---- > [!definition] Definition. ([[pushforward 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]], > - $\mathcal{F}$ a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]; > > The [[pushforward sheaf|pushforward]] **$f_{*}\mathcal{F}$** is naturally a [[sheaf]] of $\mathcal{O}_{Y}$-[[sheaf of modules|modules]], as $s \in \mathcal{O}_{Y}(U)$ [[module|acts on]] $m \in (f_{*}\mathcal{F})(U)=\mathcal{F}\big( f ^{-1}(U) \big)$ via $s \cdot m:= f^{\sharp}_{U}(s) \cdot m.$ > Thus, $f_{*}\mathcal{F}$ may be referred to as the **pushforward of the sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules** along $(f, f^{\sharp})$. > [!note] Remark. (Pushforwards and [[restriction of scalars]]) > Constructing $f_{*}\mathcal{F}$ as a (sheaf of) $\mathcal{O}_{Y}$-[[sheaf of modules|module(s)]] may also be understood as follows. Since $\mathcal{F}$ is a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]], for each open $U \subset Y$ there is a [[module|module structure]] [[ring homomorphism|homomorphism]] $\mathcal{O}_{X}\big(f ^{-1} (U) \big) \to \text{End}_{\mathsf{Ab}}\big( \mathcal{F}(f ^{-1}(U)) \big).$ > Precomposing with $f^{\sharp}_{U}: \mathcal{O}_{Y}(U) \to \mathcal{O}_{X}\big( f ^{-1}(U) \big)$ gives a [[module|structure]] [[ring homomorphism|homomorphism]] $\mathcal{O}_{Y}(U) \xrightarrow{f^{\sharp}_{U}} (f_{*}\mathcal{O}_{X})(U) \to \text{End}_{\mathsf{Ab}}\big( (f_{*}\mathcal{F})(U) \big).$ > This is precisely the discussion in [[restriction of scalars]]. > [!basicexample] Example. (Pushing forward [[sheaf associated to a module|the tilde construction]]) > - Let $A,B$ be rings, $\varphi:B \to A$ a [[ring homomorphism]] inducing $f:\text{Spec }A \to \text{Spec } B$. Let $M$ be an $A$-module, and let $M_{B}$ be the [[restriction of scalars]] to $B$, i.e., $M_{B}$ is the $B$-[[module]] with [[abelian group]] $M$ and structure morphism $B \xrightarrow{\varphi}A \to \text{End }M$. Then there is an equality of sheaves $f_{*} \widetilde{M}= \widetilde{M_{B}}.$ > > > > [!proof]- > > It is enough to check that there are isomorphisms on the basis elements $D(g)$ of $\text{Spec }B$, in other words that $(f_{*}\widetilde{M})\big( D(f) \big) \cong \widetilde{M_{B}}\big( D (g) \big)$ for all $g \in B$. Noting that $(f_{*} \widetilde{M})\big( D(g) \big)=\widetilde{M}\big( f ^{-1}(D(g)) \big)=\widetilde{M}(D(\varphi(g)))=M_{\varphi(g)}$ and $\widetilde{M_{B}}(D(g))=(M_{B})_{g}$, it is equivalent to check that there is an isomorphism of $B$-modules $M_{\varphi(g)} \cong (M_{B})_{g}$. This is true by the construction of $M_{B}$ that $b \cdot m=\varphi(b)m$ for $b \in B$. ---- #### [[spec functor]] ---- #### 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 > ```