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If a nonnegatively graded ring $S$ is generated by $S_1$ as an $S_0$-algebra, the standard affine opens of Proj $D_{+}(f)$ with $f\in S_1$ cover $\mathrm{Proj}S$. Indeed a homogeneous prime containing every degree-one element would contain the irrelevant ideal of a graded ring. This condition makes all twisting sheaves on Proj invertible and makes graded sheafification compatible with tensor products of sheaves.

In the standard graded homogeneous coordinate ring $S=k[x_0,\dots ,x_n]$ of ${\mathbb{P}}^n$, the irrelevant ideal isIt vanishes only at the origin of the corresponding affine cone, which does not represent a projective point.

It is the polynomial-ring instance of the irrelevant ideal of a graded ring.

For every homogeneous $s\in S_{+}$,Henceso $U$ is open in the Zariski topology.

If $\mathfrak{p}\in U$, then ${\phi }^{-1}(\mathfrak{p})$ is a homogeneous prime ideal that does not contain the irrelevant ideal of a graded ring $S_{+}$. Thusdefines a map $U\to \mathrm{Proj}S$. On every standard affine open of Proj the graded homomorphism inducesand therefore an affine scheme morphismThese morphisms agree after localization on overlaps, so they glue to the required morphism of schemes $f:U\to \mathrm{Proj}S$.