LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-12

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The rational-map equivalence relation is transitive

Statement

Let $X$ and $Y$ be classical affine varieties. The relation used in Rational maps between irreducible classical affine varieties is transitive: if $φ_{1} : U_{1} \to Y, φ_{2} : U_{2} \to Y, φ_{3} : U_{3} \to Y$ are representatives with $φ_{1} \sim φ_{2}$ and $φ_{2} \sim φ_{3}$ , then $φ_{1} \sim φ_{3}$ .

Facts & Assumptions

Given: Classical affine varieties $X$ and $Y$ , representatives $φ_{i} : U_{i} \to Y$ on nonempty affine opens $U_{i} \subseteq X$ , and witnesses $W_{12} \subseteq U_{1} \cap U_{2}$ and $W_{23} \subseteq U_{2} \cap U_{3}$ with $φ_{1} = φ_{2}$ on $W_{12}$ and $φ_{2} = φ_{3}$ on $W_{23}$ .

[L1]

A rational-map representative is defined on a nonempty affine open subset of $X$ , and two representatives are equivalent when they agree on a nonempty affine open subset of the overlap (Rational maps between irreducible classical affine varieties).

[L2]

Every nonempty open subset of a classical affine variety is dense (Every nonempty open subset of an affine variety is dense).

[L3]

Principal opens form a basis, and the intersection of two principal opens is again principal (Principal opens form a basis for the Zariski topology on an affine variety).

Proof

technique · direct

1.1

The sets $W_{12}$ and $W_{23}$ are nonempty open subsets of $X$ by [L1]. By [L2], both are dense in $X$ , so their intersection is nonempty.

L1L2given

2.1

The set $W_{12} \cap W_{23}$ is open in $X$ . Since both $W_{12}$ and $W_{23}$ are affine opens, [L3] identifies their intersection as another affine open subset. On that common open we have $φ_{1} = φ_{2} = φ_{3}$ .

L3step 1.1algebra

3.1

Step 2.1 gives a nonempty affine open subset of $U_{1} \cap U_{3}$ on which $φ_{1}$ and $φ_{3}$ agree. Therefore $φ_{1} \sim φ_{3}$ .

L1step 2.1∎

Depends on

Used by

The maximal domain of definition of a rational map to an affine variety Definition

Cited to discharge well-definedness by Rational maps between irreducible classical affine varieties.

Dependency tree · two levels

9 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, rational-map discussion in Chapter 5l (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, rational-map discussion in §3.2 (standard reference, not scraped)