LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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Products of affine algebraic sets have tensor-product coordinate rings

Statement

For affine algebraic sets $X \subseteq C^{m}$ and $Y \subseteq C^{n}$ , allowing empty or reducible sets, $X \times Y \subseteq C^{m + n}$ is affine algebraic and the map $C [X] \otimes C [Y] \to C [X \times Y]$ , $f \otimes h \mapsto ((x, y) \mapsto f (x) h (y))$ , is an isomorphism. Iterating gives the analogous three-factor identification. The argument is choice-free.

Facts & Assumptions

Given: Two affine algebraic sets $X, Y$ over $C$ .

[F1]

Coordinate-ring elements are precisely polynomial functions, with equality tested at all points (Polynomial functions on an affine algebraic set are its coordinate ring).

[F2]

Coordinates generate the coordinate ring (The coordinate ring of a classical affine algebraic set).

Proof

1.1givenF1F2algebra

Equations for $X$ in the first $m$ variables and for $Y$ in the last $n$ variables cut out exactly $X \times Y$ . Multiplying polynomial functions in separate variables gives the displayed algebra map. Every polynomial in the $m + n$ coordinates is a sum of products of such polynomials, so this map is surjective.

2.1step 1.1F1algebra∎

For injectivity write a kernel element as $\sum_{i = 1}^{r} f_{i} \otimes h_{i}$ with the $h_{i}$ linearly independent, by eliminating redundant terms in a finite expression. At each $x$ , the polynomial function $\sum_{i} f_{i} (x) h_{i}$ on $Y$ is zero. Linear independence in the function space therefore forces every $f_{i} (x) = 0$ . By F1 all $f_{i}$ are zero. If a factor is empty its coordinate ring and that of the product are zero, so the same conclusion holds. Iteration proves the three-factor formula. Only finite expressions and finite-dimensional elimination were used.

Depends on

Used by

Classical complex affine algebraic actions and rational modules Definition
Affine actions correspond to coordinate-ring coactions Proposition

Dependency tree · two levels

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Sources

Michel Brion, Introduction to actions of algebraic groups (2010) (standard reference, not scraped)
Philippe Gille, Introduction to reductive group schemes over rings, full notes retrieved 2026-10-02 (standard reference, not scraped)
J. S. Milne, Algebraic Groups (2022) (standard reference, not scraped)