Abstract
We want to define the product X×Y of any two prevarieties X,Y. Now we will certainly want to have An × Am ≅ An+m. But the product of the Zariski topologies in An and Am does not give the Zariski topology in An+m; in A1 × A1, for instance, the only closed sets in the product topology are finite unions of horizontal and vertical lines. The only reliable way to find the correct definition is to use the general category-theoretic definition of product.
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© 1988 Springer-Verlag Berlin Heidelberg
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Mumford, D. (1988). Products and the Hausdorff Axiom. In: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol 1358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21581-4_6
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DOI: https://doi.org/10.1007/978-3-662-21581-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50497-9
Online ISBN: 978-3-662-21581-4
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