Dimensions add under products

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

If a Noetherian space $T$ is a finite union of closed subsets $T_1,\dots ,T_m$, then $\dim T={\max }_i\dim T_i$. For $m=0$ both sides are $-\mathrm{\infty }$. (Dimension of a finite closed union).

For every open cover $T={\bigcup }_{i\in I}{U}_i$ of a Noetherian space, $\dim T={\sup }_i\dim U_i$, with empty supremum $-\mathrm{\infty }$. (Dimension can be computed on an open cover).

Let $X,Y$ be classical affine varieties over an algebraically closed field $k$. Then their affine product exists, is a classical affine variety, and has coordinate ring $k[X{\times }_{k}Y]\cong k[X]{\otimes }_{k}k[Y]\mathrm{.}$ Its projections make it a product in the classical affine-variety category. (The product of affine varieties has coordinate ring k[X] tensor_k k[Y]).

Fix the page's algebraically closed field $k$. Let $\mathcal{C}$ be the category whose objects are classical affine or projective algebraic sets over $k$ (including empty and reducible ones), and whose arrows are regular $k$-maps. Objects isomorphic to such sets are understood with their transported algebraic structure. No existence of products for arbitrary mixed affine/projective factors is asserted. For $X,Y$ in $\mathcal{C}$, a constructed product $X{\times }_{k}Y$ is an object of $\mathcal{C}$ with morphisms $p:X{\times }_{k}Y\to X$ and $q:X{\times }_{k}Y\to Y$ such that, for every object $T$ of $\mathcal{C}$ and morphisms $f:T\to X$, $g:T\to Y$, there is a unique morphism $⟨f,g⟩:T\to X{\times }_{k}Y$ satisfying $p⟨f,g⟩=f$ and $q⟨f,g⟩=g$. Thus it is the categorical product of def-products-and-coproducts in $\mathcal{C}$. The underlying set is written as pairs when a construction supplies that identification. If either factor is empty, the product set is empty. A product with the one-point affine algebraic set has the evident projection isomorphism. When $X,Y$ are varieties, this definition is used only after a construction shows that the resulting nonempty algebraic set is irreducible. (Products of classical algebraic sets and their universal property).

Let $k$ be a field and let $A$ be a nonzero finite-type $k$-algebra. Then there exist algebraically independent elements $z_1,\dots ,z_d\in A$ such that $A$ is a module-finite algebra over the polynomial ring $k[z_1,\dots ,z_d]$. (Noether normalisation yields module finiteness over a polynomial subring).