---- > [!definition] Definition. ([[morphism of ringed spaces]]) > Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$ be [[ringed space|ringed spaces]]. A **morphism $f:(X, \mathcal{O}_{X}) \to (Y, \mathcal{O}_{Y})$** is the data of a [[continuous]] map $f:X \to Y$ together with a [[morphism of (pre)sheaves|morphism of sheaves]] of [[ring|rings]] $f^{\sharp}:\mathcal{O}_{Y} \to f_{*}\mathcal{O}_{X}$, where $f_{*}\mathcal{O}_{X}$ denotes the [[pushforward sheaf|pushforward]] of $\mathcal{O}_{X}$ by $f$. > > > Explicitly, $(f_{*}\mathcal{O}_{X})(U):=\mathcal{O}_{X}\big( f ^{-1}(U) \big)$, and we require the following diagram to commute for all $U$, $V \subset U$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAHkAvgH0AmgAoAqgEoQY0uky58hFAEZyVWoxZtOPfkIajJsgGpKVa7HgJEy2-fWatEIGQDMJAFQABCa8gsLiEgAaCvJ2qiAYjppEum7UHkbefoEh3GHmltGxtsr6MFAA5vBEoL4AThBcSGQgOBBIAEwZhl4gvgB6nHB8dPVoEnIg1Ax0AEYwDAAK6k5aIPVYlXw4ygkNTUi6bR2IAMw9nmyDw6PjEtZ7dY3NiK3tR5dZHOwCUBA4PI0GD1HAPJ79F5dagfc5fPqcP4AoEgwHgsQUMRAA > \begin{tikzcd} > \mathcal{O}_Y(U) \arrow[d, "f^\sharp_U"'] \arrow[r, "\cdot \vert_V"] & \mathcal{O}_Y(V) \arrow[d, "f^\sharp_V"] \\ > (f_* \mathcal{O}_X)(U) \arrow[r, "\cdot \vert _V"] & (f_* \mathcal{O}_X)(V) > \end{tikzcd} > \end{document} > ``` > > [!note] Composition in the [[category]] $\mathsf{RingedSp}$. > How should $(g \circ f)^{\sharp}$ in the composition $\big(g \circ f, (g \circ f)^{\sharp} \big): (X, \mathcal{O}_{X}) \to (Z, \mathcal{O}_{Z})$ of two morphisms $\begin{align} (f, f^{\sharp}): (X, \mathcal{O}_{X}) \to (Y, \mathcal{O}_{Y}) \text{ and } (g, g^{\sharp}): (Y, \mathcal{O}_{Y}) \to (Z, \mathcal{O}_{Z})\end{align}$ be defined? Note that neither 'typographically obvious' choice $g^{\sharp} \circ f^{\sharp}$ or $f^{\sharp} \circ g^{\sharp}$ makes sense, because $g^{\sharp}:\mathcal{O}_{Z} \to g_{*}\mathcal{O}_{Y}$ is a [[morphism of (pre)sheaves|morphism of sheaves]] on $Z$ and $f^{\sharp}:\mathcal{O}_{Y} \to f_{*}\mathcal{O}_{Y}$ is a [[morphism of (pre)sheaves|morphism of sheaves]] on $Y$. > The answer is still natural, though: [[pushforward sheaf|push]] $f^{\sharp}$ forward along $g$ to a [[morphism of (pre)sheaves|morphism of sheaves]] on $Z$, that is, define[^1] $(g \circ f)^{\sharp}:= g_{*}f^{\sharp} \circ g^{\sharp}: \mathcal{O}_{Z} \to (g \circ f)_{*}\mathcal{O}_{Y}.$ ^note [^1]: Recall pushforward is covariantly [[covariant functor|functorial]]: $g_{*}f_{*}=(g \circ f)_{*}$. ---- #### ---- #### 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 > ```