---- > [!definition] Definition. ([[the twiddle functor]]) > > Let $A$ be a ([[commutative ring|commutative]]) [[ring]] and $(\text{Spec }A, \mathcal{O}_{\text{Spec } A})$ the corresponding [[affine scheme]]. The "[[sheaf]] [[sheaf associated to a module|associated to a module]]" construction defines a [[covariant functor]] $\widetilde{(-)}: A\text{-}\mathsf{Mod} \to \mathcal{O}_{X}\text{-}\mathsf{Mod}$ > > between the [[category]] $A\text{-}\mathsf{Mod}$ of $A$-[[module|modules]] and the [[category]] $\mathcal{O}_{X}\text{-}\mathsf{Mod}$ of $\mathcal{O}_{X}\text{-}$[[sheaf of modules|modules]] as follows. > - Objects are assigned in the obvious way: $M \mapsto \widetilde{M}$. > - An $A$-[[linear map]] $f:M \to N$ induces a [[morphism of sheaves of modules|morphism]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] $\widetilde{f}:\widetilde{M} \to \widetilde{N}$ by taking the $U$-[[natural transformation|component]] to be $\begin{align} > \widetilde{f}_{U}: \widetilde{M}(U) &\to \widetilde{N}(U) \\ > \widetilde{f}_{U}(s)(\mathfrak{p}) & := f_{\mathfrak{p}} \circ s(\mathfrak{p}), > \end{align}$ > where $f_{\mathfrak{p}}=(A-\mathfrak{p}) ^{-1} f$ is the [[localization functor]] applied to $f$,[^1] i.e., the map $\frac{n}{s} \mapsto \frac{f(n)}{s}$. [[the twiddle functor is left-adjoint to the global sections functor|The twiddle functor]] $\widetilde{(-)}$ is [[adjoint functor|left-adjoint]] to the global sections [[covariant functor|functor]]. > [!basicproperties] > The [[covariant functor|functor]] $\widetilde{(-)}$ is [[exact functor|exact]], since [[the localization functor is exact]], $(\widetilde{M})_{\mathfrak{p}}=M_\mathfrak{p}$, and [[exactness can be tested on stalks]]. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkQBfU9TXfIRQBGclVqMWbAHLdeIDNjwEiAJjHV6zVohAAZOXyWCiAZg0TtbADrWcMAB45gjugGMcXQwv7KhyUWFxLSldWwARCAB3MDoAJzjo70UBFRQyVWDJHQ4AfVsAWzocAAsAMzi6AGtgNC8eI1T-UUzNbJl86yLSiura+vkUvzVSVstQ-U7u8sqauuTfE3TSUyyrXQAKWyisWDwGWGB2LgBKKeKZvvmGn2M0gJW1ia3rHb2sA5hgaVPzntn+gs7v51Ks2usQC83jB9oc9L9ChdenMBo1hihzGMQjlbPYnC4HO4cKQAAQlGBgNwwVG3JpmR7gia4xzOVweAB09XEMCgAHN4ERQBUIAUkGQQDgIEhhDdhaLEKIJVLEKpZYl5QBWaiSpAANkZOTKf0uKO8cr12uVAHYDWxeblgIj-lcuDTzYgAByWpAATltYVeuxhH0OZS4xuRgOoDDoACMYAwAAqLNIgOJYXklHBm9W+70K8XYmyB96fYC88NOk1RkAx+NJlNCNMZrM5kVILVKpBe8Y46zYUXRuMJ5PAtjpzPZtXtxD6ruIP294sDkBD+ujum6LBgbCwNvym3z4SF9oBldrkeNtgMGBlKcULhAA > \begin{tikzcd} > M \arrow[r] & N \arrow[r] & L & \text{exact} \\ > & \Downarrow & & \\ > M_\mathfrak{p} \arrow[r, "f_\mathfrak{p}"] \arrow[d, "\sim"'] & N_\mathfrak{p} \arrow[r, "g_{\mathfrak{p}}"] \arrow[d, "\sim" description] & L_\mathfrak{p} \arrow[d, "\sim"] & {\text{exact, hence}} \\ > (\widetilde{M})_\mathfrak{p} \arrow[r, "\widetilde{f}_\mathfrak{p}"'] & (\widetilde{N})_\mathfrak{p} \arrow[r, "\widetilde{g}_\mathfrak{p}"'] & (\widetilde{L})_\mathfrak{p} & \text{exact.} > \end{tikzcd} > \end{document} > ``` > ---- #### [^1]: It is not a coincidence that the notation $f_{\mathfrak{p}}$ here looks both like a [[localization functor|localization functor map]] and a [[(pre)sheaf stalk|stalk map]]. ---- #### 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 > ```