The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.46 Comparison with SGA4

Our notation for the functors $u^ p$ and $u_ p$ from Section 7.5 and $u^ s$ and $u_ s$ from Section 7.13 is taken from [pages 14 and 42, ArtinTopologies]. Having made these choices, the notation for the functor ${}_ pu$ in Section 7.19 and ${}_ su$ in Section 7.20 seems reasonable. In this section we compare our notation with that of SGA4.

Presheaves: Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. The functor $u^ p$ is denoted $u^*$ in [Exposee I, Section 5, SGA4]. The functor $u_ p$ is denoted $u_!$ in [Exposee I, Proposition 5.1, SGA4]. The functor ${}_ pu$ is denoted $u_*$ in [Exposee I, Proposition 5.1, SGA4]. In other words, we have

\[ u_ p, u^ p, {}_ pu\quad (SP) \quad \text{versus}\quad u_!, u^*, u_*\quad (SGA4) \]

The reader should be cautioned that different notation is used for these functors in different parts of SGA4.

Sheaves and continuous functors: Suppose that $\mathcal{C}$ and $\mathcal{D}$ are sites and that $u : \mathcal{C} \to \mathcal{D}$ is a continuous functor (Definition 7.13.1). The functor $u^ s$ is denoted $u_ s$ in [Exposee III, 1.11, SGA4]. The functor $u_ s$ is denoted $u^ s$ in [Exposee III, Proposition 1.2, SGA4]. In other words, we have

\[ u_ s, u^ s\quad (SP) \quad \text{versus}\quad u^ s, u_ s\quad (SGA4) \]

When $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ (Definition 7.14.1) we see that the associated morphism of topoi (Lemma 7.15.2) is the same as that in [Exposee IV, (4.9.1.1), SGA4].

Sheaves and cocontinuous functors: Suppose that $\mathcal{C}$ and $\mathcal{D}$ are sites and that $u : \mathcal{C} \to \mathcal{D}$ is a cocontinuous functor (Definition 7.20.1). The functor ${}_ su$ (Lemma 7.20.2) is denoted $u_*$ in [Exposee III, Proposition 2.3, SGA4]. The functor $(u^ p\ )^\# $ is denoted $u^*$ in [Exposee III, Proposition 2.3, SGA4]. In other words, we have

\[ (u^ p\ )^\# , {}_ su\quad (SP) \quad \text{versus}\quad u^*, u_*\quad (SGA4) \]

Thus the morphism of topoi associated to $u$ in Lemma 7.21.1 is the same as that in [Exposee IV, 4.7, SGA4].

Morphisms of Topoi: If $f$ is a morphism of topoi given by the functors $(f^{-1}, f_*)$ then the functor $f^{-1}$ is denoted $f^*$ in [Exposee IV, Definition 3.1, SGA4]. We will use $f^{-1}$ to denote pullback of sheaves of sets or more generally sheaves of algebraic structure (Section 7.44). We will use $f^*$ to denote pullback of sheaves of modules for a morphism of ringed topoi (Modules on Sites, Definition 18.13.1).


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