Talk:Affine variety

From Wikipedia, the free encyclopedia
Jump to: navigation, search

"where \operatorname{spec} is the maximal spectrum of the ring; i.e., the set of all maximal ideals and f_1,\dots, f_r are polynomials that generate a prime ideal in k[t_1,\dots,t_n]."

no is not. spec is the set of PRIME ideals.--217.200.185.29 (talk) 12:08, 25 October 2013 (UTC)

Actually, you can do both ways. And as long as you're doing classical algebraic geometry, the distinction does not make much difference (cf. Jacobson ring). For example, that (i.e., max-spec) is how Milne does. But I do agree the notation is probably confusing. -- Taku (talk) 15:25, 25 October 2013 (UTC)
No, the IP user is right, the article is confusing. The error comes from the fact that the whole article make a confusion between an affine variety and an affine scheme. They are not the same, even if the category of affine varieties and regular maps is equivalent to a subcategory of the affine schemes. In each article on algebraic varieties, one has to choose between the classical language and the scheme language. Here clearly, the classical language is undoubtful the best one, as scheme theory introduces unnecessary complications, and is not understandable for most readers willing to learn on the subject. Moreover, the article Affine scheme exists. I have rewritten the section to make it simple and correct. D.Lazard (talk) 16:26, 25 October 2013 (UTC)
I never said the article is not confusing. Having said that, affine scheme does exist and I agree that article is a better place for the discussion on the difference between affine schemes that are varieties and affine varieties in a classical sense. (Incidentally, this important piece of fact is not mentioned in spectrum of a ring, to which affine scheme redirects.) -- Taku (talk) 18:58, 25 October 2013 (UTC)