= Coordinate ring of a product of affine varieties
{title2=$k[X\times Y]\cong k[X]\otimes_k k[Y]$}
For <affine varieties> $X,Y$ over an <algebraically closed field> $k$, their <product in a category> is affine, with <coordinate ring> $A\otimes_k B$, where $A=k[X]$ and $B=k[Y]$. Presenting $A=k[x]/J$ and $B=k[y]/L$ identifies this ring with $k[x,y]/(J,L)$. If both varieties are irreducible, this is an <integral domain>: express a nonzero tensor $F$ using linearly independent second factors. The points $x$ where its specialization $F_x$ is nonzero form a nonempty open subset of $X$. For two nonzero tensors $F,G$, choose $x$ in the intersection of these open subsets; then $F_xG_x\ne0$ in the domain $B$. The <Hilbert Nullstellensatz> identifies $(J,L)$ with the defining ideal of $X\times Y$. The universal property follows from the universal property of the <tensor product of modules> as a tensor product of commutative algebras.
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