Lemma 26.5.1. Let $R$ be a ring. Let $f \in R$.

If $g\in R$ and $D(g) \subset D(f)$, then

$f$ is invertible in $R_ g$,

$g^ e = af$ for some $e \geq 1$ and $a \in R$,

there is a canonical ring map $R_ f \to R_ g$, and

there is a canonical $R_ f$-module map $M_ f \to M_ g$ for any $R$-module $M$.

Any open covering of $D(f)$ can be refined to a finite open covering of the form $D(f) = \bigcup _{i = 1}^ n D(g_ i)$.

If $g_1, \ldots , g_ n \in R$, then $D(f) \subset \bigcup D(g_ i)$ if and only if $g_1, \ldots , g_ n$ generate the unit ideal in $R_ f$.

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