Two graded rings with the same Proj

The rings and their degree-one parts. By [F2] the ring $S=k[x,y]$ with $\mathrm{deg}x=\mathrm{deg}y=1$ is nonnegatively graded with $S_1=kx\oplus ky$, of dimension $2$, and the Veronese $T=S^{(2)}={⨁}_n{S}_{2n}$ is the graded $k$-subalgebra $k[x^2,xy,y^2]\subseteq S$ generated by the three degree-two monomials: every $S_{2n}$ is spanned by the monomials $x^a{y}^{2n-a}=(x^2{)}^{a^{\mathrm{\prime }}}(xy{)}^b(y^2{)}^c$ with $2a^{\mathrm{\prime }}+b=a$ and $b+2c=2n-a$. Hence $T_1=S_2=k{x}^2\oplus kxy\oplus k{y}^2$, of dimension $3$, so ${\dim }_k{S}_1=2$ and ${\dim }_k{T}_1=3$.

The twists are different. By [F4], a global section of ${\mathcal{O}}_S(n)$ for $n=1$ or $2$ is a pair $a(t)x^n$ on $D_{+}(x)$ and $b(t^{-1})y^n$ on $D_{+}(y)$, where $a\in k[t]$, $b\in k[t^{-1}]$, and the overlap condition is $a(t)=t^{n}b(t^{-1})$. This holds exactly when $a(t)$ has degree at most $n$, with $b(t^{-1})=t^{-n}a(t)$. Hence ${\dim }_k\mathrm{\Gamma }({\mathbb{P}}_k^1,{\mathcal{O}}_S(1))=2$ and ${\dim }_k\mathrm{\Gamma }({\mathbb{P}}_k^1,{\mathcal{O}}_S(2))=3$. An isomorphism of these sheaves would give an isomorphism of their global-section $k$-vector spaces, which is impossible. Thus the two twists are not isomorphic.

Same Proj. Under the AC premise [F5], since $T=S^{(2)}$, [F1] with $R=S$ and $d=2$ gives a canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}T$ which maps the chart $D_{+}(f)$ of $\mathrm{Proj}S$ to the chart $D_{+}(f^2)$ of $\mathrm{Proj}T$, and under which ${\mathcal{O}}_T(1)={\mathcal{O}}_{\mathrm{Proj}T}(1)$ corresponds to ${\mathcal{O}}_{\mathrm{Proj}S}(2)$; by [F3] the scheme $\mathrm{Proj}S$ is the projective line ${\mathbb{P}}_k^1$.

No graded isomorphism. Suppose $\phi :S\to T$ is an isomorphism of graded $k$-algebras, that is, a $k$-algebra isomorphism with $\phi (S_n)=T_n$ for all $n$; by [F2] it restricts to a $k$-linear isomorphism $S_1\to T_1$, so ${\dim }_k{S}_1={\dim }_k{T}_1$. By step 1.1 this would require $2=3$, which is impossible; hence $S$ and $T$ are not isomorphic as graded $k$-algebras.

Conclusion. Steps 2.1 and 2.2 exhibit the two graded $k$-algebras $S=k[x,y]$ and $T=k[x^2,xy,y^2]$ with canonically isomorphic Proj but no graded isomorphism between them; steps 2.1 and 1.2 show that the isomorphism matches ${\mathcal{O}}_T(1)$ with ${\mathcal{O}}_S(2)$ and not with ${\mathcal{O}}_S(1)$, so not even the degree-one twists correspond. Since $k[x,y]$ and its Veronese are nonnegatively graded with nonzero degree-one parts, neither Proj is empty and the invariant ${\dim }_k(-{)}_1$ is defined; the case $d=1$ of [F1] is excluded here because the two rings are then equal, while $d=2\ge 1$ is the smallest regrading for which $S^{(d)}\mathrm{\ne }S$ in this example. No choice principle is used beyond the inherited Proj construction. [F1, F2, step 1.1, step 2.1, step 2.2, step 1.2, cases: d=1 excluded and d=2 smallest] \qed