Definition:Principal Open Subset of Spectrum

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Definition

Let $A$ be a commutative ring with unity.

Let $f \in A$.


The principal open subset determined by $f$ of the spectrum $\Spec A$ is the complement of the vanishing set $\map V f$:

$\map D f = \Spec A - \map V f$

That is, it is the set of prime ideals $\mathfrak p \subseteq A$ with $f \notin \mathfrak p$.


Also denoted as

The principal open subset is also denoted $\map X f$ or $X_f$.


Also known as

A principal open subset is also known as a basic open set.


Also see


Sources