Talk:Affine variety

[Untitled]

"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)

The current linking of quasi affine variety to a wikipedia page about quasi projective variety

I am not sure that this linking is entirely helpful - the two are very different. Alainc67 (talk) 04:57, 22 August 2018 (UTC)

Is algebraic closure required?

There is a differing convention on whether or not the field over which an affine variety is defined is required to be algebraically closed. I assumed it wasn't in some edits I made yesterday, but having looked through some sources, it does seem that algebraic closure is more common (being used in Hartshorne and Fulton, as well as Humphreys' Linear Algebraic Groups), but some common references like Reid do not use it. I've decided to go with the convention requiring algebraic closure due to the usefulness of the nullstellensatz, but I was wondering if the mathematics manual of style should be updated with a sentence making this convention explicit? Donutmug (talk) 22:04, 1 June 2019 (UTC)

It is important to distinguish the field of definition of the variety (a field containing all coefficients of the defining equations} from the field in which the coordinates of the points must belong. It is the latter field that requires to be algebraically closed. This distinction is important for the common use of Nullstellensatz, as this theorem provides information on points over an algebraically closed field without needing to compute in this algebraically closed field. The article Hilbert's Nullstellensatz makes clearly this distinction. I have not yet read your edits, but, in any case, it must be clear that this convention applies only for the coordinates of the points, not for the ring in which the defining ideal is considered. A clear witness that this distinction is very common is the common use of the terms of "real point" or "rational point". D.Lazard (talk) 10:36, 2 June 2019 (UTC)

Recent revert

I have just reverted a sequence of 8 edits by Ocsenave. Contrary to my habit I have not proided any edit summary because, by mistake, I saved my revert before filling it. The reason is that these edit add original research, improper terminology and other inaccuracies. Here are some examples:

• "Fundamental set of solutions": WP:OR terminology, which is not even defined.
• "real numbers, complex numbers, or more generally members of a algebraically closed field: real numbers do not form an algebraically closed field
• "One essential characteristic of a field is which of its affine varieties can be factored into simpler parts": WP:OR and wrong
• Removing from the lead the distinction between algebraic set and algebraic variety.

There are many other issues, but listing them is a waste of time. D.Lazard (talk) 10:17, 13 October 2020 (UTC)

"Ring of regular functions" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Ring of regular functions and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 December 21 § Ring of regular functions until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 00:29, 21 December 2022 (UTC)