Graded shift convention for Proj

Let $S={⨁}_{d\ge 0}{S}_d$ be a commutative nonnegatively graded ring and let $M={⨁}_{d\in \mathbb{Z}}{M}_d$ be a graded $S$-module (Nonnegatively graded rings and modules, homogeneous elements, and twists). For an integer $n$, the shift $M(n)$ is the graded $S$-module whose underlying module is $M$ with the grading $M(n{)}_d=M_{n+d}(d\in \mathbb{Z}),$ and whose $S$-action is the original one read through these identifications: for $s\in S_i$ and $m\in M(n{)}_d=M_{n+d}$, the product $s\cdot m$ is the element of $M(n{)}_{d+i}=M_{n+d+i}$ that the given $S$-module structure assigns to the pair $(s,m)$. This is the twist $M(n)$ of Nonnegatively graded rings and modules, homogeneous elements, and twists, where the same formula $M(n{)}_d=M_{n+d}$ is recorded; the present item fixes the sign convention used throughout this page for the sheaves ${\mathcal{O}}_X(n)=\widetilde{S(n)}$ on $\mathrm{Proj}S$.

The following formal properties are immediate from the definition and are used freely below.

• Composition. $(M(m))(n)=M(m+n)$ for all $m,n\in \mathbb{Z}$, since $(M(m))(n{)}_d=M(m{)}_{n+d}=M_{m+n+d}$.
• Inverse. $M(-n)$ is a two-sided inverse on graded modules up to the canonical identification $M(n)(-n)=M$; in particular shifting is an equivalence of the category of graded $S$-modules with itself.
• Functoriality and degree. A homomorphism of graded modules of degree $e$, that is an $S$-linear map $\phi :M\to N$ with $\phi (M_d)\subseteq N_{d+e}$ for all $d$, is a homomorphism of graded modules $\phi :M(n)\to N(n+e)$ for every $n$, because it maps $M(n{)}_d=M_{n+d}$ into $N_{n+d+e}=N(n+e{)}_d$.
• Localization. If $T\subseteq S$ is a multiplicative set of homogeneous elements, then the homogeneous localizations satisfy $T^{-1}(M(n))=(T^{-1}M)(n)$ with the same grading, because degree $d$ in $T^{-1}(M(n))$ consists of fractions represented by $m\mathrm{/}f$ with $m$ homogeneous of degree $n+d+\mathrm{deg}(f)$ in $M$. Indeed the degree of $m\mathrm{/}f$ on both sides is ${\mathrm{deg}}_M(m)-\mathrm{deg}(f)-n$; the identity on fractions therefore gives the asserted graded isomorphism, including zero fractions.