101.11 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 108.57 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 101.11.1. 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 101.11.2. 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 101.11.3. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 101.11.2.

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 58.49.2 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 58.49.2 applies.

In both cases the equality $g^*g_! = \text{id}$ follows from $g^* = g^{-1}$ and the equality for abelian sheaves in Lemma 101.11.2. $\square$

Lemma 101.11.4. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemmas 101.11.2 and 101.11.3.

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 $V$.

To see that $g_!\mathcal{O}' \to \mathcal{O}$ is surjective it suffices to show that $1 \in \Gamma (\mathcal{C}, \mathcal{O})$ is locally in the image. Choose an object $U$ of $\mathcal{C}'$ corresponding to a surjective smooth morphism $U \to \mathcal{X}$. Then viewing $U$ both as an object of $\mathcal{C}'$ and $\mathcal{C}$ we see that $(U \to U, 1)$ is an element of the colimit above which maps to $1 \in \mathcal{O}(U)$. Since $U$ surjects onto the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ we conclude $g_!\mathcal{O}' \to \mathcal{O}$ is surjective.

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 101.11.5. 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 101.11.2.

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 101.11.2.

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. $\square$

The lisse-étale site is functorial for smooth morphisms of algebraic stacks and the flat-fppf site is functorial for flat morphisms of algebraic stacks.

Lemma 101.11.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

1. If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{Y}_{lisse,{\acute{e}tale}}$ which gives a morphism of ringed topoi fitting into the following commutative diagram

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \ar[r]_{g'} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{lisse,{\acute{e}tale}}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{\acute{e}tale}) }$

We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.

2. If $f$ is flat, then $f$ restricts to a continuous and cocontinuous functor $\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}$ which gives a morphism of ringed topoi fitting into the following commutative diagram

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \ar[r]_{g'} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{flat,fppf}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{fppf}) }$

We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.

Proof. The initial statement comes from the fact that if $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lies over a scheme $U$ such that $x : U \to \mathcal{X}$ is smooth (resp. flat) and if $f$ is smooth (resp. flat) then $f(x) : U \to \mathcal{Y}$ is smooth (resp. flat), see Morphisms of Stacks, Lemmas 99.33.2 and 99.25.2. The induced functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf} \to \mathcal{Y}_{flat,fppf}$) is continuous and cocontinuous by our definition of coverings in these categories. Finally, the commutativity of the diagram is a consequence of the fact that the horizontal morphisms are given by the inclusion functors (see Lemma 101.11.2) and Sites, Lemma 7.21.2.

To show that $f'_*(g')^{-1} = g^{-1}f_*$ let $\mathcal{F}$ be a sheaf on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$). There is a canonical pullback map

$g^{-1}f_*\mathcal{F} \longrightarrow f'_*(g')^{-1}\mathcal{F}$

see Sites, Section 7.45. We claim this map is an isomorphism. To prove this pick an object $y$ of $\mathcal{Y}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{Y}_{flat,fppf}$). Say $y$ lies over the scheme $V$ such that $y : V \to \mathcal{Y}$ is smooth (resp. flat). Since $g^{-1}$ is the restriction we find that

$\left(g^{-1}f_*\mathcal{F}\right)(y) = \Gamma (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})$

by Sheaves on Stacks, Equation (94.5.0.1). Let $(V \times _{y, \mathcal{Y}} \mathcal{X})' \subset V \times _{y, \mathcal{Y}} \mathcal{X}$ be the full subcategory consisting of objects $z : W \to V \times _{y, \mathcal{Y}} \mathcal{X}$ such that the induced morphism $W \to \mathcal{X}$ is smooth (resp. flat). Denote

$\text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \longrightarrow \mathcal{X}_{lisse,{\acute{e}tale}} \ (\text{resp. }\mathcal{X}_{flat,fppf})$

the restriction of the functor $\text{pr}$ used in the formula above. Exactly the same argument that proves Sheaves on Stacks, Equation (94.5.0.1) shows that for any sheaf $\mathcal{H}$ on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) we have

101.11.6.1
$$\label{stacks-cohomology-equation-pushforward-lisse-etale} f'_*\mathcal{H}(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H})$$

Since $(g')^{-1}$ is restriction we see that

$\left(f'_*(g')^{-1}\mathcal{F}\right)(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'})$

By Sheaves on Stacks, Lemma 94.22.3 we see that

$\Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'}) = \Gamma (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})$

are equal as desired; although we omit the verification of the assumptions of the lemma we note that the fact that $V \to \mathcal{Y}$ is smooth (resp. flat) is used to verify the second condition.

Finally, the equality $g'_!(f')^{-1} = f^{-1}g_!$ follows formally from the equality $f'_*(g')^{-1} = g^{-1}f_*$ by the adjointness of $f^{-1}$ and $f_*$, the adjointness of $g_!$ and $g^{-1}$, and their “primed” versions. $\square$

[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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