Associated sheaf of a graded module on Proj

Assume the Axiom of Choice as inherited from the affine scheme construction (The Axiom of Choice). Let $S={⨁}_{d\ge 0}{S}_d$ be a commutative nonnegatively graded ring with $\mathrm{Proj}S$ a scheme whose charts $D_{+}(f)=\mathrm{Spec}{S}_{(f)},S_{(f)}=(S[f^{-1}]{)}_0,$ are affine opens for every homogeneous $f\in S_{+}$ of positive degree (Proj carries a scheme structure), and let $M={⨁}_{d\in \mathbb{Z}}{M}_d$ be a $\mathbb{Z}$-graded $S$-module. For a homogeneous $f\in S_{+}$ of positive degree put $M_{(f)}=(M[f^{-1}]{)}_0,$ the degree-zero part of the homogeneous localisation of $M$ at $f$: the set of fractions $m\mathrm{/}{f}^k$ with $m\in M$ homogeneous of degree $kd$. This is a module over the ring $S_{(f)}$ by the same formula as the ring case, and it is the module denoted $M_{(f)}$ in Nonnegatively graded rings and modules, homogeneous elements, and twists.

Construction. On the affine chart $D_{+}(f)=\mathrm{Spec}{S}_{(f)}$ let ${\tilde{M}}_{(f)}$ be the associated sheaf of the $S_{(f)}$-module $M_{(f)}$ (Module sheaf on an affine scheme), which exists as a quasi-coherent sheaf by The associated module sheaf exists. The associated sheaf $\tilde{M}$ of $M$ on $\mathrm{Proj}S$ is the unique sheaf of ${\mathcal{O}}_{\mathrm{Proj}S}$-modules whose restriction to $D_{+}(f)$ is ${\tilde{M}}_{(f)}$ under the identification $D_{+}(f)=\mathrm{Spec}{S}_{(f)}$, for every homogeneous $f\in S_{+}$ of positive degree.

Well-definedness. The data glue. Since the standard opens $D_{+}(f)$ form a basis of the topology of $\mathrm{Proj}S$ (Standard opens of Proj), it suffices to specify the sheaf on this basis and to give compatible restriction isomorphisms. For homogeneous $f,g\in S_{+}$ of positive degrees $d=\mathrm{deg}f$, $e=\mathrm{deg}g$, the element ${\tau }_{f,g}=\frac{g^d}{f^e}\in S_{(f)}$ is well defined, and $D_{+}(fg)=D_{+}(f)\cap D_{+}(g)$ is the distinguished open $D({\tau }_{f,g})$ of $\mathrm{Spec}{S}_{(f)}$ (Prime correspondence on a Proj chart). Localising the fraction description of $M_{(f)}$ at ${\tau }_{f,g}$ performs exactly the further localisation inverting $g$: $(M_{(f)}{)}_{{\tau }_{f,g}}=((M[f^{-1}]{)}_0{)}_{{\tau }_{f,g}}=(M[f^{-1},g^{-1}]{)}_0=M_{(fg)},$ the middle equality because a fraction of fractions $m\mathrm{/}{f}^k$ divided by a power of ${\tau }_{f,g}$ is a fraction with denominator a power of $fg$, and conversely every class in $M[f^{-1},g^{-1}]$ of degree zero can be written with denominator a power of $fg$ and numerator of matching degree. The two composite identifications of $M_{(fg)}$ obtained from $M_{(f)}$ and from $M_{(g)}$ agree, because both are the canonical localisation maps into the localisation at the product $fg$; the same computation on triple overlaps $D_{+}(fgh)$ shows the cocycle condition, and gluing the affine localisations of a module along a basis with compatible restrictions is the standard module-sheaf gluing. Consequently $\tilde{M}$ is well defined, the identifications are isomorphisms of $S_{(f)}$-modules, and no choice of charts or trivialisations enters: the only appeal to AC is the inherited one through the associated-module-sheaf construction of The associated module sheaf exists. Functoriality. A homomorphism of graded $S$-modules of degree zero, $\phi :M\to N$ with $\phi (M_d)\subseteq N_d$ for all $d$ (Nonnegatively graded rings and modules, homogeneous elements, and twists), induces $S_{(f)}$-linear maps $M_{(f)}\to N_{(f)}$ and hence morphisms of associated sheaves on each chart, compatible with the identifications above; they glue to a morphism $\tilde{\phi }:\tilde{M}\to \tilde{N}$ of ${\mathcal{O}}_{\mathrm{Proj}S}$-modules. The construction is additive and respects composition and identities, so $M\mapsto \tilde{M}$ is a functor from graded $S$-modules with degree-zero maps to ${\mathcal{O}}_{\mathrm{Proj}S}$-modules.

• Twists. Applying the construction to the shifted module $S(n)$ of Graded shift convention for Proj defines the sheaves ${\mathcal{O}}_X(n)=\widetilde{S(n)}$ studied at Twisting sheaf on Proj; the sign convention is the one fixed there.
• No invertibility claim. For an arbitrary nonnegatively graded ring the sheaf ${\mathcal{O}}_X(1)$ need not be invertible; invertibility is proved in Invertible twists for degree-one generated rings under the hypothesis that $S$ is generated in degree one over $S_0$.
• Torsion is not seen. The construction only uses the localisations $M_{(f)}$; elements of $M$ annihilated by a power of the irrelevant ideal localise to zero on every chart and therefore give the zero sheaf. The precise statement is in Empty Proj and irrelevant torsion.