The Stacks project

37.43 Application to the fppf topology

We can use the above étale localization techniques to prove the following result describing the fppf topology as being equal to the topology “generated by” Zariski coverings and by coverings of the form $\{ f : T \to S\} $ where $f$ is surjective finite locally free.

Lemma 37.43.1. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exist

  1. an étale covering $\{ S'_ a \to S\} $,

  2. surjective finite locally free morphisms $V_ a \to S'_ a$,

such that the fppf covering $\{ V_ a \to S\} $ refines the given covering $\{ S_ i \to S\} $.

Proof. We may assume that each $S_ i \to S$ is locally quasi-finite, see Lemma 37.21.6.

Fix a point $s \in S$. Pick an $i \in I$ and a point $s_ i \in S_ i$ mapping to $s$. Choose an elementary étale neighbourhood $(S', s) \to (S, s)$ such that there exists an open

\[ S_ i \times _ S S' \supset V \]

which contains a unique point $v \in V$ mapping to $s \in S'$ and such that $V \to S'$ is finite, see Lemma 37.36.1. Then $V \to S'$ is finite locally free, because it is finite and because $S_ i \times _ S S' \to S'$ is flat and locally of finite presentation as a base change of the morphism $S_ i \to S$, see Morphisms, Lemmas 29.20.4, 29.24.8, and 29.46.2. Hence $V \to S'$ is open, and after shrinking $S'$ we may assume that $V \to S'$ is surjective finite locally free. Since we can do this for every point of $S$ we conclude that $\{ S_ i \to S\} $ can be refined by a covering of the form $\{ V_ a \to S\} _{a \in A}$ where each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free. $\square$

Lemma 37.43.2. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exist

  1. a Zariski open covering $S = \bigcup U_ j$,

  2. surjective finite locally free morphisms $W_ j \to U_ j$,

  3. Zariski open coverings $W_ j = \bigcup _ k W_{j, k}$,

  4. surjective finite locally free morphisms $T_{j, k} \to W_{j, k}$

such that the fppf covering $\{ T_{j, k} \to S\} $ refines the given covering $\{ S_ i \to S\} $.

Proof. Let $\{ V_ a \to S\} _{a \in A}$ be the fppf covering found in Lemma 37.43.1. In other words, this covering refines $\{ S_ i \to S\} $ and each $V_ a \to S$ factors as $V_ a \to S'_ a \to S$ with $S'_ a \to S$ étale and $V_ a \to S'_ a$ surjective finite locally free.

By Remark 37.35.3 there exists a Zariski open covering $S = \bigcup U_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to U_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\} $ such that the family $\{ W_{j, k} \to S\} $ refines the étale covering $\{ S'_ a \to S\} $, i.e., for each pair $j, k$ there exists an $a(j, k)$ and a factorization $W_{j, K} \to S'_ a \to S$ of the morphism $W_{j, K} \to S$. Set $T_{j, k} = W_{j, k} \times _{S'_ a} V_ a$ and everything is clear. $\square$

Lemma 37.43.3. Let $S$ be a scheme. If $U \subset S$ is open and $V \to U$ is a surjective integral morphism, then there exists a surjective integral morphism $\overline{V} \to S$ with $\overline{V} \times _ S U$ isomorphic to $V$ as schemes over $U$.

Proof. Let $V' \to S$ be the normalization of $S$ in $U$, see Morphisms, Section 29.51. By construction $V' \to S$ is integral. By Morphisms, Lemmas 29.51.6 and 29.51.12 we see that the inverse image of $U$ in $V'$ is $V$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$

Lemma 37.43.4. Let $S$ be a quasi-compact and quasi-separated scheme. If $U \subset S$ is a quasi-compact open and $V \to U$ is a surjective finite morphism, then there exists a surjective finite morphism $\overline{V} \to S$ with $\overline{V} \times _ S U$ isomorphic to $V$ as schemes over $U$.

Proof. By Zariski's Main Theorem (Lemma 37.38.3) we can assume $V$ is a quasi-compact open in a scheme $V'$ finite over $S$. After replacing $V'$ by the scheme theoretic image of $V$ we may assume that $V$ is dense in $V'$. It follows that $V' \times _ S U = V$ because $V \to V' \times _ S U$ is closed as $V$ is finite over $U$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$

Lemma 37.43.5. Let $S$ be a scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exists a surjective integral morphism $S' \to S$ and an open covering $S' = \bigcup U'_\alpha $ such that for each $\alpha $ the morphism $U'_\alpha \to S$ factors through $S_ i \to S$ for some $i$.

Proof. Choose $S = \bigcup U_ j$, $W_ j \to U_ j$, $W_ j = \bigcup W_{j, k}$, and $T_{j, k} \to W_{j, k}$ as in Lemma 37.43.2. By Lemma 37.43.3 we can extend $W_ j \to U_ j$ to a surjective integral morphism $\overline{W}_ j \to S$. After this we can extend $T_{j, k} \to W_{j, k}$ to a surjective integral morphism $\overline{T}_{j, k} \to \overline{W}_ j$. We set $\overline{T}_ j$ equal to the product of all the schemes $\overline{T}_{j, k}$ over $\overline{W}_ j$ (Limits, Lemma 32.3.1). Then we set $S'$ equal to the product of all the schemes $\overline{T}_ j$ over $S$. If $x \in S'$, then there is a $j$ such that the image of $x$ in $S$ lies in $U_ j$. Hence there is a $k$ such that the image of $x$ under the projection $S' \to \overline{W}_ j$ lies in $W_{j, k}$. Hence under the projection $S' \to \overline{T}_ j \to \overline{T}_{j, k}$ the point $x$ ends up in $T_{j, k}$. And $T_{j, k} \to S$ factors through $S_ i$ for some $i$. Finally, the morphism $S' \to S$ is integral and surjective by Limits, Lemmas 32.3.3 and 32.3.2. $\square$

Lemma 37.43.6. Let $S$ be a quasi-compact and quasi-separated scheme. Let $\{ S_ i \to S\} _{i \in I}$ be an fppf covering. Then there exists a surjective finite morphism $S' \to S$ of finite presentation and an open covering $S' = \bigcup U'_\alpha $ such that for each $\alpha $ the morphism $U'_\alpha \to S$ factors through $S_ i \to S$ for some $i$.

Proof. Let $Y \to X$ be the integral surjective morphism found in Lemma 37.43.5. Choose a finite affine open covering $Y = \bigcup V_ j$ such that $V_ j \to X$ factors through $S_{i(j)}$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_\lambda $ with $Y_\lambda \to X$ finite and of finite presentation, see Limits, Lemma 32.7.2. For large enough $\lambda $ we can find affine opens $V_{\lambda , j} \subset Y_\lambda $ whose inverse image in $Y$ recovers $V_ j$, see Limits, Lemma 32.4.11. For even larger $\lambda $ the morphisms $V_ j \to S_{i(j)}$ over $X$ come from morphisms $V_{\lambda , j} \to S_{i(j)}$ over $X$, see Limits, Proposition 32.6.1. Setting $S' = Y_\lambda $ for this $\lambda $ finishes the proof. $\square$

Proof. Let $\{ T_ i \to T\} $ be an fppf covering of schemes, see Topologies, Definition 34.7.1. Observe that $T_ i \to T$ is locally of finite type. Let $U \subset T$ be an affine open. It suffices to show that $\{ T_ i \times _ T U \to U\} $ can be refined by a standard ph covering, see Topologies, Definition 34.8.4. This follows immediately from Lemma 37.43.6 and the fact that a finite morphism is proper (Morphisms, Lemma 29.42.11). $\square$

Remark 37.43.8. As a consequence of Lemma 37.43.7 we obtain a comparison morphism

\[ \epsilon : (\mathit{Sch}/S)_{ph} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

This is the morphism of sites given by the identity functor on underlying categories (with suitable choices of sites as in Topologies, Remark 34.11.1). The functor $\epsilon _*$ is the identity on underlying presheaves and the functor $\epsilon ^{-1}$ associated to an fppf sheaf its ph sheafification. By composition we can in addition compare the ph topology with the syntomic, smooth, étale, and Zariski topologies.


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