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*}

6.26 Morphisms of ringed spaces and modules

We have now introduced enough notation so that we are able to define the pullback and pushforward of modules along a morphism of ringed spaces.

Definition 6.26.1. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces.

  1. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We define the pushforward of $\mathcal{F}$ as the sheaf of $\mathcal{O}_ Y$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ of the module structure given in Lemma 6.24.5.

  2. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$-modules. We define the pullback $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_ X$-modules defined by the formula

    \[ f^*\mathcal{G} = \mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ Y} f^{-1}\mathcal{G} \]

    where the ring map $f^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X$ is the map corresponding to $f^\sharp $, and where the module structure is given by Lemma 6.24.6.

Thus we have defined functors

\begin{eqnarray*} f_* : \textit{Mod}(\mathcal{O}_ X) & \longrightarrow & \textit{Mod}(\mathcal{O}_ Y) \\ f^* : \textit{Mod}(\mathcal{O}_ Y) & \longrightarrow & \textit{Mod}(\mathcal{O}_ X) \end{eqnarray*}

The final result on these functors is that they are indeed adjoint as expected.

Lemma 6.26.2. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$-modules. There is a canonical bijection

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(f^*\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Y}(\mathcal{G}, f_*\mathcal{F}). \]

In other words: the functor $f^*$ is the left adjoint to $f_*$.

Proof. This follows from the work we did before:

\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(f^*\mathcal{G}, \mathcal{F}) & = & \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O}_ X)}( \mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ Y} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{Mod}(f^{-1}\mathcal{O}_ Y)}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_ Y}) \\ & = & \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Y}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}

Here we use Lemmas 6.20.2 and 6.24.7. $\square$

Lemma 6.26.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. The functors $(g \circ f)_*$ and $g_* \circ f_*$ are equal. There is a canonical isomorphism of functors $(g \circ f)^* \cong f^* \circ g^*$.

Proof. The result on pushforwards is a consequence of Lemma 6.21.2 and our definitions. The result on pullbacks follows from this by the same argument as in the proof of Lemma 6.21.6. $\square$

Given a morphism of ringed spaces $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$, and a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$, a sheaf of $\mathcal{O}_ Y$-modules $\mathcal{G}$ on $Y$, the notion of an $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ of sheaves of modules makes sense. We can just define it as an $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ of abelian sheaves such that for all open $V \subset Y$ the map

\[ \mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V) \]

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 $f^\sharp _ V : \mathcal{O}_ Y(V) \to \mathcal{O}_ X(f^{-1}V)$. The set of $f$-maps between $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection with the sets $\mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O}_ X)}(f^*\mathcal{G}, \mathcal{F})$ and $\mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O}_ Y)}(\mathcal{G}, f_*\mathcal{F})$. See above.

Composition of $f$-maps is defined in exactly the same manner as in the case of $f$-maps of sheaves of sets. In addition, given an $f$-map $\mathcal{G} \to \mathcal{F}$ as above, and $x \in X$ the induced map on stalks

\[ \varphi _ x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_ x \]

is an $\mathcal{O}_{Y, f(x)}$-module map where the $\mathcal{O}_{Y, f(x)}$-module structure on $\mathcal{F}_ x$ comes from the $\mathcal{O}_{X, x}$-module structure via the map $f^\sharp _ x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$. Here is a related lemma.

Lemma 6.26.4. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$-modules. Let $x \in X$. Then

\[ (f^*\mathcal{G})_ x = \mathcal{G}_{f(x)} \otimes _{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x} \]

as $\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\sharp _ x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.

Proof. This follows from Lemma 6.20.3 and the identification of the stalks of pullback sheaves at $x$ with the corresponding stalks at $f(x)$. See the formulae in Section 6.23 for example. $\square$


Comments (1)

Comment #1121 by OlgaD on

Hi! I'm a little confused. In section 6.25 is determined by mapping Here


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0094. Beware of the difference between the letter 'O' and the digit '0'.