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*}

7.27 More localization

In this section we prove a few lemmas on localization where we impose some additional hypotheses on the site on or the object we are localizing at.

Lemma 7.27.1. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. If the topology on $\mathcal{C}$ is subcanonical, see Definition 7.12.2, and if $\mathcal{G}$ is a sheaf on $\mathcal{C}/U$, then

\[ j_{U!}(\mathcal{G})(V) = \coprod \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U), \]

in other words sheafification is not necessary in Lemma 7.25.2.

Proof. Let $\mathcal{V} = \{ V_ i \to V\} _{i \in I}$ be a covering of $V$ in the site $\mathcal{C}$. We are going to check the sheaf condition for the presheaf $\mathcal{H}$ of Lemma 7.25.2 directly. Let $(s_ i, \varphi _ i)_{i \in I} \in \prod _ i \mathcal{H}(V_ i)$, This means $\varphi _ i : V_ i \to U$ is a morphism in $\mathcal{C}$, and $s_ i \in \mathcal{G}(V_ i \xrightarrow {\varphi _ i} U)$. The restriction of the pair $(s_ i, \varphi _ i)$ to $V_ i \times _ V V_ j$ is the pair $(s_ i|_{V_ i \times _ V V_ j/U}, \text{pr}_1 \circ \varphi _ i)$, and likewise the restriction of the pair $(s_ j, \varphi _ j)$ to $V_ i \times _ V V_ j$ is the pair $(s_ j|_{V_ i \times _ V V_ j/U}, \text{pr}_2 \circ \varphi _ j)$. Hence, if the family $(s_ i, \varphi _ i)$ lies in $\check{H}^0(\mathcal{V}, \mathcal{H})$, then we see that $\text{pr}_1 \circ \varphi _ i = \text{pr}_2 \circ \varphi _ j$. The condition that the topology on $\mathcal{C}$ is weaker than the canonical topology then implies that there exists a unique morphism $\varphi : V \to U$ such that $\varphi _ i$ is the composition of $V_ i \to V$ with $\varphi $. At this point the sheaf condition for $\mathcal{G}$ guarantees that the sections $s_ i$ glue to a unique section $s \in \mathcal{G}(V \xrightarrow {\varphi } U)$. Hence $(s, \varphi ) \in \mathcal{H}(V)$ as desired. $\square$

Lemma 7.27.2. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume $\mathcal{C}$ has products of pairs of objects. Then

  1. the functor $j_ U$ has a continuous right adjoint, namely the functor $v(X) = X \times U / U$,

  2. the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}$ whose associated morphism of topoi equals $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

  3. we have $j_{U*}\mathcal{F}(X) = \mathcal{F}(X \times U/U)$.

Proof. The functor $v$ being right adjoint to $j_ U$ means that given $Y/U$ and $X$ we have

\[ \mathop{Mor}\nolimits _\mathcal {C}(Y, X) = \mathop{Mor}\nolimits _{\mathcal{C}/U}(Y/U, X \times U/U) \]

which is clear. To check that $v$ is continuous let $\{ X_ i \to X\} $ be a covering of $\mathcal{C}$. By the third axiom of a site (Definition 7.6.2) we see that

\[ \{ X_ i \times _ X (X \times U) \to X \times _ X (X \times U)\} = \{ X_ i \times U \to X \times U\} \]

is a covering of $\mathcal{C}$ also. Hence $v$ is continuous. The other statements of the lemma follow from Lemmas 7.22.1 and 7.22.2. $\square$

Lemma 7.27.3. Let $\mathcal{C}$ be a site. Let $U \to V$ be a morphism of $\mathcal{C}$. Assume $\mathcal{C}$ has fibre products. Let $j$ be as in Lemma 7.25.8. Then

  1. the functor $j : \mathcal{C}/U \to \mathcal{C}/V$ has a continuous right adjoint, namely the functor $v : (X/V) \mapsto (X \times _ V U/U)$,

  2. the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}/V$ whose associated morphism of topoi equals $j$, and

  3. we have $j_*\mathcal{F}(X/V) = \mathcal{F}(X \times _ V U/U)$.

Proof. Follows from Lemma 7.27.2 since $j$ may be viewed as a localization functor by Lemma 7.25.8. $\square$

A fundamental property of an open immersion is that the restriction of the pushforward and the restriction of the extension by the empty set produces back the original sheaf. This is not always true for the functors associated to $j_ U$ above. It is true when $U$ is a “subobject of the final object”.

Lemma 7.27.4. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume that every $X$ in $\mathcal{C}$ has at most one morphism to $U$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}/U$. The canonical maps $\mathcal{F} \to j_ U^{-1}j_{U!}\mathcal{F}$ and $j_ U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are isomorphisms.

Proof. If $\mathcal{C}$ has fibre products, then this is a special case of Lemma 7.21.7. In general we have the following direct proof.

Let $X/U$ be an object over $U$. In Lemmas 7.20.2 and 7.21.5 we have seen that sheafification is not necessary for the functors $j_ U^{-1} = (u^ p\ )^\# $ and $j_{U*} = ({}_ pu )^\# $. We may compute $(j_ U^{-1}j_{U*}\mathcal{F})(X/U) = j_{U*}\mathcal{F}(X) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}(Y/U)$. Here the limit is over the category of pairs $(Y/U, Y \to X)$ where the morphisms $Y \to X$ are not required to be over $U$. By our assumption however we see that they are automatically morphisms over $U$ and we deduce that the limit is the value on $\text{id}_ X$, i.e., $\mathcal{F}(X/U)$. This proves that $j_ U^{-1}j_{U*}\mathcal{F} = \mathcal{F}$.

On the other hand, $(j_ U^{-1}j_{U!}\mathcal{F})(X/U) = j_{U!}\mathcal{F}(X) = (u_ p\mathcal{F})^\# (X)$, and $u_ p\mathcal{F}(X) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}(Y/U)$. Here the colimit is over the category of pairs $(Y/U, X \to Y)$ where the morphisms $X \to Y$ are not required to be over $U$. By our assumption however we see that they are automatically morphisms over $U$ and we deduce that the colimit is the value on $\text{id}_ X$, i.e., $\mathcal{F}(X/U)$. This shows that the sheafification is not necessary (since any object over $X$ is automatically in a unique way an object over $U$) and the result follows. $\square$

Lemma 7.27.5. Let $\mathcal{C}$ be a site. Let

\[ \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V } \]

be a commutative diagram of $\mathcal{C}$. The morphisms of Lemma 7.25.8 produce commutative diagrams

\[ \vcenter { \xymatrix{ \mathcal{C}/U' \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \mathcal{C}/U \ar[d]^{j_{U/V}} \\ \mathcal{C}/V' \ar[r]^{j_{V'/V}} & \mathcal{C}/V } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U') \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[d]^{j_{U/V}} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V') \ar[r]^{j_{V'/V}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) } } \]

of continuous and cocontinuous functors and of topoi. Moreover, if the initial diagram of $\mathcal{C}$ is cartesian, then we have $j_{V'/V}^{-1} \circ j_{U/V, *} = j_{U'/V', *} \circ j_{U'/U}^{-1}$.

Proof. The commutativity of the left square in the first statement of the lemma is immediate from the definitions. It implies the commutativity of the diagram of topoi by Lemma 7.21.2. Assume the diagram is cartesian. By the uniqueness of adjoint functors, to show $j_{V'/V}^{-1} \circ j_{U/V, *} = j_{U'/V', *} \circ j_{U'/U}^{-1}$ is equivalent to showing $j_{U/V}^{-1} \circ j_{V'/V!} = j_{U'/U!} \circ j_{U'/V'}^{-1}$. Via the identifications of Lemma 7.25.4 we may think of our diagram of topoi as

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U'}^\# \ar[d] \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{V'}^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# } \]

and we know how to interpret the functors $j^{-1}$ and $j_!$ by Lemma 7.25.9. Thus we have to show given $\mathcal{F} \to h_{V'}^\# $ that

\[ \mathcal{F} \times _{h_{V'}^\# } h_{U'}^\# = \mathcal{F} \times _{h_ V^\# } h_ U^\# \]

as sheaves with map to $h_ U^\# $. This is true because $h_{U'} = h_{V'} \times _{h_ V} h_ U$ and hence also

\[ h_{U'}^\# = h_{V'}^\# \times _{h_ V^\# } h_ U^\# \]

as sheafification is exact. $\square$


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