Degree-one generation condition for Proj
= Degree-one generation condition for Proj
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 $\operatorname{Proj}S$. 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>.