Commutative Algebra

Let $S = \bigoplus_{d\geq0} S_d$ be a graded ring. Recall that an ideal $I$ of $S$ is called homogeneous if $I$ is generated by homogeneous elements. (For instance, the principal ideal of $k[s,t]$ generated by $s+t$ would be a homogeneous ideal.)

For any ideal $I$ of $S$, define the following sets of primes:

• $\operatorname{Spec} S$ denotes the set of prime ideals of $S$;
• $V(I) = \{ \wp\in \operatorname{Spec} S: \wp\supseteq I\}$ denotes the set of prime ideals $\wp$ of $S$ containing $I$;
• $D(I)$ denotes the complement of $V(I)$ in $\operatorname{Spec} S$;
• $\operatorname{Proj} S$ denotes the set of homogeneous prime ideals $\wp$ of $S$ not containing the irrelevant ideal $S_+ = \bigoplus_{d\geq1} S_d$ of $S$;
• $V_+(I) = \{ \wp\in \operatorname{Proj} S: \wp\supseteq I\}$ denotes the the intersection $V(I)\cap \operatorname{Proj} S$;
• $D_+(I)$ denotes the complement of $V_+(I)$ in $\operatorname{Proj} S$, or equivalently the intersection $D(I)\cap \operatorname{Proj} S$.

Remark: In practice, one usually only cares about $V_+(I)$ and $D_+(I)$ for homogeneous ideals $I$.