From 522a3c6606021f83ab087dd939cbab47463c47a8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 8 Oct 2025 14:53:28 +0200 Subject: [PATCH 1/8] Added "Pulling back sections" section in modules.tex --- modules.tex | 113 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 113 insertions(+) diff --git a/modules.tex b/modules.tex index 91c0227d..ccfd74c2 100644 --- a/modules.tex +++ b/modules.tex @@ -458,7 +458,120 @@ \section{Sections of sheaves of modules} +\section{Pulling back sections} +\label{section-pullback-section} +\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 two sections in $f^*G(f^{-1}(V))$ +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 $f^*$, and +\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 Section \ref{section-sections}). +\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 result 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 $g^*$. +\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} + +\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 +Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}, +we can identify $(f^*s)_x=s_{f(x)}\otimes 1$. +\end{lemma} + +\begin{proof} +Without loss of generality, assume $V=Y$. +Let $\mathcal{O}_Y\to\mathcal{G}$ be the +$\mathcal{O}_Y$-linear map sending the global section $1$ to $s$. +The result follows by studying the map on stalks at +$x$ induced by $f^*\mathcal{O}_Y\to f^*\mathcal{G}$. +\end{proof} + +\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 +$(g\circ f)^*\cong f^*\circ g^*$ +(Sheaves, Lemma \ref{sheaves-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} From 6f4691774887532f3bdb6ef0c41c78de0d7f5f8a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 12:11:54 +0200 Subject: [PATCH 2/8] Moved it to sheaves.tex - Moved everything from modules.tex to sheaves.tex. After all, the results we use are from the latter chapter, so it seemed to fit better there. - Defined the inverse image section, proven its functorial properties, and made the connection with the pullback section. --- modules.tex | 113 ----------------------------- sheaves.tex | 202 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 202 insertions(+), 113 deletions(-) diff --git a/modules.tex b/modules.tex index ccfd74c2..91c0227d 100644 --- a/modules.tex +++ b/modules.tex @@ -458,120 +458,7 @@ \section{Sections of sheaves of modules} -\section{Pulling back sections} -\label{section-pullback-section} -\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 two sections in $f^*G(f^{-1}(V))$ -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 $f^*$, and -\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 Section \ref{section-sections}). -\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 result 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 $g^*$. -\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} - -\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 -Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}, -we can identify $(f^*s)_x=s_{f(x)}\otimes 1$. -\end{lemma} - -\begin{proof} -Without loss of generality, assume $V=Y$. -Let $\mathcal{O}_Y\to\mathcal{G}$ be the -$\mathcal{O}_Y$-linear map sending the global section $1$ to $s$. -The result follows by studying the map on stalks at -$x$ induced by $f^*\mathcal{O}_Y\to f^*\mathcal{G}$. -\end{proof} - -\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 -$(g\circ f)^*\cong f^*\circ g^*$ -(Sheaves, Lemma \ref{sheaves-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} diff --git a/sheaves.tex b/sheaves.tex index 5339e008..95238a26 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3264,7 +3264,209 @@ \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} + +\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} + +\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} + +\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} From fce1e6706e586707923ff99c81be97912e7dabd1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 14:12:25 +0200 Subject: [PATCH 3/8] Pullback of sections of structure sheaf is as expected --- sheaves.tex | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/sheaves.tex b/sheaves.tex index 95238a26..37817ec5 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3404,6 +3404,11 @@ \section{Pulling back sections} $f^*\mathcal{G}(f^{-1}(V))$ 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 $f^*s=f^\sharp s$ for a local section $s$ of the structure sheaf $\mathcal{O}_Y$. + \begin{lemma} \label{lemma-pullback-section-germ} Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of From 3de3b74f4c1299074d96ec64ec045f784a3d235b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 14:39:14 +0200 Subject: [PATCH 4/8] Made the connection with f-maps for linearity --- sheaves.tex | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/sheaves.tex b/sheaves.tex index 37817ec5..ff2d6671 100644 --- a/sheaves.tex +++ b/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 @@ -3407,7 +3409,10 @@ \section{Pulling back sections} 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 $f^*s=f^\sharp s$ for a local section $s$ of the structure sheaf $\mathcal{O}_Y$. +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 $\mathcal{G}\to f^*\mathcal{G}$ +(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} From fc006b412bea8de2926aa98780a545f0ef83c0b6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 9 Oct 2025 14:44:16 +0200 Subject: [PATCH 5/8] deleted indentation --- sheaves.tex | 171 ++++++++++++++++++++++++++-------------------------- 1 file changed, 86 insertions(+), 85 deletions(-) diff --git a/sheaves.tex b/sheaves.tex index ff2d6671..887dfae0 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3282,28 +3282,28 @@ \section{Pulling back sections} \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}. +\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}. +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} \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 $f^{-1}(\varphi(s))=(f^{-1}\varphi)(f^{-1}s)$. +\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} @@ -3311,8 +3311,8 @@ \section{Pulling back sections} the following square commutes: $$ \xymatrix{ - \mathcal{G} \ar@{->}[r] \ar@{->}[d] & \mathcal{G}' \ar@{->}[d] \\ - f_*f^{-1}\mathcal{G} \ar@{->}[r] & f_*f^{-1}\mathcal{G}' +\mathcal{G} \ar@{->}[r] \ar@{->}[d] & \mathcal{G}' \ar@{->}[d] \\ +f_*f^{-1}\mathcal{G} \ar@{->}[r] & f_*f^{-1}\mathcal{G}' } $$ In the square, if we map $s$ first down then right, @@ -3321,22 +3321,22 @@ \section{Pulling back sections} \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 $(g\circ f)^{-1}t=f^{-1}(g^{-1}t)$. +\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}, - $((g\circ f)^{-1}t)_x=t_{g(f(x))}=(g^{-1}t)_{f(x)}=f^{-1}(g^{-1}t)_x$. +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} @@ -3397,85 +3397,86 @@ \section{Pulling back sections} \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}. +\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 $\mathcal{G}\to f^*\mathcal{G}$ +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 $(f^*s)_x=s_{f(x)}\otimes 1$. +\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}. +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} \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 $f^*(\varphi(s))=(f^*\varphi)(f^*s)$. +\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 $f^*(\varphi(s))$. +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 $(g\circ f)^*t=f^*(g^*t)$. +\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}. +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} From b7268cffce13dd80593ffe058ac7daea2a99f223 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 10 Oct 2025 10:12:44 +0200 Subject: [PATCH 6/8] Added linearity of inverse image --- sheaves.tex | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/sheaves.tex b/sheaves.tex index 887dfae0..9474da0b 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3296,6 +3296,13 @@ \section{Pulling back sections} it follows from the commutative square to be found before Lemma \ref{lemma-compose-f-maps-stalks}. \end{proof} +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. From f46b95d2c3d1b8effef07ff634ed5e9faf31b6f3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 21 Oct 2025 09:49:18 +0200 Subject: [PATCH 7/8] Added adjunction compatibility --- sheaves.tex | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) diff --git a/sheaves.tex b/sheaves.tex index 9474da0b..c4869f8c 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3281,6 +3281,18 @@ \section{Pulling back sections} adjunction between $f_*$ and $f^{-1}$ (see Section \ref{section-presheaves-functorial}). \end{definition} +The inverse image section is compatible with the adjunction +between $f_*$ and $f^{-1}$ 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 $\varphi^\flat:f^{-1}\mathcal{G}\to \mathcal{F}$. +Then, for $s\in \mathcal{G}(V)$, we have $\varphi^\sharp_V(s)=\varphi^\flat_{f^{-1}(V)}(f^{-1}s)$. +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 $f_*$. + \begin{lemma} \label{lemma-inverse-image-section-germ} Let $f:X\to Y$ be a continuous map of topological spaces. @@ -3422,6 +3434,18 @@ \section{Pulling back sections} (see Section \ref{section-ringed-spaces-functoriality-modules}); thus $f^*(as)=f^\sharp(a)f^*(s)$ for $a\in\mathcal{O}_Y(V)$ and $s\in\mathcal{G}(V)$. +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 $\varphi^\sharp_V(s)=\varphi^\flat_{f^{-1}(V)}(f^*s)$. +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 $f_*$. + \begin{lemma} \label{lemma-pullback-section-germ} Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of From 82bfc35c26f30092c540fa1c260ffa05e7a910e9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Wed, 12 Nov 2025 12:14:26 +0100 Subject: [PATCH 8/8] Corrected what were the involved objects --- sheaves.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/sheaves.tex b/sheaves.tex index c4869f8c..f86e8410 100644 --- a/sheaves.tex +++ b/sheaves.tex @@ -3283,8 +3283,8 @@ \section{Pulling back sections} The inverse image section is compatible with the adjunction between $f_*$ and $f^{-1}$ 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 +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 $\varphi^\sharp_V(s)=\varphi^\flat_{f^{-1}(V)}(f^{-1}s)$. This follows from the fact that