---- > [!definition] Definition. ([[sheaf associated to a module]]) > > - Let $A$ be a ([[commutative ring|commutative]]) [[ring]] and $M$ an $A$-[[module]]. > - Consider the [[ringed space]] $(\text{Spec }A, \mathcal{O}_{\text{Spec }A})$. > - Recall that for each $\mathfrak{p} \in \text{Spec }A$, we have the [[localization]] of $M$ at $\mathfrak{p}$, $M_{\mathfrak{p}}=\left\{ \frac{m}{a}: m \in M, a \in A - \mathfrak{p} \right\}$. > > An entirely analogous construction to the [[structure sheaf on a ring spectrum|structure sheaf]] $\mathcal{O}_{\text{Spec }A}$ on the [[prime ideal|ring spectrum]] $\text{Spec } A$ produces a [[sheaf]] $\widetilde{M}=\widetilde{M}_{A}$ of $\mathcal{O}_{\text{Spec }A}$-[[sheaf of modules|modules]]. Specifically, we define for $U \subset X$ > > $\widetilde{M}(U)=\left\{ s:U \to \coprod_{\mathfrak{p} \in U} M_{\mathfrak{p}} : \begin{align} > &\textcolor{Thistle}{(1) \ s(\mathfrak{p}) \in M_{\mathfrak{p}} \text{ for all }\mathfrak{p}} \\ > & \textcolor{Skyblue}{(2) \ \forall \mathfrak{p} \in U, \exists \text{nbhd } \mathfrak{p} \in V \subset U } \\ > & \quad \quad \textcolor{Skyblue}{\text{ and } m \in M, a \in A \text{ s.t. } \forall \mathfrak{q} \in V, } \\ > & \textcolor{Skyblue}{\quad \quad a \notin \mathfrak{q} \text{ and } s(\mathfrak{q})=\frac{m}{a} \in M_{\mathfrak{q}}} > \end{align} \right\}.$ > $\widetilde{M}(U)$ evidently carries ([[sheaf|sheaf of]]) $\mathcal{O}_{\text{Spec }A}$-[[module]] structure, because each $M_{\mathfrak{p}}$ is an $A_{\mathfrak{p}}$-[[module]]. Explicitly, given $r \in \mathcal{O}_{\text{Spec }A}(U)$, $s \in \widetilde{M}(U)$, one takes $(r \cdot s)(\mathfrak{p}) := r(\mathfrak{p})s(\mathfrak{p})$ > where the latter multiplication comes from the $A_{\mathfrak{p}}$-[[module]] structure on $M_{\mathfrak{p}}$. > > Taking twiddles is a [[covariant functor|functor]] $\widetilde{(-)}:A\text{-}\mathsf{Mod} \to \mathcal{O}_{X}\text{-}\mathsf{Mod}$. See [[the twiddle functor]]. > [!specialization] > Putting $M=A$ recovers the familiar [[structure sheaf on a ring spectrum]] $\tilde{A}=\mathcal{O}_{\text{Spec } A}$. ^specialization > [!basicproperties] > 1. $(\widetilde{M})_{\mathfrak{p}}=M_{\mathfrak{p}}$; > 2. $\widetilde{M}\big( D(f) \big)=M_{f}$. Special case: $\widetilde{M}(\text{Spec }A)=M$. > > [!proof] Proof. > > The proofs in [[structure sheaf on a ring spectrum|structure sheaf]] go straight through, replacing the ring elements in numerators with module elements. > [!basicexample] > If $M=A^{\oplus I}$ is a [[free module]], then $\widetilde{M}=\bigoplus_{i \in I} \mathcal{O}_{\text{Spec } A}$ > is a [[free sheaf of modules|free]] $\mathcal{O}_{\text{Spec } A}$-[[sheaf of modules|module]]. > > Indeed, [[localization commutes with intersections, sums, and quotients|since localization commutes with taking direct sums]], we have $\coprod_{\mathfrak{p} \in U}(A^{\oplus I})_{\mathfrak{p}}=\coprod_{\mathfrak{p} \in U}(A_{\mathfrak{p}})^{\oplus I}.$ > This means that a section $s \in \widetilde{M}(U)$ is of the form $s:U \to \coprod_{\mathfrak{p} \in U}\bigoplus_{i \in I}A_{\mathfrak{p}}$, which is precisely the same data as specifying component maps $s_{i}:U \to \coprod_{\mathfrak{p} \in U}A_{\mathfrak{p}}$ for $i \in I$, i.e., as specifying a section section $(s_{i})_{i \in I} \in( \bigoplus_{i \in I} \mathcal{O}_{\text{Spec } A})(U)$. > ---- #### ---- #### 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 > ```