Standard opens of Proj

Let $S={⨁}_{d\ge 0}{S}_d$ be a commutative nonnegatively graded ring with projective spectrum $\mathrm{Proj}S$ as in Points of Proj of a graded ring. For a homogeneous element $f\in S_{+}$ of positive degree set $D_{+}(f)=\{\mathfrak{p}\in \mathrm{Proj}S:f\mathrm{\notin }\mathfrak{p}\}\mathrm{.}$ These are the standard opens of $\mathrm{Proj}S$.

• Openness. Since $f$ is homogeneous, $(f)=fS$ is a homogeneous ideal, and $V_{+}((f))=\{\mathfrak{p}\in \mathrm{Proj}S:f\in \mathfrak{p}\}$; hence $D_{+}(f)$ is the complement of the closed set $V_{+}((f))$ and is open. If $f,g\in S_{+}$ are homogeneous then $fg\in S_{+}$ is homogeneous of positive degree and $D_{+}(f)\cap D_{+}(g)=D_{+}(fg),$ because a prime $\mathfrak{p}$ contains $fg$ if and only if it contains $f$ or $g$. In particular the standard opens are closed under nonempty finite intersections.
• Basis. The family $\{D_{+}(f){\}}_{f\in S_{+},f\text{ homogeneous}}$ is a basis for the topology of $\mathrm{Proj}S$: every open set is a union of standard opens. Indeed let $U=\mathrm{Proj}S\setminus V_{+}(I)$ be open with $I$ homogeneous, and let $\mathfrak{p}\in U$. Since $I\subseteq ̸\mathfrak{p}$ and $\mathfrak{p}$ is homogeneous, some homogeneous $h\in I$ satisfies $h\mathrm{\notin }\mathfrak{p}$; and since $\mathfrak{p}\in \mathrm{Proj}S$ we may choose a homogeneous $g\in S_{+}$ with $g\mathrm{\notin }\mathfrak{p}$. Put $f=gh\in S_{+}$. Then $f\mathrm{\notin }\mathfrak{p}$, so $\mathfrak{p}\in D_{+}(f)$; and for any $\mathfrak{q}\in D_{+}(f)$ one has $h\mathrm{\notin }\mathfrak{q}$ (else $f\in \mathfrak{q}$) with $h\in I$, so $I\subseteq ̸\mathfrak{q}$ and $\mathfrak{q}\in U$. Hence $\mathfrak{p}\in D_{+}(f)\subseteq U$. The case $U=\mathrm{\varnothing }$ is the empty union. The same computation with $h=0$ is never needed since $h\mathrm{\notin }\mathfrak{p}$ forces $h\mathrm{\ne }0$.
• Nilpotent generators give nothing. If $f$ is nilpotent then $f$ lies in every prime, hence $D_{+}(f)=\mathrm{\varnothing }$. No converse is asserted here; the equivalence between emptiness of $\mathrm{Proj}S$ and nilpotence of the irrelevant ideal, under its stated hypothesis, is proved in Empty Proj and irrelevant torsion.