Sheaves of Modules

Let \(X=\text{Spec}(R)\) be an affine scheme. Let \(X'\) be an affine \(X\) scheme i.e., \(X'=\text{Spec}(R')\) for some ring \(R'\) with a morphism of schemes \(\pi: X'\rightarrow X\). This \(\pi\) comes with morphism of global sections \(R=\Gamma(X,\mathcal{O}_X)\rightarrow \Gamma(X',\mathcal{O}_X')=R'\) giving \(R'\), structure of an \(R\) algebra. So, any affine scheme over \(X=\text{Spec}(R)\) is simply the specturm of an \(R\) algebra. Conversely, given an \(R\) algebra say \(R'\), we have an affine scheme \(X'=\text{Spec}(R')\) over \(X\) with morphism \(\pi:X'\rightarrow X\). Being a morphism of affine schemes, \(\pi: X\rightarrow X\) is an affine morphism. Let \(X\) be an arbitrary scheme. We want to associate an \(X\) scheme \(X'\) such that the structure morphism \(\pi:X'\rightarrow X\) is an affine morphism. To do this in case of \(X=\text{Spec}(R)\) we have fixed an \(R\) algebra and then associated an affine scheme for this. In case of an arbitrary scheme \(X\) unlike the case of affine scheme \(X=\text{Spec}(R)\) there is no single ring that has all information about the scheme \(X\). It is only natural to consider the collection \(\{\mathcal{O}_X(U): U\subseteq X\}\) varying over all open subsets of \(X\) to get information about the scheme \(X\). Choosing an \(\mathcal{O}_X(U)\) algebra \(\mathcal{F}(U)\) for each open \(U\subseteq X\) we associate an \(X\) scheme \(X'\) for this collection \(\{\mathcal{F}(U)\}\) of \(\mathcal{O}_X(U)\) algebras. It is only natural to put a condition that this collection \(\{\mathcal{F}(U)\}\) to be compatible with structure sheaf \(\mathcal{O}_X\) i.e., we want \(U\mapsto \mathcal{F}(U)\) to give a structure of sheaf of \(\mathcal{O}_X\) algebras on \(X\). So, given an arbitrary scheme \(X\) and a sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras we associate an \(X\) scheme \(X'\) such that the structure map \(X'\rightarrow X\) is an affine morphism. It is not obvious at this point but we also want \(\mathcal{F}\) to be a quasicoherent sheaf of \(\mathcal{O}_X\) modules. We call this \(X'\), Global spec or Relative spec of sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\) algebras over \(X\) denoted by \(\textbf{Spec} (\mathcal{F})\). Here we make two important remarks : The \(\textbf{Spec}\) construction gives an important way to understand affine morphisms. Note that \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism. The converse is also true. If \(f:X\rightarrow Y\) is an affine morphism then \(\mathcal{A}=f_*\mathcal{O}_X\) is a quasi coherent sheaf of \(\mathcal{O}_Y\) algebras and \(X\cong \text{Spec} (\mathcal{A})\). The \(\textbf{Spec}\) construction is used to assign a geometric vector bundle on a scheme \(Y\) to each locally free sheaf \(\mathcal{E}\) of rank \(n\) on a scheme \(Y\) which gives a bijection between isomorphism classes of locall free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) on \(Y\). To define \(\textbf{Spec}(\mathcal{F})\) we do not need \(\mathcal{F}\) to be quasi coherent, but \(\mathcal{F}\) quasi coherent implies the structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\) is an affine morphism. Now, we try to construct \(\textbf{Spec}(\mathcal{F})\) and structure map \(\textbf{Spec}(\mathcal{F})\rightarrow X\). One way to do this is gluing the schemes \(\text{Spec}(\mathcal{F}(U))\) over all open subsets \(U\subseteq X\). Another way is to use universal property of \(\text{Spec}\) of a ring. We have following result : Let \(A\) be a ring and let \((X,\mathcal{O}_X)\) be a scheme. Given a morphism \(f:X\rightarrow \text{Spec}(A)\) we have an associated map on sheaves \(f^{\#}:\mathcal{O}_{\text{Spec}(A)}\rightarrow f_*\mathcal{O}_X\). Taking global sections, we obtain a ring homomorphism \(A\rightarrow \mathcal{O}_X(X)\). Thus there is a natural map \(\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow\text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).\) ...

Let \(A\) be a ring and \(M\) be an \(A\) module. We associate a sheaf of modules \(\widetilde{M}\) on \(X= \text{Spec(A)}\) with this module \(M\). These modules are our models for quasi-coherent sheaves. For an open subset \(U\subseteq \text{Spec(A)}\) we definewhere the condition \(\dagger\) says that given \(p\in U\) we have \(s(p)\in M_p\) and that \(s\) is locally a fraction i.e., given \(p\in U\) there exists an open set \(U(p)\subseteq U\) and \(m\in M, f\in A\) such that \(s(q)=\frac{m}{f}\in M_q\) for all \(q\in U(p)\). With obvious restriction maps this defines a sheaf \(\widetilde{M}\) on \(X= \text{Spec(A)}\) called the sheaf associated with \(M\). This should remind you something similar we have done before. We have defined structure sheaf on \(X=\text{Spec(A)}\) in exactly same way where \(M\) in this definition is replaced by ring \(A\). Just to confirm, we havewhere the condition \(\dagger\) says that given \(p\in U\) we have \(s(p)\in A_p\) and that \(s\) is locally a fraction i.e., given \(p\in U\) there exists an open set \(U(p)\subseteq U\) and \(a\in A, f\in A\) such that \(s(q)=\frac{a}{f}\in A_q\) for all \(q\in U(p)\). This should suggest some relation between \(\widetilde{M}(U)\) and \(\mathcal{O}_X(U)\). It turns out that \(\widetilde{M}(U)\) is an \(\mathcal{O}_X(U)\) module for every open \(U\subseteq \text{Spec(A)}\). So, \(\widetilde{M}\) is a sheaf of \(\mathcal{O}_X\) modules. In case of structure sheaf \(\mathcal{O}_X\) on \(X=\text{Spec (A)}\) we have following proposition. Proposition : Let \(A\) be a ring, and \((\text{Spec A},\mathcal{O})\) its spectrum. For any \(\mathfrak{p}\in \text{Spec A}\), the stalk \(\mathcal{O}_{\mathfrak{p}}\) of the sheaf \(\mathcal{O}_{}\) is isomorphic to the local ring \(A_{\mathfrak{p}}\) i.e., \(\mathcal{O}_{\mathfrak{p}}\cong A_{\mathfrak{p}}\). For any element \(f\in A\), the ring \(\mathcal{O}(D(f))\) is isomorphic to the localized ring \(A_f\) i.e., \(\mathcal{O}(D(f))\cong A_f\). In particular, \(\Gamma(\text{Spec A}, \mathcal{O})\cong A\). In case of associated sheaf \(\widetilde{M}\) on \(X=\text{Spec A}\) we have same results with \(A\) replaced by \(M\) and \(\mathcal{O}\) replaced by \(\widetilde{M}\). Precisely, we have following proposition. Proposition : Let \(A\) be a ring, \(M\) be an \(A\) module and and \(\widetilde{M}\) be the associated sheaf on \(X=\text{Spec A}\). For any \(\mathfrak{p}\in \text{Spec A}\), the stalk \(\widetilde{M}_{\mathfrak{p}}\) of the sheaf \(\widetilde{<}\) is isomorphic to the localization \(M_{\mathfrak{p}}\) i.e., \(\widetilde{M}_{\mathfrak{p}}\cong M_{\mathfrak{p}}\). For any element $f\in A$, the ring \(\widetilde{M}(D(f))\) is isomorphic to the localization \(M_f\) i.e., \(\widetilde{M}(D(f))\cong M_f\). In particular, \(\Gamma(\text{Spec A}, \widetilde{M})\cong M\). Proposition : Let \(A\) be a ring and let \(X=\text{Spec}(A)\). Also let \(A\rightarrow B\) be a ring homomorphism, and let \(f:\text{Spec}(B)\rightarrow \text{Spec}(A)\) be the correpsonding morphism of spectra. Then : the map \(M\mapsto \widetilde{M}\) gives an exact, fully faithful functor from category of \(A\) modules to the category of \(\mathcal{O}_X\) modules. \(\widetilde{M\otimes_A N}=\widetilde{M}\otimes_{\mathcal{O}_X}\widetilde{N}\). \(\widetilde{\bigoplus M_i}=\bigoplus \widetilde{M_i}\). For a \(B\) module \(N\), we have \(f_*(\widetilde{N})=\widetilde{~_A N}\) where \(~_A N\) is \(N\) considered as an \(A\) module. For a \(A\) module \(M\), we have \(f^*(\widetilde{M})=\widetilde{M\otimes_A B}\).