96.7 Sheaves of modules
Since we have a structure sheaf we have modules.
Definition 96.7.1. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.
A presheaf of modules on $\mathcal{X}$ is a presheaf of $\mathcal{O}_\mathcal {X}$-modules. The category of presheaves of modules is denoted $\textit{PMod}(\mathcal{O}_\mathcal {X})$.
We say a presheaf of modules $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module, or more precisely a sheaf of $\mathcal{O}_\mathcal {X}$-modules if $\mathcal{F}$ is an fppf sheaf. The category of $\mathcal{O}_\mathcal {X}$-modules is denoted $\textit{Mod}(\mathcal{O}_\mathcal {X})$.
These (pre)sheaves of modules occur in the literature as (pre)sheaves of $\mathcal{O}_\mathcal {X}$-modules on the big fppf site of $\mathcal{X}$. We will occasionally use this terminology if we want to distinguish these categories from others. We will also encounter presheaves of modules which are sheaves in the Zariski, étale, smooth, or syntomic topologies (without necessarily being sheaves). If need be these will be denoted $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and similarly for the other topologies.
Next, we address functoriality – first for presheaves of modules. Let
\[ \xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd]_ p & & \mathcal{Y} \ar[ld]^ q \\ & (\mathit{Sch}/S)_{fppf} } \]
be a $1$-morphism of categories fibred in groupoids. The functors $f^{-1}$, $f_*$ on abelian presheaves extend to functors
96.7.1.1
\begin{equation} \label{stacks-sheaves-equation-functoriality-presheaves-modules} f^{-1} : \textit{PMod}(\mathcal{O}_\mathcal {Y}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal {X}) \quad \text{and}\quad f_* : \textit{PMod}(\mathcal{O}_\mathcal {X}) \longrightarrow \textit{PMod}(\mathcal{O}_\mathcal {Y}) \end{equation}
This is immediate for $f^{-1}$ because $f^{-1}\mathcal{G}(x) = \mathcal{G}(f(x))$ which is a module over $\mathcal{O}_\mathcal {Y}(f(x)) = \mathcal{O}(q(f(x))) = \mathcal{O}(p(x)) = \mathcal{O}_\mathcal {X}(x)$. Alternatively it follows because $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$ and because $f^{-1}$ commutes with limits (on presheaves). Since $f_*$ is a right adjoint it commutes with all limits (on presheaves) in particular products. Hence we can extend $f_*$ to a functor on presheaves of modules as in the proof of Modules on Sites, Lemma 18.12.1. We claim that the functors (96.7.1.1) form an adjoint pair of functors:
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O}_\mathcal {X})}( f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O}_\mathcal {Y})}( \mathcal{G}, f_*\mathcal{F}). \]
As $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$ this follows from Modules on Sites, Lemma 18.12.3 by endowing $\mathcal{X}$ and $\mathcal{Y}$ with the chaotic topology.
Next, we discuss functoriality for modules, i.e., for sheaves of modules in the fppf topology. Denote by $f$ also the induced morphism of ringed topoi, see Lemma 96.6.2 (for the fppf topologies right now). Note that the functors $f^{-1}$ and $f_*$ of (96.7.1.1) preserve the subcategories of sheaves of modules, see Lemma 96.4.4. Hence it follows immediately that
96.7.1.2
\begin{equation} \label{stacks-sheaves-equation-functoriality-sheaves-modules} f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal {Y}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {X}) \quad \text{and}\quad f_* : \textit{Mod}(\mathcal{O}_\mathcal {X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {Y}) \end{equation}
form an adjoint pair of functors:
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_\mathcal {X})}( f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}_\mathcal {Y})}( \mathcal{G}, f_*\mathcal{F}). \]
By uniqueness of adjoints we conclude that $f^* = f^{-1}$ where $f^*$ is as defined in Modules on Sites, Section 18.13 for the morphism of ringed topoi $f$ above. Of course we could have seen this directly because $f^*(-) = f^{-1}(-) \otimes _{f^{-1}\mathcal{O}_\mathcal {Y}} \mathcal{O}_\mathcal {X}$ and because $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$.
Similarly for sheaves of modules in the Zariski, étale, smooth, syntomic topology.
Comments (2)
Comment #3557 by Félix B Boudreau on
Comment #3682 by Johan on