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
- 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $1$: Rings and Ideals: Exercise 17