= Dense principal open inside a dense open subset
{title2=$D(s)\subseteq U,\quad s\text{ a non-zero-divisor}$}
In an <affine variety> with <coordinate ring> $A$, every dense open subset $U$ contains a dense <principal open subset>. Write its closed complement as $V(I)$. Density says that $I$ is contained in none of the finitely many <minimal prime ideals> of the <reduced ring> $A$. <Prime avoidance> supplies $s\in I$ outside all these primes. Thus $D(s)\subseteq U$, and $s$ is a <non-zero-divisor>. <Noetherian Zariski topology> also makes $U$ a finite union of principal opens; one can add $D(s)$ to this cover. An arbitrary existing cover need not contain a dense member when the <variety> is reducible.
Back to article page
