LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Regular functions on a classical affine variety form a sheaf

Statement

Let $X$ be a classical affine variety. The assignment $U ⟼ O_{X} (U)$ from Zariski-open subsets of $X$ to rings of regular functions is a sheaf of $k$ -algebras on $X$ .

Facts & Assumptions

Given: A classical affine variety $X$ , an open subset $U \subseteq X$ , and an open cover $U = ⋃_{i \in I} U_{i}$ .

[L1]

A function on an open subset is regular exactly when each point has a neighbourhood on which the function is a quotient $a / b$ of elements of $k [X]$ with $b$ nowhere zero there (Regular functions on open subsets of a classical affine variety).

Proof

technique · direct

1.1

If $f \in O_{X} (U)$ and $V \subseteq U$ is open, then every point of $V$ has an open neighbourhood $W \subseteq U$ on which [L1] gives a quotient formula $f = a / b$ . Then $W \cap V$ is an open neighbourhood of that point inside $V$ , the denominator $b$ is still nowhere zero there, and $f ∣_{V} = a / b$ on $W \cap V$ . Hence restriction maps are well defined.

L1given

1.2

If $f, g \in O_{X} (U)$ and $f ∣_{U_{i}} = g ∣_{U_{i}}$ for every $i \in I$ , then for each $x \in U$ some $U_{i}$ contains $x$ , so $f (x) = g (x)$ . Therefore $f = g$ on $U$ . This is the uniqueness clause.

given

1.3

Suppose for each $i \in I$ we are given $f_{i} \in O_{X} (U_{i})$ , and suppose $f_{i} = f_{j}$ on $U_{i} \cap U_{j}$ for all $i, j$ . Define $f : U \to k$ by $f (x) = f_{i} (x)$ whenever $x \in U_{i}$ . This is well defined by the overlap hypothesis.

givenconstruct

2.1

Fix $x \in U$ . Choose $i$ with $x \in U_{i}$ . Since $f_{i}$ is regular on $U_{i}$ , [L1] gives an open neighbourhood $W \subseteq U_{i}$ of $x$ and elements $a, b \in k [X]$ with $b$ nowhere zero on $W$ and $f_{i} = a / b$ on $W$ . On that same neighbourhood, the glued function $f$ equals $f_{i}$ , so $f = a / b$ on $W$ . By [L1], $f$ is regular at $x$ .

L1step 1.3choose

3.1

Step 1.1 gives restriction, step 1.2 gives uniqueness of gluing, and steps 1.3 and 2.1 give existence of gluing. Therefore $U \mapsto O_{X} (U)$ is a sheaf of $k$ -algebras on $X$ .

step 1.1step 1.2step 1.3step 2.1∎

Depends on

Regular functions on open subsets of a classical affine variety

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, Proposition 3.9 (standard reference, not scraped)
Michael Artin, Notes for a Course in Algebraic Geometry, 2.5 (standard reference, not scraped)