The Stacks project

103.3 Notation

Different topologies. If we indicate an algebraic stack by a calligraphic letter, such as $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$, then the notation $\mathcal{X}_{Zar}, \mathcal{X}_{\acute{e}tale}, \mathcal{X}_{smooth}, \mathcal{X}_{syntomic}, \mathcal{X}_{fppf}$ indicates the site introduced in Sheaves on Stacks, Definition 96.4.1. (Think “big site”.) Correspondingly the structure sheaf of $\mathcal{X}$ is a sheaf on $\mathcal{X}_{fppf}$. On the other hand, algebraic spaces and schemes are usually indicated by roman capitals, such as $X, Y, Z$, and in this case $X_{\acute{e}tale}$ indicates the small étale site of $X$ (as defined in Topologies, Definition 34.4.8 or Properties of Spaces, Definition 66.18.1). It seems that the distinction should be clear enough.

The default topology is the fppf topology. Hence we will sometimes say “sheaf on $\mathcal{X}$” or “sheaf of $\mathcal{O}_\mathcal {X}$-modules” when we mean sheaf on $\mathcal{X}_{fppf}$ or object of $\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$.

If $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks, then the functors $f_*$ and $f^{-1}$ defined on presheaves preserves sheaves for any of the topologies mentioned above. In particular when we discuss the pushforward or pullback of a sheaf we don't have to mention which topology we are working with. The same isn't true when we compute cohomology groups and/or higher direct images. In this case we will always mention which topology we are working with.

Suppose that $f : X \to \mathcal{Y}$ is a morphism from an algebraic space $X$ to an algebraic stack $\mathcal{Y}$. Let $\mathcal{G}$ be a sheaf on $\mathcal{Y}_\tau $ for some topology $\tau $. In this case $f^{-1}\mathcal{G}$ is a sheaf for the $\tau $ topology on $\mathcal{S}_ X$ (the algebraic stack associated to $X$) because (by our conventions) $f$ really is a $1$-morphism $f : \mathcal{S}_ X \to \mathcal{Y}$. If $\tau = {\acute{e}tale}$ or stronger, then we write $f^{-1}\mathcal{G}|_{X_{\acute{e}tale}}$ to denote the restriction to the étale site of $X$, see Sheaves on Stacks, Section 96.22. If $\mathcal{G}$ is an $\mathcal{O}_\mathcal {X}$-module we sometimes write $f^*\mathcal{G}$ and $f^*\mathcal{G}|_{X_{\acute{e}tale}}$ instead.