The Stacks project

103.14 The lisse-étale and the flat-fppf sites

In the book [LM-B] many of the results above are proved using the lisse-étale site of an algebraic stack. We define this site here. In Examples, Section 110.59 we show that the lisse-étale site isn't functorial. We also define its analogue, the flat-fppf site, which is better suited to the development of algebraic stacks as given in the Stacks project (because we use the fppf topology as our base topology). Of course the flat-fppf site isn't functorial either.

Definition 103.14.1. Let $\mathcal{X}$ be an algebraic stack.

  1. The lisse-étale site of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{lisse,{\acute{e}tale}}$1 of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is smooth. A covering of $\mathcal{X}_{lisse,{\acute{e}tale}}$ is a family of morphisms $\{ x_ i \to x\} _{i \in I}$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$ which forms a covering of $\mathcal{X}_{\acute{e}tale}$.

  2. The flat-fppf site of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{flat,fppf}$ of $\mathcal{X}$ whose objects are those $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is flat. A covering of $\mathcal{X}_{flat,fppf}$ is a family of morphisms $\{ x_ i \to x\} _{i \in I}$ of $\mathcal{X}_{flat,fppf}$ which forms a covering of $\mathcal{X}_{fppf}$.

We denote $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ the restriction of $\mathcal{O}_\mathcal {X}$ to the lisse-étale site and similarly for $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$. The relationship between the lisse-étale site and the étale site is as follows (we mainly stick to “topological” properties in this lemma).

Lemma 103.14.2. Let $\mathcal{X}$ be an algebraic stack.

  1. The inclusion functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{X}_{\acute{e}tale}$ is fully faithful, continuous and cocontinuous. It follows that

    1. there is a morphism of topoi

      \[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}) \]

      with $g^{-1}$ given by restriction,

    2. the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,

    3. the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms,

    4. the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,

    5. the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and

    6. we have $g^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$.

  2. The inclusion functor $\mathcal{X}_{flat,fppf} \to \mathcal{X}_{fppf}$ is fully faithful, continuous and cocontinuous. It follows that

    1. there is a morphism of topoi

      \[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \]

      with $g^{-1}$ given by restriction,

    2. the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,

    3. the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms,

    4. the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,

    5. the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and

    6. we have $g^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$.

Proof. In both cases it is immediate that the functor is fully faithful, continuous, and cocontinuous (see Sites, Definitions 7.13.1 and 7.20.1). Hence properties (a), (b), (c) follow from Sites, Lemmas 7.21.5 and 7.21.7. Parts (d), (e) follow from Modules on Sites, Lemmas 18.16.2 and 18.16.4. Part (f) is immediate. $\square$

Lemma 103.14.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.14.2.

  1. For an abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$ we have

    1. $H^ p(\mathcal{X}_{\acute{e}tale}, \mathcal{F}) = H^ p(\mathcal{X}_{lisse,{\acute{e}tale}}, g^{-1}\mathcal{F})$, and

    2. $H^ p(x, \mathcal{F}) = H^ p(\mathcal{X}_{lisse,{\acute{e}tale}}/x, g^{-1}\mathcal{F})$ for any object $x$ of $\mathcal{X}_{lisse,{\acute{e}tale}}$.

    The same holds for sheaves of modules.

  2. For an abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{fppf}$ we have

    1. $H^ p(\mathcal{X}_{fppf}, \mathcal{F}) = H^ p(\mathcal{X}_{flat,fppf}, g^{-1}\mathcal{F})$, and

    2. $H^ p(x, \mathcal{F}) = H^ p(\mathcal{X}_{flat,fppf}/x, g^{-1}\mathcal{F})$ for any object $x$ of $\mathcal{X}_{flat,fppf}$.

    The same holds for sheaves of modules.

Proof. Part (1)(a) follows from Sheaves on Stacks, Lemma 96.23.3 applied to the inclusion functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{X}_{\acute{e}tale}$. Part (1)(b) follows from part (1)(a). Namely, if $x$ lies over the scheme $U$, then the site $\mathcal{X}_{\acute{e}tale}/x$ is equivalent to $(\mathit{Sch}/U)_{\acute{e}tale}$ and $\mathcal{X}_{lisse,{\acute{e}tale}}$ is equivalent to $U_{lisse,{\acute{e}tale}}$. Part (2) is proved in the same manner. $\square$

Lemma 103.14.4. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.14.2.

  1. There exists a functor

    \[ g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}}) \]

    which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

  2. There exists a functor

    \[ g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \longrightarrow \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}}) \]

    which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

Proof. In both cases, the existence of the functor $g_!$ follows from Modules on Sites, Lemma 18.41.1. To see that $g_!$ agrees with the functor on abelian sheaves we will show the maps Modules on Sites, Equation (18.41.2.1) are isomorphisms.

Lisse-étale case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ smooth. Consider the induced fully faithful functor

\[ g' : \mathcal{X}_{lisse,{\acute{e}tale}}/x \longrightarrow \mathcal{X}_{\acute{e}tale}/x \]

The right hand side is identified with $(\mathit{Sch}/U)_{\acute{e}tale}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is smooth. Thus Étale Cohomology, Lemma 59.49.2 applies.

Flat-fppf case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{flat,fppf})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ flat. Consider the induced fully faithful functor

\[ g' : \mathcal{X}_{flat,fppf}/x \longrightarrow \mathcal{X}_{fppf}/x \]

The right hand side is identified with $(\mathit{Sch}/U)_{fppf}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is flat. Thus Étale Cohomology, Lemma 59.49.2 applies.

In both cases the equality $g^*g_! = \text{id}$ follows from $g^* = g^{-1}$ and the equality for abelian sheaves in Lemma 103.14.2. $\square$

Lemma 103.14.5. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemmas 103.14.2 and 103.14.4.

  1. We have $g_!\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}} = \mathcal{O}_\mathcal {X}$.

  2. We have $g_!\mathcal{O}_{\mathcal{X}_{flat, fppf}} = \mathcal{O}_\mathcal {X}$.

Proof. In this proof we write $\mathcal{C} = \mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{C} = \mathcal{X}_{fppf}$) and we denote $\mathcal{C}' = \mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{C}' = \mathcal{X}_{flat, fppf}$). Then $\mathcal{C}'$ is a full subcategory of $\mathcal{C}$. In this proof we will think of objects $V$ of $\mathcal{C}$ as schemes over $\mathcal{X}$ and objects $U$ of $\mathcal{C}'$ as schemes smooth (resp. flat) over $\mathcal{X}$. Finally, we write $\mathcal{O} = \mathcal{O}_\mathcal {X}$ and $\mathcal{O}' = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O}' = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). In the notation above we have $\mathcal{O}(V) = \Gamma (V, \mathcal{O}_ V)$ and $\mathcal{O}'(U) = \Gamma (U, \mathcal{O}_ U)$. Consider the $\mathcal{O}$-module homomorphism $g_!\mathcal{O}' \to \mathcal{O}$ adjoint to the identification $\mathcal{O}' = g^{-1}\mathcal{O}$.

Recall that $g_!\mathcal{O}'$ is the sheaf associated to the presheaf $g_{p!}\mathcal{O}'$ given by the rule

\[ V \longmapsto \mathop{\mathrm{colim}}\nolimits _{V \to U} \mathcal{O}'(U) \]

where the colimit is taken in the category of abelian groups (Modules on Sites, Definition 18.16.1). Below we will use frequently that if

\[ V \to U \to U' \]

are morphisms and if $f' \in \mathcal{O}'(U')$ restricts to $f \in \mathcal{O}'(U)$, then $(V \to U, f)$ and $(V \to U', f')$ define the same element of the colimit. Also, $g_!\mathcal{O}' \to \mathcal{O}$ maps the element $(V \to U, f)$ simply to the pullback of $f$ to $V$.

Let us prove that $g_!\mathcal{O}' \to \mathcal{O}$ is surjective. Let $h \in \mathcal{O}(V)$ for some object $V$ of $\mathcal{C}$. It suffices to show that $h$ is locally in the image. Choose an object $U$ of $\mathcal{C}'$ corresponding to a surjective smooth morphism $U \to \mathcal{X}$. Since $U \times _\mathcal {X} V \to V$ is surjective smooth, after replacing $V$ by the members of an étale covering of $V$ we may assume there exists a morphism $V \to U$, see Topologies on Spaces, Lemma 73.4.4. Using $h$ we obtain a morphism $V \to U \times \mathbf{A}^1$ such that writing $\mathbf{A}^1 = \mathop{\mathrm{Spec}}(\mathbf{Z}[t])$ the element $t \in \mathcal{O}(U \times \mathbf{A}^1)$ pulls back to $h$. Since $U \times \mathbf{A}^1$ is an object of $\mathcal{C}'$ we see that $(V \to U \times \mathbf{A}^1, t)$ is an element of the colimit above which maps to $h \in \mathcal{O}(V)$ as desired.

Suppose that $s \in g_!\mathcal{O}'(V)$ is a section mapping to zero in $\mathcal{O}(V)$. To finish the proof we have to show that $s$ is zero. After replacing $V$ by the members of a covering we may assume $s$ is an element of the colimit

\[ \mathop{\mathrm{colim}}\nolimits _{V \to U} \mathcal{O}'(U) \]

Say $s = \sum (\varphi _ i, s_ i)$ is a finite sum with $\varphi _ i : V \to U_ i$, $U_ i$ smooth (resp. flat) over $\mathcal{X}$, and $s_ i \in \Gamma (U_ i, \mathcal{O}_{U_ i})$. Choose a scheme $W$ surjective étale over the algebraic space $U = U_1 \times _\mathcal {X} \ldots \times _\mathcal {X} U_ n$. Note that $W$ is still smooth (resp. flat) over $\mathcal{X}$, i.e., defines an object of $\mathcal{C}'$. The fibre product

\[ V' = V \times _{(\varphi _1, \ldots , \varphi _ n), U} W \]

is surjective étale over $V$, hence it suffices to show that $s$ maps to zero in $g_!\mathcal{O}'(V')$. Note that the restriction $\sum (\varphi _ i, s_ i)|_{V'}$ corresponds to the sum of the pullbacks of the functions $s_ i$ to $W$. In other words, we have reduced to the case of $(\varphi , s)$ where $\varphi : V \to U$ is a morphism with $U$ in $\mathcal{C}'$ and $s \in \mathcal{O}'(U)$ restricts to zero in $\mathcal{O}(V)$. By the commutative diagram

\[ \xymatrix{ V \ar[rr]_-{(\varphi , 0)} \ar[rrd]_\varphi & & U \times \mathbf{A}^1 \\ & & U \ar[u]_{(\text{id}, 0)} } \]

we see that $((\varphi , 0) : V \to U \times \mathbf{A}^1, \text{pr}_2^*x)$ represents zero in the colimit above. Hence we may replace $U$ by $U \times \mathbf{A}^1$, $\varphi $ by $(\varphi , 0)$ and $s$ by $\text{pr}_1^*s + \text{pr}_2^*x$. Thus we may assume that the vanishing locus $Z : s = 0$ in $U$ of $s$ is smooth (resp. flat) over $\mathcal{X}$. Then we see that $(V \to Z, 0)$ and $(\varphi , s)$ have the same value in the colimit, i.e., we see that the element $s$ is zero as desired. $\square$

The lisse-étale and the flat-fppf sites can be used to characterize parasitic modules as follows.

Lemma 103.14.6. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module with the flat base change property on $\mathcal{X}_{\acute{e}tale}$. The following are equivalent

    1. $\mathcal{F}$ is parasitic, and

    2. $g^*\mathcal{F} = 0$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ is as in Lemma 103.14.2.

  2. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module on $\mathcal{X}_{fppf}$. The following are equivalent

    1. $\mathcal{F}$ is parasitic, and

    2. $g^*\mathcal{F} = 0$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is as in Lemma 103.14.2.

Proof. Part (2) is immediate from the definitions (this is one of the advantages of the flat-fppf site over the lisse-étale site). The implication (1)(a) $\Rightarrow $ (1)(b) is immediate as well. To see (1)(b) $\Rightarrow $ (1)(a) let $U$ be a scheme and let $x : U \to \mathcal{X}$ be a surjective smooth morphism. Then $x$ is an object of the lisse-étale site of $\mathcal{X}$. Hence we see that (1)(b) implies that $\mathcal{F}|_{U_{\acute{e}tale}} = 0$. Let $V \to \mathcal{X}$ be an flat morphism where $V$ is a scheme. Set $W = U \times _\mathcal {X} V$ and consider the diagram

\[ \xymatrix{ W \ar[d]_ p \ar[r]_ q & V \ar[d] \\ U \ar[r] & \mathcal{X} } \]

Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_{\acute{e}tale}} \cong q_{small}^*\mathcal{F}|_{V_{\acute{e}tale}}$. Thus if $\mathcal{F}$ is in the kernel of $g^*$, then $\mathcal{F}|_{V_{\acute{e}tale}} = 0$ as desired. $\square$

[1] In the literature the site is denoted $\text{Lis-ét}(\mathcal{X})$ or $\text{Lis-Et}(\mathcal{X})$ and the associated topos is denoted $\mathcal{X}_{\text{lis-é}t}$ or $\mathcal{X}_{\text{lis-et}}$. In the Stacks project our convention is to name the site and denote the corresponding topos by $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

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