Proj is invariant under Veronese regrading

The matching chart covers. Contraction sends a homogeneous prime $\mathfrak{p}\subseteq S$ avoiding $S_{+}$ to a homogeneous prime $\mathfrak{q}=\mathfrak{p}\cap S^{(d)}$ avoiding $S_{+}^{(d)}$: choose homogeneous $f\in S_{+}\setminus \mathfrak{p}$, and then $f^d\in S_{+}^{(d)}\setminus \mathfrak{q}$. This defines a map $c:\mathrm{Proj}S\to \mathrm{Proj}{S}^{(d)}$ with $c^{-1}(D_{+}(f^d))=D_{+}(f)$, since $f^d\in \mathfrak{p}$ if and only if $f\in \mathfrak{p}$. The opens $D_{+}(f)$ cover $\mathrm{Proj}S$. Their matches $D_{+}(f^d)$ cover $\mathrm{Proj}{S}^{(d)}$: if $\mathfrak{q}$ avoids the Veronese irrelevant ideal, choose homogeneous $h\in S_{+}^{(d)}\setminus \mathfrak{q}$ and take $f=h$ viewed as an element of $S_{+}$; then $f^d=h^d\mathrm{\notin }\mathfrak{q}$. The chart-ring comparison below proves that $c$ is a bijection and an isomorphism of schemes.

Same chart rings. For homogeneous $f\in S_{+}$ of positive ordinary degree $d_0$ we compare the two chart rings inside the common localisation $S[f^{-1}]$. The chart ring of $\mathrm{Proj}S$ is $S_{(f)}=\{a\mathrm{/}{f}^k:a\in S_{k{d}_0},k\ge 0\}$; the chart ring of $\mathrm{Proj}{S}^{(d)}$ at $f^d$ is $S_{(f^d)}^{(d)}=\{b\mathrm{/}(f^d{)}^k:b\in S_{k{d}_0}^{(d)}=S_{dk{d}_0},k\ge 0\}$. Every $b\mathrm{/}(f^d{)}^k=b\mathrm{/}{f}^{dk}$ with $b\in S_{dk{d}_0}$ lies in $S_{(f)}$. Conversely, given $a\mathrm{/}{f}^k$ with $a\in S_{k{d}_0}$, choose an integer $s\ge 0$ with $ds\ge k$ and rewrite $a\mathrm{/}{f}^k=(a{f}^{ds-k})\mathrm{/}(f^d{)}^s$, where $a{f}^{ds-k}\in S_{k{d}_0+(ds-k)d_0}=S_{ds{d}_0}=S_{s{d}_0}^{(d)}$; hence $a\mathrm{/}{f}^k\in S_{(f^d)}^{(d)}$. Therefore $S_{(f)}=S_{(f^d)}^{(d)}$ as subrings of $S[f^{-1}]$, in agreement with the identification $D_{+}(f)=D_{+}(f^d)$ of step 1.1.

The scheme isomorphism. By steps 1.1 and 2.1 the standard affine charts of the two schemes have identical coordinate rings inside the corresponding graded localizations. The overlap $D_{+}(f)\cap D_{+}(g)=D_{+}(fg)$ corresponds to $D_{+}(f^d)\cap D_{+}(g^d)=D_{+}((fg{)}^d)$, both with coordinate ring $S_{(fg)}=S_{((fg{)}^d)}^{(d)}$. The transition maps of [F2] are the same canonical localization maps under these ring identities. Thus the identity maps on matching charts glue to an isomorphism of schemes $\mathrm{Proj}S\to \mathrm{Proj}{S}^{(d)}$. To identify its point map, let $\mathfrak{p}\in D_{+}(f)$ and put $\mathfrak{q}=\mathfrak{p}\cap S^{(d)}$. For $a\mathrm{/}{f}^{\mathrm{\ell }}\in S_{(f)}$, choose $k$ with $dk\ge \mathrm{\ell }$ and rewrite it as $(a{f}^{dk-\mathrm{\ell }})\mathrm{/}(f^d{)}^k$ as in step 2.1. Its chart-prime membership for $\mathfrak{p}$ is $a\in \mathfrak{p}$; on the Veronese side it is $a{f}^{dk-\mathrm{\ell }}\in \mathfrak{q}$, equivalent to $a\in \mathfrak{p}$ because $f\mathrm{\notin }\mathfrak{p}$. Thus the glued map on points is the contraction map $c$ of step 1.1, and $c$ is bijective. The construction is canonical, and for $d=1$ it is the identity.

The twists on a chart. Fix homogeneous $f\in S_{+}$ of ordinary degree $e>0$ and let $V=D_{+}(f)$ correspond to $D_{+}(f^d)$. The degree-zero part of the shifted localization is $S(d{)}_{(f)}=\{a\mathrm{/}{f}^{\mathrm{\ell }}:\mathrm{\ell }\ge 0,a\in S_{\mathrm{\ell }e+d}\}$, viewed as the degree-$d$ component of $S[f^{-1}]$. On the Veronese chart, the shift has $S^{(d)}(1{)}_{(f^d)}=\{b\mathrm{/}{f}^{dk}:k\ge 0,b\in S_{d(ke+1)}\}$, a submodule of that same degree-$d$ component. Conversely, given $a\mathrm{/}{f}^{\mathrm{\ell }}$ in the first set, choose $k$ with $dk\ge \mathrm{\ell }$ and rewrite it as $(a{f}^{dk-\mathrm{\ell }})\mathrm{/}{f}^{dk}$; the numerator has degree $d(ke+1)$, so the fraction lies in the second set. The reverse inclusion is immediate. Hence the two local modules are canonically equal inside $(S[f^{-1}]{)}_d$, with the common scalar ring $S_{(f)}=S_{(f^d)}^{(d)}$ of step 2.1. This gives the desired local isomorphism without asserting either twist is free.

The twists glue. For homogeneous $f,g\in S_{+}$ of positive degree, the identifications of step 3.2 on $D_{+}(f)$ and $D_{+}(g)$ restrict to the same identification over $D_{+}(fg)$: each is the identity on the degree-$d$ component after the canonical graded localization into $S[(fg{)}^{-1}]$. Hence these local isomorphisms glue along the chart cover of [F2] to an isomorphism of ${\mathcal{O}}_{\mathrm{Proj}{S}^{(d)}}$-modules ${\mathcal{O}}_{\mathrm{Proj}{S}^{(d)}}(1)\cong {\mathcal{O}}_{\mathrm{Proj}S}(d)$ under the scheme isomorphism of step 3.1.

Conclusion. Step 3.1 gives the canonical scheme isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$ with matching charts, and step 4.1 identifies $\mathcal{O}(1)$ with $\mathcal{O}(d)$. For $d=1$ the map is the identity; if either Proj is empty, the isomorphism makes the other empty. No invertibility of ${\mathcal{O}}_{S^{(d)}}(1)$ or ${\mathcal{O}}_S(d)$ is used or asserted. The Axiom of Choice [A1] is inherited from the Proj sheaf construction [F2]; no further choice is made. [A1, F2, step 3.1, step 4.1, cases: d=1 and empty Proj] \qed