The sheaf attached to a module on an affine scheme

$P_M$ is a presheaf of ${\mathcal{O}}_X$-modules. The restriction $M{\otimes }_B{\mathcal{O}}_X(U)\to M{\otimes }_B{\mathcal{O}}_X(V)$ is ${\mathrm{i}\mathrm{d}}_M\otimes \rho$ for $V\subseteq U$, it is additive and functorial, and $a\cdot (m\otimes a^{\mathrm{\prime }})=m\otimes a{a}^{\mathrm{\prime }}$ shows that it is ${\mathcal{O}}_X(U)$-linear after restricting scalars along ${\mathcal{O}}_X(U)\to {\mathcal{O}}_X(V)$; the module structure on $a{P}_M$ is constructed as follows. For a presheaf of ${\mathcal{O}}_X$-modules $P$, a scalar $a\in {\mathcal{O}}_X(U)$ acts on a plus-section represented by $(U_i,s_i)$ by $(U_i,a{\mathrm{\mid }}_{U_i}{s}_i)$. This is independent of the representative because equality of germs is preserved by multiplication. Addition is defined on the common refinement of two covers. All module laws and compatibility with restriction follow on these local representatives from the corresponding laws in $P$. Apply this construction twice to obtain the module structure on $a{P}_M$; its unit map is linear. Moreover every section of $aP$ is locally represented by a section of $P$, by refining twice the presentations in [F1].

Morphisms of presheaves of ${\mathcal{O}}_X$-modules $\phi \text{\hspace{0.17em}⁣}:P_M\to \mathcal{F}$ with $\mathcal{F}$ a sheaf are the compatible families of ${\mathcal{O}}_X(U)$-linear maps ${\phi }_U\text{\hspace{0.17em}⁣}:M{\otimes }_B{\mathcal{O}}_X(U)\to \mathcal{F}(U)$, and these are in canonical bijection with $B$-linear maps $g\text{\hspace{0.17em}⁣}:M\to \mathcal{F}(X)$: the map $g$ is recovered as ${\phi }_X(-\otimes 1)$, while for a given $g$ the formulas ${\psi }_U(m\otimes a):=a\cdot g(m){\mathrm{\mid }}_U$ define a family that is well defined and $B$-bilinear in $(m,a)$, hence ${\mathcal{O}}_X(U)$-linear by [F4], and compatible with restrictions by the compatibility of the restrictions of $\mathcal{F}$. The two assignments are inverse because the values on the elements $m\otimes 1$ determine an ${\mathcal{O}}_X(U)$-linear map on all of $M{\otimes }_B{\mathcal{O}}_X(U)$.

By [F2] a linear presheaf map $P_M\to \mathcal{F}$ extends uniquely as a morphism of sheaves. This extension is linear: locally write a section as $\eta (s)$ using step 1.1, and then $\bar{\phi }(a\eta (s))=\bar{\phi }(\eta (as))=\phi (as)=a\phi (s)$; additivity is checked on a common local presentation in the same way. Equality of sheaf sections is local. Conversely precomposition of a linear sheaf map with the linear unit is linear. Thus [F2] restricts to the module morphisms, and step 1.2 gives a bijection ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\tilde{M},\mathcal{F})\cong {\mathrm{Hom}}_B(M,\mathcal{F}(X))$; under it, $\phi$ corresponds to ${\phi }_X\circ {\eta }_{P_M,X}\circ {\iota }_M={\phi }_X\circ \epsilon$, using the identification ${\iota }_M$ from [F5]. The displayed formula is the component of the presheaf map $P_M\to \mathcal{F}$, hence the composite of the induced sheaf morphism with $P_M(U)\to \tilde{M}(U)$. Naturality in $M$ and in $\mathcal{F}$ is immediate from the formula $m\otimes a\mapsto a\cdot g(m){\mathrm{\mid }}_U$, and a $B$-linear $u\text{\hspace{0.17em}⁣}:M\to N$ induces $u\otimes \mathrm{i}\mathrm{d}\text{\hspace{0.17em}⁣}:P_M\to P_N$ and hence a morphism $\tilde{M}\to \tilde{N}$.

For $M=B$ one has $P_B(U)=B{\otimes }_B{\mathcal{O}}_X(U)\cong {\mathcal{O}}_X(U)$, so $\tilde{B}\cong a{\mathcal{O}}_X={\mathcal{O}}_X$ because ${\mathcal{O}}_X$ is already a sheaf, and $\mathrm{\Gamma }(X,\tilde{B})\cong B$ by [F5]. For an exact sequence $M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$, [F6] makes the corresponding sequence of presheaves $P_{M^{\mathrm{\prime }}}\to P_M\to P_{M^{\mathrm{\prime }\mathrm{\prime }}}\to 0$ objectwise right exact; sheafification, as the left adjoint supplied by [F2], preserves its cokernel and gives ${\tilde{M}}^{\mathrm{\prime }}\to \tilde{M}\to {\tilde{M}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ exact.