Added "Pulling back sections" section in modules.tex

sheaves.tex

@@ -3218,6 +3218,8 @@ \section{Morphisms of ringed spaces and modules}
 is an $\mathcal{O}_Y(V)$-module map. Here we think of
 $\mathcal{F}(f^{-1}V)$ as an $\mathcal{O}_Y(V)$-module
 via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V)$.
+In other words, the formula $\varphi_V(as)=f^\sharp_V(a)\varphi_V(s)$ holds,
+for $a\in\mathcal{O}_Y(V)$ and $s\in\mathcal{G}(V)$.
 The set of $f$-maps between
 $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection
 with the sets
@@ -3264,7 +3266,266 @@ \section{Morphisms of ringed spaces and modules}
 for example.
 \end{proof}
 
+\section{Pulling back sections}
+\label{section-pullback-section}
 
+\begin{definition}
+\label{definition-inverse-image-section}
+Let $f:X\to Y$ be a continuous map of topological spaces.
+Let $\mathcal{G}$ be a sheaf of sets over $Y$.
+Let $V\subset Y$ be an open subset.
+Let $s\in\mathcal{G}(V)$ be a section.
+The \emph{inverse image section of $s$ along $f$}, denoted $f^{-1}s$,
+is the section in $f^{-1}\mathcal{G}(f^{-1}(V))$ given by the
+image of $s$ along the unit $\mathcal{G}\to f_*f^{-1}\mathcal{G}$ of the
+adjunction between $f_*$ and $f^{-1}$ (see Section \ref{section-presheaves-functorial}).
+\end{definition}
+
+\begin{lemma}
+\label{lemma-inverse-image-section-germ}
+Let $f:X\to Y$ be a continuous map of topological spaces.
+Let $\mathcal{G}$ be a sheaf of sets over $Y$.
+Let $V\subset Y$ be an open subset.
+Let $s\in\mathcal{G}(V)$ be a section.
+Let $x\in X$ be a point.
+Then $(f^{-1}s)_x=s_{f(x)}$ via the isomorphism of Lemma \ref{lemma-stalk-pullback}.
+\end{lemma}
+
+\begin{proof}
+Since $\mathcal{G}\to f_*f^{-1}\mathcal{G}$ is an $f$-map,
+it follows from the commutative square to be found before Lemma \ref{lemma-compose-f-maps-stalks}.
+\end{proof}
+
+The inverse image section is compatible with the adjunction
+between $f_*$ and $f^{-1}$ in the following sense.
+Let $\mathcal{F}$, $\mathcal{G}$ be sheaves respectively over $X$ and $Y$,
+and let $\varphi^\sharp:\mathcal{G}\to f_*\mathcal{F}$ be a morphism of sheaves
+whose adjunct is $\varphi^\flat:f^{-1}\mathcal{G}\to \mathcal{F}$.
+Then, for $s\in \mathcal{G}(V)$, we have
+\begin{equation}
+\label{equation-adjunct-and-inverse-image-section}
+\varphi^\sharp_V(s)=\varphi^\flat_{f^{-1}(V)}(f^{-1}s).
+\end{equation}
+This follows from the fact that
+$$
+\varphi^\sharp:\mathcal{G}\to f_*f^{-1}\mathcal{G}\xrightarrow{f_*\varphi^\flat}f_*\mathcal{F},
+$$
+where the first morphism is the unit of the adjunction between $f^{-1}$ and $f_*$.
+Since a morphism of sheaves is determined by the induced morphisms on stalks,
+from Lemma \ref{lemma-inverse-image-section-germ} it follows that $\varphi^\flat$
+is the unique map satisfying \eqref{equation-adjunct-and-inverse-image-section}
+for every $s\in\mathcal{G}(V)$ and every open $V\subset Y$.
+
+Suppose $\mathcal{O}$ is a sheaf of rings over $Y$ and $\mathcal{G}$ is an $\mathcal{O}$-module.
+From Lemma \ref{lemma-inverse-image-section-germ} it follows that
+$f^{-1}(as)=f^{-1}(a)f^{-1}(s)$
+for $a\in\mathcal{O}(V)$ and $s\in\mathcal{G}(V)$,
+where we are using the $f^{-1}\mathcal{O}$-module structure on
+$f^{-1}\mathcal{G}$ coming from Lemma \ref{lemma-pullback-module}.
+
+
+\begin{lemma}
+\label{lemma-inverse-image-section-morphism}
+Let $f :X\to Y$ be a continuous map of topological spaces.
+Let $\varphi:\mathcal{G}\to\mathcal{G}'$ be a morphism of
+sheaves of sets over $Y$.
+Let $V\subset Y$ be an open subset.
+Let $s\in\mathcal{G}(V)$ be a section.
+Then $f^{-1}(\varphi(s))=(f^{-1}\varphi)(f^{-1}s)$.
+\end{lemma}
+
+\begin{proof}
+This follows from the naturality of the unit, i.e.,
+the following square commutes:
+$$
+\xymatrix{
+\mathcal{G} \ar@{->}[r] \ar@{->}[d] & \mathcal{G}' \ar@{->}[d] \\
+f_*f^{-1}\mathcal{G} \ar@{->}[r] & f_*f^{-1}\mathcal{G}'
+}
+$$
+In the square, if we map $s$ first down then right,
+we get $(f^{-1}\varphi)(f^{-1}s)$. If we map $s$ first right then down,
+we get $f^{-1}(\varphi(s))$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-inverse-image-section-functorial}
+Let
+$X\xrightarrow{f}Y\xrightarrow{g}Z$ be continuous maps of topological spaces.
+Let $\mathcal{H}$ be a sheaf of sets over $Z$.
+Let $W\subset Z$ be an open subset.
+Let $t\in\mathcal{G}(W)$ be a section.
+By the means of the isomorphism of functors
+$(g\circ f)^{-1}\cong f^{-1}\circ g^{-1}$
+of Lemma \ref{lemma-pullback-composition},
+we can identify $(g\circ f)^{-1}t=f^{-1}(g^{-1}t)$.
+\end{lemma}
+
+\begin{proof}
+It suffices to see that $(g\circ f)^{-1}t$ and $f^{-1}(g^{-1}t)$ have same stalk at $x\in X$.
+By Lemma \ref{lemma-inverse-image-section-germ},
+$((g\circ f)^{-1}t)_x=t_{g(f(x))}=(g^{-1}t)_{f(x)}=f^{-1}(g^{-1}t)_x$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-pullback-section}
+Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
+ringed spaces. Let $\mathcal{G}$ be a sheaf of
+$\mathcal{O}_Y$-modules. Let $V\subset Y$ be an open subset. Let
+$s\in\mathcal{G}(V)$ be a section. Denote $g:f^{-1}(V)\to V$ to the
+restriction of $f$. The following sections in $f^*\mathcal{G}(f^{-1}(V))$
+are the same:
+\begin{enumerate}
+\item The image of $s$ along the unit
+$\mathcal{G}\to f_*f^*\mathcal{G}$ of the adjunction between
+$f_*$ and $f^*$ (Lemma \ref{lemma-adjoint-pullback-pushforward-modules}),
+\item The global section of $g^*(\mathcal{G}|_V)$ associated with
+$\mathcal{O}_{f^{-1}(V)}\cong
+g^*\mathcal{O}_V\to g^*(\mathcal{G}|_V)$,
+where $\mathcal{O}_V\to\mathcal{G}|_V$ is the map associated with $s$
+(see first paragraph of Modules, Section \ref{modules-section-sections}), and
+\item The image of $1\otimes f^{-1}s$ (see Definition \ref{definition-inverse-image-section}) along the morphism
+$
+\mathcal{O}_X\otimes_{p,f^{-1}\mathcal{O}_Y}f^{-1}\mathcal{G}
+\to
+\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}\mathcal{G}=f^*\mathcal{G}$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Naturality of the unit of the adjunction between $g_*$ and $g^*$ gives a
+commutative diagram
+$$
+\xymatrix{
+\mathcal{O}_V \ar@{->}[r] \ar@{->}[d] & \mathcal{G}|_V \ar@{->}[d] \\
+g_*\mathcal{O}_{f^{-1}(V)}\cong g_*g^*\mathcal{O}_V \ar@{->}[r]
+& g_*g^*(\mathcal{G}|_V)
+}
+$$
+The equality between (1) and (2) follows then from the fact that the restriction of the unit
+$\mathcal{G}\to f_*f^*\mathcal{G}$ of the adjunction between $f_*$ and $f^*$ to $V$
+equals $\mathcal{G}|_V\to g_*g^*(\mathcal{G}|_V)$,
+the unit of the adjunction between $g_*$ and $g^*$.
+
+On the other hand, from the proof of Lemma \ref{lemma-adjoint-pullback-pushforward-modules}
+it follows that the unit of the adjunction between $f_*$ and $f^*$ at $\mathcal{G}$ is the composite
+$$
+\mathcal{G}
+\xrightarrow{\eta^\mathrm{ip}_\mathcal{G}}
+f_*f^{-1}\mathcal{G}
+\xrightarrow{f_*(\eta^\mathrm{er}_{f^{-1}\mathcal{G}})}
+f_*(\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}\mathcal{G})
+$$
+where $\eta^\mathrm{ip}$ is the unit of the adjunction between
+inverse image and pushforward functors from Lemma \ref{lemma-adjoint-push-pull-modules} and
+$\eta^\mathrm{er}$ is the unit of the adjunction between
+extension and restriction of scalars functors from Lemma \ref{lemma-adjointness-tensor-restrict}.
+Then $s$ maps along this composite as $s\mapsto f^{-1}s\mapsto 1\otimes f^{-1}s$,
+which shows the equality between (1) and (3).
+\end{proof}
+
+\begin{definition}
+\label{definition-pullback-section}
+Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
+ringed spaces. Let $\mathcal{G}$ be a sheaf of
+$\mathcal{O}_Y$-modules. Let $V\subset Y$ be an open subset. Let
+$s\in\mathcal{G}(V)$ be a section. The {\it pullback section of $s$
+along $f$}, denoted $f^*s$, is defined to be the section in
+$f^*\mathcal{G}(f^{-1}(V))$ from Lemma \ref{lemma-pullback-section}.
+\end{definition}
+
+Note that when $\mathcal{G}=\mathcal{O}_Y$,
+the unit $\mathcal{O}_Y\to f_*f^*\mathcal{O}_Y\cong f_*\mathcal{O}_X$
+is just the map $f^\sharp$ on structure sheaves.
+Hence $f^*a=f^\sharp a$ for $a\in\mathcal{O}_Y(V)$.
+Note that the unit $\mathcal{G}\to f_*f^*\mathcal{G}$ can be understood as
+an $f$-map $\mathcal{G}\to f^*\mathcal{G}$ of sheaves of modules
+(see Section \ref{section-ringed-spaces-functoriality-modules});
+thus $f^*(as)=f^\sharp(a)f^*(s)$ for $a\in\mathcal{O}_Y(V)$ and $s\in\mathcal{G}(V)$.
+
+\begin{lemma}
+\label{lemma-pullback-section-germ}
+Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
+ringed spaces. Let $\mathcal{G}$ be a sheaf of
+$\mathcal{O}_Y$-modules.
+Let $V\subset Y$ be an open subset.
+Let $s\in\mathcal{G}(V)$ be a section.
+Let $x\in f^{-1}(V)$ be a point.
+By means of the isomorphism in
+Lemma \ref{lemma-stalk-pullback-modules},
+we can identify $(f^*s)_x=s_{f(x)}\otimes 1$.
+\end{lemma}
+
+\begin{proof}
+Via the isomorphism of Lemma \ref{lemma-stalk-pullback-modules}, we have
+$(f^*s)_x=(f^{-1}s\otimes 1)_x=(f^{-1}s)_x\otimes 1=s_{f(x)}\otimes 1$,
+by Lemma \ref{lemma-inverse-image-section-germ}.
+\end{proof}
+
+The pullback section is compatible with the adjunction
+between $f_*$ and $f^*$ in the following sense.
+Let $\mathcal{F}$ be an $\mathcal{O}_X$-module, $\mathcal{G}$ be an $\mathcal{O}_Y$-module
+and $\varphi^\sharp:\mathcal{G}\to f_*\mathcal{F}$ be a morphism of $\mathcal{O}_Y$-modules
+whose adjunct is $\varphi^\flat:f^*\mathcal{G}\to \mathcal{F}$.
+Then, for $s\in \mathcal{G}(V)$, we have
+\begin{equation}
+\label{equation-adjunct-and-pullback-section}
+\varphi^\sharp_V(s)=\varphi^\flat_{f^{-1}(V)}(f^*s)
+\end{equation}
+This follows from the fact that
+$$
+\varphi^\sharp:\mathcal{G}\to f_*f^*\mathcal{G}\xrightarrow{f_*\varphi^\flat}f_*\mathcal{F},
+$$
+where the first morphism is the unit of the adjunction between $f^*$ and $f_*$.
+Since a morphism of sheaves is determined by the induced morphisms on stalks,
+from Lemma \ref{lemma-pullback-section-germ} it follows that $\varphi^\flat$
+is the unique map satisfying \eqref{equation-adjunct-and-pullback-section}
+for every $s\in\mathcal{G}(V)$ and every open $V\subset Y$.
+
+\begin{lemma}
+\label{lemma-pullback-section-morphism}
+Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
+be a morphism of ringed spaces.
+Let $\varphi:\mathcal{G}\to\mathcal{G}'$ be a morphism of
+sheaves of $\mathcal{O}_Y$-modules.
+Let $V\subset Y$ be an open subset.
+Let $s\in\mathcal{G}(V)$ be a section.
+Then $f^*(\varphi(s))=(f^*\varphi)(f^*s)$.
+\end{lemma}
+
+\begin{proof}
+This follows from the naturality of the unit, i.e.,
+the following square commutes:
+$$
+\xymatrix{
+\mathcal{G} \ar@{->}[r] \ar@{->}[d] & \mathcal{G}' \ar@{->}[d] \\
+f_*f^*\mathcal{G} \ar@{->}[r] & f_*f^*\mathcal{G}'
+}
+$$
+In the square, if we map $s$ first down then right,
+we get $(f^*\varphi)(f^*s)$. If we map $s$ first right then down,
+we get $f^*(\varphi(s))$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-pullback-section-functorial}
+Let
+$(X,\mathcal{O}_X)
+\xrightarrow{f}(Y,\mathcal{O}_Y)
+\xrightarrow{g}(Z,\mathcal{O}_Z)$ be morphisms of ringed spaces.
+Let $\mathcal{H}$ be a sheaf of $\mathcal{O}_Z$-modules.
+Let $W\subset Z$ be an open subset.
+Let $t\in\mathcal{G}(W)$ be a section.
+By the means of the isomorphism of functors
+$(g\circ f)^*\cong f^*\circ g^*$
+of Lemma \ref{lemma-push-pull-composition-modules},
+we can identify $(g\circ f)^*t=f^*(g^*t)$.
+\end{lemma}
+
+\begin{proof}
+This follows by the description of the pullback section
+in point (2) of Lemma \ref{lemma-pullback-section}.
+\end{proof}
 
 \section{Skyscraper sheaves and stalks}
 \label{section-skyscraper-sheaves}