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Principal opens form a basis for the Zariski topology on an affine variety

Statement

Let $X$ be a classical affine variety over an algebraically closed field $k$ . Then the principal opens $D_{X} (\overline{f})$ form a basis for the Zariski topology on $X$ . In particular, $D_{X} (\overline{f}) \cap D_{X} (\overline{g}) = D_{X} (\overline{f} \overline{g})$ for all $\overline{f}, \overline{g} \in k [X]$ .

Facts & Assumptions

Given: A classical affine variety $X$ over an algebraically closed field $k$ .

[L1]

For $\overline{f} \in k [X]$ , the principal open is $D_{X} (\overline{f}) = {x \in X : \overline{f} (x) \neq 0}$ (A principal open subset of a classical affine variety).

[L2]

Closed subsets of affine space are zero loci of sets of polynomials (Zero loci in affine space are the closed sets of the classical Zariski topology).

Proof

technique · direct

1.1

For any $\overline{f} \in k [X]$ , the set $D_{X} (\overline{f})$ is open because it is the complement in $X$ of the closed subset where $\overline{f}$ vanishes. Also a point lies in $D_{X} (\overline{f}) \cap D_{X} (\overline{g})$ exactly when both $\overline{f}$ and $\overline{g}$ are nonzero there, equivalently when $\overline{f} \overline{g}$ is nonzero there. Thus $D_{X} (\overline{f}) \cap D_{X} (\overline{g}) = D_{X} (\overline{f} \overline{g})$ .

L1givenalgebra

1.2

Let $U \subseteq X$ be Zariski-open and $x \in U$ . Then $X ∖ U$ is closed in $X$ , so there is a set $S$ of polynomials on the ambient affine space with $X ∖ U = X \cap V (S)$ by [L2]. Since $x \notin V (S)$ , choose $f \in S$ with $f (x) \neq 0$ . Then $x \in D_{X} (\overline{f}) \subseteq U$ . Hence every open set is a union of principal opens.

L1L2choose

2.1

Steps 1.1 and 1.2 are exactly the basis criterion.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, Chapter 3c (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, §1.3 and localization discussion (standard reference, not scraped)