What is a simple definition of the pullback of a section?

Question

I am simply asking for a definition for something everyone uses but nobody defines. Really, this is used in class and in Hartshorne, and I have tried to look for a definition in Hartshorne, Qing Liu, Wikipedia, nothing comes up, so I am wondering whether somebody on this planet knows a definition of this.

Let $X,Y$ be topological spaces and $F, G$ be sheaves of modules over $X,Y$ respectively.

The pull-back of a sheaf is very well-documented and defined everywhere with high precision:

If $f:X\rightarrow Y$ is a continuous map, then

$f^*G=f^{-1}G\otimes_{f^{-1}O_Y} O_X$

So I know what $f^*G$ and what $(f^*G)(U)$ are (with $U \subset X$).

But what is $f^*s$ if $s\in G(Y)$, or more generally $s \in G(U)$ where $U \subset Y$ is some open subset of $Y$?

I know there is already a discussion in this thread and apparently the definition is given in a comment for affine schemes (it is just the image by the induced ring map), but I don't find it particularly enlightening. Could somebody please provide a straightforward definition for the pull-back of a section of a sheaf of modules on a general scheme? Can it be defined in a simple way (with e.g. a formula) without using high-powered, unintelligible stuff? In particular I don't know what adjunction correspondance is...

Answer 1

There is a pretty geometric answer, if $X, Y$ are schemes and $G = \mathcal{O}_Y$ is the structure sheaf. In that case, a global section $s \in \Gamma(Y, \mathcal{O}_Y)$ is the same as a morphism $Y \to \mathbb{A}^1_\mathbb{Z}$. The pull-back $f^*s$ then corresponds to the composed morphism $$X \xrightarrow{f} Y \to \mathbb{A}^1_\mathbb{Z},$$ and defines in that way a global section $f^*s \in \Gamma(X, \mathcal{O}_X)$.

For arbitrary ringed spaces $X,Y$, and sheaf $G$, verify that a global section $s \in \Gamma(Y, G)$ is the same as a $\mathcal{O}_Y$-module homomorphism $s: \mathcal{O}_Y \to G$. Pulling-back that morphism yields $$f^*s: \mathcal{O}_X = f^*\mathcal{O}_Y \to f^*G,$$ which defines the global section $f^*s \in \Gamma(X, f^*G)$. Here it is important to verify, that $f^*$ is a functor $\operatorname{Mod}(Y) \to \operatorname{Mod}(X)$.

Also verify that this yields the first construction in the case $G = \mathcal{O}_Y$.

Answer 2

red_trumpet's answer is quite categorical, choosing to talk about morphisms and apply functoriality in place of talking about elements as much as possible. This is a good perspective, but it can also be useful to understand what's going on directly at the level of elements, so I thought I'd add another answer. In fact, there's a fairly satisfying "only thing it could be" definition of $f^*s$.

Suppose we have: $f\colon X\to Y$ a morphism of schemes, $G$ a sheaf of $\mathcal{O}_Y$-modules, $U\subseteq Y$ open, and $s\in G(U)$.

First of all, what should $f^*s$ be? The most natural answer is that we should have $f^*s \in f^*G(f^{-1}U)$. Ok, now let's unpack the definitions. Set $V = f^{-1}(U)$.

First, $f^*G = f^{-1}G\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$. This tensor product is the sheafification of the tensor product presheaf $f^{-1}G\otimes^{\text{p}}_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$ (that p is for "presheaf"). And there is a map of presheaves from any presheaf to its sheafification. So to give an element of $f^*G(V)$, it suffices to give an element of $$(f^{-1}G\otimes_{f^{-1}\mathcal{O}_Y} ^{\text{p}}\mathcal{O}_X)(V) = f^{-1}G(V)\otimes_{f^{-1}\mathcal{O}_Y(V)}\mathcal{O}_X(V).$$ Now given an $R$-module $M$ and an extension of rings $R\to S$, there is a natural map $m\mapsto m\otimes_R 1_S$ from $M$ to the "base changed" $S$-module $M\otimes_R S$ So to give an element of the module above, it suffices to give an element of the module $f^{-1}G(V)$.

The sheaf $f^{-1}G$ is the sheafification of the presheaf $f^{-1}_\text{p}G$ defined by $$f^{-1}_\text{p}G(V) = \text{colim}_{W\supseteq f(V)} G(W).$$ Again, there is a map from any presheaf to its sheafification, so to give an element of $f^{-1}G(V)$, it suffices to give an element of $f^{-1}_\text{p}G(V)$.

We have $U\supseteq f(f^{-1}U) = f(V)$, so $G(U)$ is one of the modules included in the colimit diagram. That is, we have a map $$G(U) \to \text{colim}_{W\supseteq f(V)} G(W) = f^{-1}_\text{p}G(V).$$

The image of $s\in G(U)$ under this long chain of maps $$G(U)\to f^{-1}_\text{p}G(V) \to f^{-1}G(V) \to f^{-1}G(V)\otimes_{f^{-1}\mathcal{O}_Y(V)} \mathcal{O}_X(V)\to f^*G(V)$$ is the desired element $f^*s\in f^*G(V)$.

Answer 3

We treat pullbacks of global sections, nothing changes if we restrict to some other open set. We will be using the fact that tensor product of sheaves commutes with restriction to open sets.

Call $f: X \rightarrow Y$ our morphism of schemes. Let $\mathscr{G}$ be an $\mathcal{O}_Y$-module. There is a canonical morphism $\mathscr{G} \rightarrow f_* f^* \mathscr{G}$ of $\mathcal{O}_Y$-modules. This induces a morphism of global sections $\psi:\Gamma(Y,\mathscr{G}) \rightarrow \Gamma(X,f^*\mathscr{G})$. The question is how to describe concretely the image of $s \in \Gamma(Y,\mathscr{G})$ under this map.

It is instructive to begin with the special case $\mathscr{G}=\mathcal{O}_Y$. Consider the following rings $A=\Gamma(Y,\mathcal{O}_Y), \, B = \Gamma(X,f^{-1} \mathcal{O}_Y), \, C = \Gamma(X,\mathcal{O}_X)$. The data of $f$ include a morphism $\mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. This induces a ring homomorphism $\phi_1: A \rightarrow C$ on global sections. Now the canonical morphism $\mathcal{O}_Y \rightarrow f_* f^{*} \mathcal{O}_Y$ induces a ring homomorphism $\phi_1':A \rightarrow \Gamma(X,f^* \mathscr{G})$. The key is that $\phi_1$ and $\phi_1'$ have the same image. To see this note that we have a ring homomorphism $\phi_2: A \rightarrow B$ induced by $\mathcal{O}_Y \rightarrow f_* f^{-1} \mathcal{O}_Y$. We also have a $\phi_3 : B \rightarrow C$ induced by $f^{-1} \mathcal{O}_Y \rightarrow f^{-1} f_* \mathcal{O}_X \rightarrow \mathcal{O}_X$. Moreover $\phi_1 = \phi_3 \circ \phi_2$. Now $\Gamma(X,f^* \mathcal{O}_Y) = B \otimes_B C$. Finally by checking the definition of $\phi_1'$ we can see that $\phi_1'$ sends $s$ to $\phi_2(s) \otimes 1 \in B \otimes_B C$ which we can identify with $\phi_3(\phi_2(s))=\phi_1(s)$.

For general $\mathscr{G}$ the picture is analogous with the note that $\Gamma(X,f^*\mathscr{G}) = \Gamma(X,f^{-1}\mathscr{G}) \otimes_B C$. That is $\psi$ sends global sections of $\mathscr{G}$ to their canonical image in the $B$-module $\Gamma(X,f^{-1} \mathscr{G})$ and then to their image in the $C$-module $\Gamma(X,f^{-1} \mathscr{G}) \otimes_B C$, the latter being an extended module by scalars according to $\phi_3: B \rightarrow C$.

Answer 4

Let's give geometric motivation to the definition of the pullback of a section. Suppose $f:X\to Y$ is a morphism of topological manifolds (resp., of smooth manifolds, complex manifolds, or classical algebraic $k$-varieties), where $X$ and $Y$ are regarded as locally ringed spaces with their natural sheaves of real-valued continuous (resp., real-valued smooth, complex-valued holomorphic, $k$-valued regular) functions. Let $U\subset Y$ be open. The pullback of a section $s\in\mathcal{O}_Y(U)$ is defined to be $f^*s=s\circ f|_{f^{-1}(U)}\in\mathcal{O}_X(f^{-1}(U))$. That is, $f^*s$ is the image of $s$ along the map $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$. In general:

Definition 1. Let $f:X\to Y$ be a morphism of ringed spaces, let $G$ be a sheaf of $\mathcal{O}_Y$-modules and let $U\subset Y$ be open. The pullback of a section $s\in G(U)$ is the image of $s$ along the unit $G\to f_*f^*G$ of $f^*\dashv f_*$.

In the particular case $G=\mathcal{O}_Y$, one can check that the unit equals $f^\sharp$, so that our general definition is compatible with the particular case above. On the other hand, naturality of the unit implies equivalence with red_trumpet's definition, i.e., the following diagram commutes $$ \require{AMScd} \begin{CD} \mathcal{O}_Y@>>> G\\ @VVV@VVV\\ f_*\mathcal{O}_X=f_*f^*\mathcal{O}_Y@>>> f_*f^*G \end{CD} $$ where the left map is $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$.

Naturality of the unit also allows to prove the following: if $\varphi:G\to G'$ is a morphism of sheaves of $\mathcal{O}_Y$-modules and $s$ is a local section of $G$, then $f^*(\varphi (s))=(f^*\varphi)(f^*s)$. (Write the naturality square of the unit of $f^*\dashv f_*$ for the morphism $\varphi$ and apply Definition 1.)

Note that from red_trumpet's definition it is easy to show that pullingback sections is a contravariantly functorial operation: if $X\xrightarrow{f}Y\xrightarrow{g}Z$ are morphisms of ringed spaces, $H$ is a sheaf of $\mathcal{O}_Z$-modules and $t\in\mathcal{O}_Z(W)$, by using the isomorphism of functors $(g\circ f)^*\cong f^*\circ g^*$ one gets $(g\circ f)^*(t)=f^*(g^*(t))$.

For a very detailed description of the action of the unit $G\to f_*f^*G$ on sections, see Alex Kruckman's answer (it's the composite at the end of his answer). On the level of stalks, the morphism $G(U)\to f^*G(f^{-1}(U))$ is the unique map such that for all $x\in f^{-1}(U)$ the following diagram commutes: $$ \require{AMScd} \begin{CD} G(U)@>>> f^*G(f^{-1}(U))\\ @VVV@VVV\\ G_{f(x)}@>>>(f^*G)_x\cong G_{f(x)}\otimes_{\mathcal{O}_{Y,f(x)}}\mathcal{O}_{X,x} \end{CD} $$ where the bottom map sends $s_{f(x)}$ to $s_{f(x)}\otimes 1\in G_{f(x)}\otimes_{\mathcal{O}_{Y,f(x)}}\mathcal{O}_{X,x}$. That is, $\operatorname{germ}_x^{f^*G}(f^*s)=s_{f(x)}\otimes 1$. To prove this, consider the map $\tilde{s}:\mathcal{O}_Y\to G$ coming from $s\in\Gamma(G,Y)$ and study the map induced by $f^*\tilde{s}$ on stalks at $x\in X$ (and use 0098).

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Warning for the differential geometers (there is some terminological overload): given a smooth manifold $M$, denote $C^\infty_M$ to the sheaf of $\mathbb{R}$-algebras of smooth real-valued functions on $M$. Given a vector bundle $\pi:E\to M$, denote $\Sigma_E$ to the sheaf of $C^\infty_M$-modules of smooth sections of $\pi$. A smooth map of smooth manifolds $f:M\to N$ induces the pullback of covector fields $\varphi:\Sigma_{T^*N}\to f_*\Sigma_{T^*M}$ (that in turns induces the pullback of differential $k$-forms $\Omega_N^k\to f_*\Omega_M^k$), which is a morphism of $C_N^\infty$-modules. The map $\varphi$ should not be confused with the unit $\Sigma_{T^*N}\to f_*f^*\Sigma_{T^*N}$. Actually, $\varphi$ equals $$ \Sigma_{T^*N} \to f_*f^*\Sigma_{T^*N} \to f_*\Sigma_{T^*M}, $$ where $f^*\Sigma_{T^*N}\to \Sigma_{T^*M}$ is the adjoint map to $\Sigma_{T^*N}\to f_*\Sigma_{T^*M}$.