Module sheaf on an affine scheme

Let $A$ be a commutative ring with $1$, let $M$ be an $A$-module, and put $X=\mathrm{Spec}A$ with its distinguished-open basis $D(f)=\{\mathfrak{p}:f\mathrm{\notin }\mathfrak{p}\}$ (The underlying space of an affine spectrum). For $f\in A$ write $M_f$ for the localisation of $M$ at the multiplicative subset $\{1,f,f^2,\dots \}$, as in Localisation of a module at a multiplicative subset. The distinguished-open data of $M$ assign $\tilde{M}(D(f)):=M_f\mathrm{.}$

Restriction maps. Suppose $D(g)\subseteq D(f)$. Then $g\in \sqrt{(f)}$, so $g^n=fh$ for some $n\ge 1$ and some $h\in A$; hence the image of $f$ in $A_g$ is a unit and $M_g$ is an $A_f$-module. The localisation map ${\lambda }_g:M\to M_g$ is $A$-linear, so the universal property of module localisation (Universal property of localisation for modules) gives a unique $A_f$-linear map ${\rho }_{fg}:M_f⟶M_g,{\rho }_{fg}(m\mathrm{/}{f}^n)=m\mathrm{/}{f}^n,$ with ${\rho }_{fg}\circ {\lambda }_f={\lambda }_g$.

Well-definedness. The map ${\rho }_{fg}$ depends only on the pair of open sets: it is characterised as the unique $A_f$-linear map compatible with the canonical maps from $M$. Consequently ${\rho }_{ff}={\mathrm{id}}_{M_f}$, and for distinguished opens $D(h)\subseteq D(g)\subseteq D(f)$ one has ${\rho }_{fh}={\rho }_{gh}\circ {\rho }_{fg}$, because both sides are $A_f$-linear maps $M_f\to M_h$ that agree after composition with ${\lambda }_f$. Moreover, if $D(f)=D(f^{\mathrm{\prime }})$ then the maps ${\rho }_{f{f}^{\mathrm{\prime }}}$ and ${\rho }_{f^{\mathrm{\prime }}f}$ are mutually inverse; the two inclusions $D(f)\subseteq D(f^{\mathrm{\prime }})$ and $D(f^{\mathrm{\prime }})\subseteq D(f)$ are available, and the composition rule gives ${\rho }_{f^{\mathrm{\prime }}f}\circ {\rho }_{f{f}^{\mathrm{\prime }}}={\rho }_{ff}=\mathrm{id}$ and ${\rho }_{f{f}^{\mathrm{\prime }}}\circ {\rho }_{f^{\mathrm{\prime }}f}={\rho }_{f^{\mathrm{\prime }}{f}^{\mathrm{\prime }}}=\mathrm{id}$. Thus the data are functorial on distinguished opens, canonically up to these identifications.

Module structure. On $D(f)$ the structure sheaf has ${\mathcal{O}}_X(D(f))=A_f$ and the ring restriction for $D(g)\subseteq D(f)$ is the canonical localisation $A_f\to A_g$ (Sections and restrictions on distinguished opens of an affine scheme). The map ${\rho }_{fg}$ is $A_f$-linear by construction, so the data form an ${\mathcal{O}}_X$-module on the distinguished-open basis: sections on $D(f)$ are the $A_f$-modules $M_f$ and restrictions are compatible with the ring restrictions.

Completion to a sheaf. A later result on this page proves that these distinguished-open data satisfy the sheaf conditions on the basis and extend uniquely to an ${\mathcal{O}}_X$-module, still denoted $\tilde{M}$, whose sections on $D(f)$ are the given modules $M_f$; the notation $\tilde{M}$ always refers to that ${\mathcal{O}}_X$-module.

Degenerate cases. For $f=0$ one has $D(0)=\mathrm{\varnothing }$ and $M_0=0$, the localisation in which $0$ is inverted; this is the zero module. For a unit $f$ one has $D(f)=X$ and $M_f=M$. If $A=0$ then $X=\mathrm{\varnothing }$ and $\tilde{M}$ is the zero sheaf on the empty space, whose module of sections on $\mathrm{\varnothing }$ is $0$. The construction is functorial in $M$: an $A$-linear map $M\to N$ induces $A_f$-linear maps $M_f\to N_f$ commuting with all restriction maps.