} Y_ j$ such that $\{ Y_ j \to X\} _{j = 1, \ldots , m}$ refines $\{ X_ i \to X\} _{i \in I}$. This means that for each $j$ there is an index $i_ j \in I$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ over $X$. See Definition 38.34.2 and Topologies, Definition 34.8.4. The problem is that we don't know that $Y \to X$ is of finite presentation. By Limits, Lemma 32.13.2 we can write

as a directed limit of schemes $Y_\lambda $ proper and of finite presentation over $X$ such that the morphisms $Y \to Y_\lambda $ and the the transition morphisms are closed immersions. Observe that each $Y_\lambda \to X$ is surjective. By Limits, Lemma 32.4.11 we can find a $\lambda $ and quasi-compact opens $Y_{\lambda , j} \subset Y_\lambda $, $j = 1, \ldots , m$ covering $Y_\lambda $ and restricting to $Y_ j$ in $Y$. Then $Y_ j = \mathop{\mathrm{lim}}\nolimits Y_{\lambda , j}$. After increasing $\lambda $ we may assume $Y_{\lambda , j}$ is affine for all $j$, see Limits, Lemma 32.4.13. Finally, since $X_ i \to X$ is locally of finite presentation we can use the functorial characterization of morphisms which are locally of finite presentation (Limits, Proposition 32.6.1) to find a $\lambda $ such that for each $j$ there is a morphism $h_{\lambda , j} : Y_{\lambda , j} \to X_{i_ j}$ whose restriction to $Y_ j$ is the morphism $h_ j$ chosen above. Thus $\{ Y_{\lambda , j} \to X\} $ refines $\{ X_ i \to X\} $ and the proof is complete. $\square$

Proof. This is true because in an fppf covering the morphisms are required to be locally of finite presentation and because fppf coverings are ph covering, see More on Morphisms, Lemma 37.48.7. The second statement follows from the first and Topologies, Lemma 34.7.2. $\square$

Proof. Combine Topologies, Lemmas 34.10.10 and 34.8.6. $\square$

Proof. This follows from the analogous statement for ph coverings (Topologies, Lemma 34.8.7) or from the analogous statement for V coverings (Topologies, Lemma 34.10.8). $\square$

Proof. Follows immediately from the corresponding statement for either ph or V coverings (Topologies, Lemma 34.8.8 or 34.10.9) and the fact that the class of morphisms which are locally of finite presentation is preserved under base change and composition. $\square$

Proof. Omitted. Hint: this is exactly the same as the proof of Topologies, Lemma 34.8.10. $\square$

Proof. Reasoning as in the proof of Topologies, Lemma 34.4.9 it suffices to show that the collection of standard h coverings satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard h covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ a standard h covering $\{ T_{ij} \to T_ i\} _{j \in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a h covering (Lemma 38.34.9), $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. Thus $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard h covering. $\square$

Proof. For $\mathit{Sch}_ h$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. The fibre product $V \times _ U W$ in $\mathit{Sch}_ h$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_ h$. This proves the result for $(\mathit{Sch}/S)_ h$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_ h$. $\square$

Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_ h \to (\mathit{Sch}/S)_ h$. Being cocontinuous follows because any h covering of $T/S$, $T$ affine, can be refined by a standard h covering for example by Lemma 38.34.5. Hence (1) holds. We see $u$ is continuous simply because a standard h covering is a h covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering (which is a h covering). $\square$

Proof. We will show that if (1) and (2) hold, then $\mathcal{F}$ is sheaf. Let $\{ T_ i \to T\} $ be a covering in $(\mathit{Sch}/S)_ h$. We will verify the sheaf condition for this covering. Let $s_ i \in \mathcal{F}(T_ i)$ be sections which restrict to the same section over $T_ i \times _ T T_{i'}$. We will show that there exists a unique section $s \in \mathcal{F}(T)$ restricting to $s_ i$ over $T_ i$. Let $T = \bigcup U_ j$ be an affine open covering. By property (1) it suffices to produce sections $s_ j \in \mathcal{F}(U_ j)$ which agree on $U_ j \cap U_{j'}$ in order to produce $s$. Consider the coverings $\{ T_ i \times _ T U_ j \to U_ j\} $. Then $s_{ji} = s_ i|_{T_ i \times _ T U_ j}$ are sections agreeing over $(T_ i \times _ T U_ j) \times _{U_ j} (T_{i'} \times _ T U_ j)$. Choose a proper surjective morphism $V_ j \to U_ j$ of finite presentation and a finite affine open covering $V_ j = \bigcup V_{jk}$ such that $\{ V_{jk} \to U_ j\} $ refines $\{ T_ i \times _ T U_ j \to U_ j\} $. See Lemma 38.34.5. If $s_{jk} \in \mathcal{F}(V_{jk})$ denotes the pullback of $s_{ji}$ to $V_{jk}$ by the implied morphisms, then we find that $s_{jk}$ glue to a section $s'_ j \in \mathcal{F}(V_ j)$. Using the agreement on overlaps once more, we find that $s'_ j$ is in the equalizer of the two maps $\mathcal{F}(V_ j) \to \mathcal{F}(V_ j \times _{U_ j} V_ j)$. Hence by (2) we find that $s'_ j$ comes from a unique section $s_ j \in \mathcal{F}(U_ j)$. We omit the verification that these sections $s_ j$ have all the desired properties.

Proof of the equivalence of (2) and (2') in the presence of (1). Suppose $V \to U$ is a morphism of $(\mathit{Sch}/S)_ h$ which is proper, surjective, and of finite presentation. Choose an affine open covering $U = \bigcup U_ i$ and set $V_ i = V \times _ U U_ i$. Then we see that $\mathcal{F}(U) \to \mathcal{F}(V)$ is injective because we know $\mathcal{F}(U_ i) \to \mathcal{F}(V_ i)$ is injective by (2') and we know $\mathcal{F}(U) \to \prod \mathcal{F}(U_ i)$ is injective by (1). Finally, suppose that we are given an $t \in \mathcal{F}(V)$ in the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times _ U V)$. Then $t|_{V_ i}$ is in the equalizer of the two maps $\mathcal{F}(V_ i) \to \mathcal{F}(V_ i \times _{U_ i} V_ i)$ for all $i$. Hence we obtain a unique section $s_ i \in \mathcal{F}(U_ i)$ mapping to $t|_{V_ i}$ for all $i$ by (2'). We omit the verification that $s_ i|_{U_ i \cap U_ j} = s_ j|_{U_ i \cap U_ j}$ for all $i, j$; this uses the uniqueness property just shown. By the sheaf property for the covering $U = \bigcup U_ i$ we obtain a section $s \in \mathcal{F}(U)$. We omit the proof that $s$ maps to $t$ in $\mathcal{F}(V)$. $\square$

Proof. The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\mathop{\mathrm{Mor}}\nolimits _ S(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$

Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 38.34.18. $\square$