---- > [!definition] Definition. ([[pullback of a sheaf of modules]]) > Given > - $(f, f^{\sharp}): (X , \mathcal{O}_{X}) \to (Y, \mathcal{O}_{Y})$ a [[morphism of ringed spaces|morphism]] of [[ringed space|ringed spaces]], > - $\mathcal{G}$ a [[sheaf]] of $\mathcal{O}_{Y}$-[[sheaf of modules|modules]]; > > The [[pullback sheaf|pullback]] $f ^{-1}\mathcal{G}$ is naturally a [[sheaf]] of $f ^{-1}\mathcal{O}_{Y}$-[[sheaf of modules|modules]],[^2] but is not naturally a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]].[^1] > > Recall, however, that [[sheaf pullback and pushforward are adjoint functors]]. The [[adjoint functor|adjoint]] [[morphism of (pre)sheaves|morphism]] to $f^{\sharp}$ (again denoted $f^{\sharp}$) is a [[morphism of (pre)sheaves|sheaf morphism]] $f^{\sharp}: f ^{-1} \mathcal{O}_{Y} \to \mathcal{O}_{X}$ > inducing $f ^{-1} \mathcal{O}_{Y}$-[[sheaf of modules|module]] structure on $\mathcal{O}_{X}$. Using this, we can define the **pullback** $f^{*}\mathcal{G}$ (note the modified notation) to be the [[tensor product sheaf of modules|tensor product sheaf]] *of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]* $f^{*}\mathcal{G}=f ^{-1} \mathcal{G} \otimes_{f ^{-1} \mathcal{O}_{Y}}\mathcal{O}_{X}$ > by employing [[extension of scalars]] per the note below. > > [!note] Pullbacks and [[extension of scalars]]. > At the level of [[module|modules]] (let $U \subset X)$, we have the following situation. There is a [[ring homomorphism]] $f^{\sharp}_{U}: (f ^{-1} \mathcal{O}_{Y})(U) \to \mathcal{O}_{X}(U)$ endowing $\mathcal{O}_{X}(U)$ with $(f ^{-1} \mathcal{O}_{Y})(U)$-[[module]] structure [[bimodule|in addition to]] its native $\mathcal{O}_{X}(U)$-[[module]] structure. This allows us to form the [[tensor product of modules]] $( f ^{-1} \mathcal{G})(U) \otimes_{(f ^{-1} \mathcal{O})(U)} \mathcal{O}_{X}(U),$ viewed as an $\mathcal{O}_{X}(U)$-[[module]] via the natural action on pure tensors $r(s \otimes t) := s \otimes (rt)$. This is precisely the methodology discussed in [[extension of scalars]]: in order to get our sheaf to 'have scalars in $\mathcal{O}_{X}, we tensor. ^note [^1]: Optional motivation: in [[pushforward of a sheaf of modules]] we saw that $f^{\sharp}: \mathcal{O}_{Y} \to f_{*}\mathcal{O}_{X}$ lets us [[restriction of scalars|restrict scalars]] to go from an $f_{*}\mathcal{O}_{X}$-module to an $\mathcal{O}_{Y}$-module. Presumably, we would want to do the 'adjoint thing' here, i.e. to *extend* scalars to go from an $f ^{-1}\mathcal{O}_{Y}$-module to an $\mathcal{O}_{X}$-module. That'd require a [[morphism of ringed spaces]] $f^{\sharp}: f ^{-1}\mathcal{O}_{Y} \to \mathcal{O}_{X}$, which is naturally given by its adjoint morphism under [[sheaf pullback and pushforward are adjoint functors]]. > [!basicexample] Pulling Back a Line Bundle. > The situation simplifies in the case that $\mathcal{G}=\mathcal{L}$ is a [[locally free sheaf|line bundle]] on $Y$. > Letting $\{ U_{i} \}$ denote a trivializing cover for $\mathcal{L}$, put $X_{i}:=f ^{-1}(U_{i})$. We have restricted morphisms $f_{i}: f |_{X_{i}} \to U_{i}$. It is easy to pull back $\mathcal{L} |_{U_{i}}$ under $f_{i}$: since $\mathcal{L} |_{U_{i}} \cong \mathcal{O}_{U_{i}}$, $f_{i}^{*}(\mathcal{L} |_{U_{i}})=f_{i}^{*}(\mathcal{O}_{U_{i}})= (f ^{-1} \mathcal{O}_{U_{i}}) \otimes_{f ^{-1} \mathcal{O}_{U_{i}}} \mathcal{O}_{X_{i}} \cong \mathcal{O}_{X_{i}},$ where we have used the general fact $R \otimes_{R} M \cong M$. This means that the trivializing cover $\{ U_{i} \}$ for $\mathcal{L}$ pulls back to a trivializing cover $\{ X_{i} \}$ for $f^{*}\mathcal{L}$, making it into a line bundle. Furthermore, the transition functions for $f^{*}\mathcal{L}$ are given by $f^{\sharp}(g_{ij})$. ^basic-example - [ ] be more thorough in the part about transition functions > [!basicexample] Example. (Pulling back [[sheaf associated to a module|the tilde construction]]) > Let $A$ and $B$ be [[ring|rings]], $\varphi:B \to A$ a [[ring homomorphism]] inducing a [[morphism of locally ringed spaces|morphism]] $f:\text{Spec }A \to \text{Spec }B$. Let $N$ be a $B$-module. $\varphi$ turns $A$ into a $(B,A)$-[[bimodule]], whence we may [[extension of scalars|extend scalars]] to naturally obtain an $A$-[[module]] $N \otimes_{B} A$. Then $f^{*} \widetilde{N}= \widetilde{N \otimes_{B} A}$ > as an equality of $\mathcal{O}_{\text{Spec }A}$-modules. ^basic-example Check on stalks. For $\mathfrak{p} \in \text{Spec }A$, $\mathfrak{q}=f(\mathfrak{p})=\varphi ^{-1}(\mathfrak{p})$: on the one hand$\begin{align} (f^{*} \widetilde{N}) _{\mathfrak{p}} &= (f ^{-1} \widetilde{N} \otimes_{f ^{-1} \mathcal{O}_{\text{Spec } B} } \mathcal{O}_{\text{Spec } A})_{\mathfrak{p}} \\ &= (f ^{-1} \widetilde{N})_{\mathfrak{p}} \otimes_{B_{f(\mathfrak{p})}} \mathcal{O}_{\text{Spec }A, \mathfrak{p}} \\ &= \widetilde{N}_{f(\mathfrak{p})} \otimes_{B_{f(\mathfrak{p})}}A_{\mathfrak{p}} \\ &= {N}_{\mathfrak{q}} \otimes_{B_{\mathfrak{q}}} A_{\mathfrak{p}}. \end{align}$ On the other, $\begin{align} \widetilde{(N \otimes_{B} A)}_{\mathfrak{p}} &= (N \otimes_{B} A)_{\mathfrak{p}} \\ &= N_{\mathfrak{q}} \otimes_{B_{\mathfrak{q}}}A_{\mathfrak{p}} \end{align}$ as required. ---- #### [^2]: Indeed, for $U \subset X$ the action of $(f^{-1}\mathcal{O}_{Y})(U)$ on $(f ^{-1} \mathcal{G})(U)$ is $\underbrace{ [W,s] }_{ \in (f ^{-1}\mathcal{O}_{Y})(U) } \cdot \underbrace{ [V, m] }_{ \in (f ^{-1} \mathcal{G})(U) }:= s |_{W \cap V} \cdot m |_{W \cap V}$. This is well-defined by construction of the $\mathcal{O}_{X}$-module definition. [[lines bundles and transition functions]] ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```