## 94.4 Sheaves

We first make an observation that is important and trivial (especially for those readers who do not worry about set theoretical issues).

Consider a big fppf site $\mathit{Sch}_{fppf}$ as in Topologies, Definition 34.7.6 and denote its underlying category $\mathit{Sch}_\alpha$. Besides being the underlying category of a fppf site, the category $\mathit{Sch}_\alpha$ can also can serve as the underlying category for a big Zariski site, a big étale site, a big smooth site, and a big syntomic site, see Topologies, Remark 34.11.1. We denote these sites $\mathit{Sch}_{Zar}$, $\mathit{Sch}_{\acute{e}tale}$, $\mathit{Sch}_{smooth}$, and $\mathit{Sch}_{syntomic}$. In this situation, since we have defined the big Zariski site $(\mathit{Sch}/S)_{Zar}$ of $S$, the big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$ of $S$, the big smooth site $(\mathit{Sch}/S)_{smooth}$ of $S$, the big syntomic site $(\mathit{Sch}/S)_{syntomic}$ of $S$, and the big fppf site $(\mathit{Sch}/S)_{fppf}$ of $S$ as the localizations (see Sites, Section 7.25) $\mathit{Sch}_{Zar}/S$, $\mathit{Sch}_{\acute{e}tale}/S$, $\mathit{Sch}_{smooth}/S$, $\mathit{Sch}_{syntomic}/S$, and $\mathit{Sch}_{fppf}/S$ of these (absolute) big sites we see that all of these have the same underlying category, namely $\mathit{Sch}_\alpha /S$.

It follows that if we have a category $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ fibred in groupoids, then $\mathcal{X}$ inherits a Zariski, étale, smooth, syntomic, and fppf topology, see Stacks, Definition 8.10.2.

Definition 94.4.1. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

1. The associated Zariski site, denoted $\mathcal{X}_{Zar}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{Zar}$.

2. The associated étale site, denoted $\mathcal{X}_{\acute{e}tale}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{\acute{e}tale}$.

3. The associated smooth site, denoted $\mathcal{X}_{smooth}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{smooth}$.

4. The associated syntomic site, denoted $\mathcal{X}_{syntomic}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{syntomic}$.

5. The associated fppf site, denoted $\mathcal{X}_{fppf}$, is the structure of site on $\mathcal{X}$ inherited from $(\mathit{Sch}/S)_{fppf}$.

This definition makes sense by the discussion above. If $\mathcal{X}$ is an algebraic stack, the literature calls $\mathcal{X}_{fppf}$ (or a site equivalent to it) the big fppf site of $\mathcal{X}$ and similarly for the other ones. We may occasionally use this terminology to distinguish this construction from others.

Remark 94.4.2. We only use this notation when the symbol $\mathcal{X}$ refers to a category fibred in groupoids, and not a scheme, an algebraic space, etc. In this way we will avoid confusion with the small étale site of a scheme, or algebraic space which is denoted $X_{\acute{e}tale}$ (in which case we use a roman capital instead of a calligraphic one).

Now that we have these topologies defined we can say what it means to have a sheaf on $\mathcal{X}$, i.e., define the corresponding topoi.

Definition 94.4.3. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.

1. We say $\mathcal{F}$ is a Zariski sheaf, or a sheaf for the Zariski topology if $\mathcal{F}$ is a sheaf on the associated Zariski site $\mathcal{X}_{Zar}$.

2. We say $\mathcal{F}$ is an étale sheaf, or a sheaf for the étale topology if $\mathcal{F}$ is a sheaf on the associated étale site $\mathcal{X}_{\acute{e}tale}$.

3. We say $\mathcal{F}$ is a smooth sheaf, or a sheaf for the smooth topology if $\mathcal{F}$ is a sheaf on the associated smooth site $\mathcal{X}_{smooth}$.

4. We say $\mathcal{F}$ is a syntomic sheaf, or a sheaf for the syntomic topology if $\mathcal{F}$ is a sheaf on the associated syntomic site $\mathcal{X}_{syntomic}$.

5. We say $\mathcal{F}$ is an fppf sheaf, or a sheaf, or a sheaf for the fppf topology if $\mathcal{F}$ is a sheaf on the associated fppf site $\mathcal{X}_{fppf}$.

A morphism of sheaves is just a morphism of presheaves. We denote these categories of sheaves $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{Zar})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{smooth})$, $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{syntomic})$, and $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$.

Of course we can also talk about sheaves of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc. The category of abelian sheaves, i.e., sheaves of abelian groups, is denoted $\textit{Ab}(\mathcal{X}_{fppf})$ and similarly for the other topologies. If $\mathcal{X}$ is an algebraic stack, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is equivalent (modulo set theoretical problems) to what in the literature would be termed the category of sheaves on the big fppf site of $\mathcal{X}$. Similar for other topologies. We may occasionally use this terminology to distinguish this construction from others.

Since the topologies are listed in increasing order of strength we have the following strictly full inclusions

$\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{syntomic}) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{smooth}) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{Zar}) \subset \textit{PSh}(\mathcal{X})$

We sometimes write $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{X})$ and $\textit{Ab}(\mathcal{X}_{fppf}) = \textit{Ab}(\mathcal{X})$ in accordance with our terminology that a sheaf on $\mathcal{X}$ is an fppf sheaf on $\mathcal{X}$.

With this setup functoriality of these topoi is straightforward, and moreover, is compatible with the inclusion functors above.

Lemma 94.4.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$. The functors ${}_ pf$ and $f^ p$ of (94.3.1.1) transform $\tau$ sheaves into $\tau$ sheaves and define a morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$.

Proof. This follows immediately from Stacks, Lemma 8.10.3. $\square$

In other words, pushforward and pullback of presheaves as defined in Section 94.3 also produces pushforward and pullback of $\tau$-sheaves. Having said all of the above we see that we can write $f^ p = f^{-1}$ and ${}_ pf = f_*$ without any possibility of confusion.

Definition 94.4.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We denote

$f = (f^{-1}, f_*) : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_{fppf})$

the associated morphism of fppf topoi constructed above. Similarly for the associated Zariski, étale, smooth, and syntomic topoi.

As discussed in Sites, Section 7.44 the same formula (on the underlying sheaf of sets) defines pushforward and pullback for sheaves (for one of our topologies) of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc.

Comment #5421 by on

In the sentence It follows that if we have a category ... fppf topology, see Stacks, Definition 06NV the link on 06NV works well if I remain on the webpage, but directs me wrongly if I click on the same link in the pdf file https://stacks.math.columbia.edu/download/stacks-sheaves.pdf. In the latter case, it directs me to Tag 073N instead of 06NV (which is Lemma 10.2 of the same PDF file whereas it should send me to Definition 10.2 of another pdf file).

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