----- > [!proposition] Proposition. ([[the twiddle functor is left-adjoint to the global sections functor]]) > Let $A$ be a ([[commutative ring|commutative]]) [[ring]]. Let $A\text{-}\mathsf{Mod}$ denote the [[category]] of $A$-[[module|modules]] and $\mathcal{O}_{\text{Spec }A}\text{-}\mathsf{Mod}$ denote the [[category]] of [[sheaf|sheaves]] of $\mathcal{O}_{\text{Spec }A}\text{-}$[[sheaf of modules|modules]]. > > [[the twiddle functor|The twiddle functor]] $\widetilde{(-)}:A\text{-}\mathsf{Mod} \to \mathcal{O}_{\text{Spec } A}\text{-}\mathsf{Mod}$ is [[adjoint functor|left-adjoint]] to the global sections [[covariant functor|functor]] $\Gamma(\text{Spec }A, -): \mathcal{O}_{\text{Spec } A}\text{-}\mathsf{Mod} \to A\text{-}\mathsf{Mod}$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACzgAzYAEEAvpxwwAHjmABaCd14DhAWWhiQY0uky58hFAEZyVWoxZtOPfgGNGwAPJiA+sEky5AZTQw7AARi4p6yCkq2qsAaUFpi5jBQAObwRKCCAE4QXEhkIDgQSKYgAEYwYFBIAMx59MysiBzsAO5YsHgMsMAAFPIAlHG6IJnZRdQFudRlFdW1lg1NAOJ0XDzdoT5+gcGkAfIBfdoUYkA > \begin{tikzcd} > {A}\text{-}\mathsf{Mod} \arrow[r, "\widetilde{(-)}", bend left] & \mathcal{O}_{\text{Spec }A}\text{-}\mathsf{Mod} \arrow[l, "{\Gamma(\text{Spec }A, - )}", bend left] > \end{tikzcd} > \end{document} > ``` > > That is, we have a [[natural transformation|natural]] [[bijection]] $\text{Hom}_{A\text{-}\mathsf{Mod}}\big( M, \Gamma(\text{Spec } A, \mathcal{F}) \big) \leftrightarrow \text{Hom}_{\mathcal{O}_{\text{Spec } A}\text{-}\mathsf{Mod}}(\widetilde{M}, \mathcal{F})$ > for all objects $M$ of $A\text{-}\mathsf{Mod}$ and $\mathcal{F}$ of $\mathcal{O}_{\text{Spec } A}\text{-}\mathsf{Mod}$. > [!proof]- Proof. ([[the twiddle functor is left-adjoint to the global sections 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 > ```