Proj forgets irrelevant torsion and grading scale

Assume the Axiom of Choice (The Axiom of Choice) for the cited twisting-sheaf construction and the example below.

The scheme $\mathrm{Proj}S$ does not determine the graded ring $S={⨁}_{d\ge 0}{S}_d$ up to graded isomorphism; two independent losses occur, and the twist data must be tracked separately.

Irrelevant torsion is forgotten. By Empty Proj and irrelevant torsion, an element $m$ of a graded $S$-module $M$ that is annihilated by a power of $S_{+}$ has zero image in the full localisation $M_f$ for every homogeneous $f\in S_{+}$ of positive degree; every degree-zero fraction formed from it therefore vanishes in $M_{(f)}=\mathrm{\Gamma }(D_{+}(f),\tilde{M})$. Applied to $M=S$ or to a homogeneous quotient, this says that passing from a graded module to $\tilde{M}$ kills all $S_{+}$-power torsion: quotients by ideals differing only in such torsion, and in particular the passage $I\mapsto I^{\mathrm{s}\mathrm{a}\mathrm{t}}$ of Saturation detected on projective charts, are invisible to the associated sheaves.

Positive Veronese regradings are invisible. By Proj is invariant under Veronese regrading, for every $d\ge 1$ the Veronese regrading $S^{(d)}={⨁}_{n\ge 0}{S}_{dn}$ satisfies $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$, and under this isomorphism the degree-one twist of the Veronese corresponds canonically to the $d$-th twist of $S$: ${\mathcal{O}}_{\mathrm{Proj}{S}^{(d)}}(1)$ is identified with ${\mathcal{O}}_{\mathrm{Proj}S}(d)$. These twists may also happen to be isomorphic to ${\mathcal{O}}_{\mathrm{Proj}S}(1)$ in special cases. So even when the underlying graded rings are not isomorphic, the spaces agree as schemes.

Consequences for twist data. Because of Twisting sheaf on Proj, the twisting sheaves are constructed from the shifted modules $S(n)$; for an arbitrary nonnegatively graded ring the sheaf ${\mathcal{O}}_X(1)$ need not be invertible and the multiplication maps ${\mathcal{O}}_X(m)\otimes {\mathcal{O}}_X(n)\to {\mathcal{O}}_X(m+n)$ need not be isomorphisms. Degree-one generation gives an invertibility criterion in Invertible twists for degree-one generated rings. Recovering a chosen graded presentation from the geometry therefore requires retaining extra graded coordinate data; the graded ring itself is not a function of the scheme $\mathrm{Proj}S$ alone.

The example Two graded rings with the same Proj ↗ exhibits the smallest instance: for a field $k$, the graded $k$-algebras $k[x,y]$ with $\mathrm{deg}x=\mathrm{deg}y=1$ and its second Veronese $k[x^2,xy,y^2]$ have isomorphic Proj but degree-one parts of dimensions $2$ and $3$.