Talk:Chow group

Mathematics B‑class Mid‑priority

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Intersection theory

Untitled

I believe that, so far, the intersection theory on X is only defined when X is smooth. 117.28.251.191 (talk) 08:58, 26 October 2014 (UTC)[reply]

Do you mean to say X must be smooth throughout the article? I don't think so. As far as I can tell from Fulton's "intersection theory", the minimum requirement is a regular embedding; that X should be regularly embedded into some ambient scheme. This is more general than requiring X and the ambient one to be smooth. This generality is needed also to cover the complete intersection case. -- Taku (talk) 18:50, 27 June 2015 (UTC)[reply]

Ah, but for the "ring structure", "smooth" is indispensable? Not sure. -- Taku (talk) 18:51, 27 June 2015 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Chow group/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

In accordance with general principles, this article needs references. Fulton's Intersection Theory might work. I would also like to see history: this is certainly a subject with a past, and at the very least one wants to know why it's named after Chow, who (if not him; probably not, knowing math) originated it, and with some more ambition, to whom the main theorems are due (in particular, if they have "classical" and "post-Grothendieck" versions). And in even more generality, I think this article might benefit from the broad vision of someone who knows that the subject is about: I know it as little more than a collection of strange definitions and some basic theorems, but a real algebraic geometer could give it a great deal more context within mathematics. The article is way too complete to be just a "Start", which is why I've awarded it a "B"; it doesn't seem to merit a "B+" given these criticisms. Ryan Reich 21:27, 15 May 2007 (UTC)[reply]

Last edited at 21:27, 15 May 2007 (UTC). Substituted at 01:52, 5 May 2016 (UTC)

Much needed examples

We really need to add some more examples to this page. There should be the following:

chow ring for grassmannians
chow ring for flag varieties
chow ring for blowups, do for projective spaces and maybe for G(2,4)
chow ring of an algebraic curve
chow rings of complements of points in one of these spaces
arithmetic examples, such as the chow ring of the ring of integers for a number field
chow ring of an abelian variety

Also, we should write down examples of proper pushforward and flat pullback using families of varieties. There are some interesting examples using morphisms of relative dimension one, but it would be nice to have some explicit illustrations/diagrams showing how these morphisms on the chow rings work.