Talk:Chow group

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)

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)

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

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 The comments may be irrelevant or outdated; if so, please feel free to remove this section.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)

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. — Preceding unsigned comment added by 97.122.179.164 (talk) 03:18, 5 May 2017 (UTC)

This is probably a good idea but I would suggest some detailed computations to appear on object articles as opposed to invariant articles; i.e., discuss Chow ring of a flag variety in flag variety. This is because, over C, the answer is the same as that of cohomology (and cohomology is more familiar). The complications can also be discussed in the object article (not here). See also projective bundle for what I mean. -- Taku (talk) 23:33, 8 August 2017 (UTC)

Fulton's intersection theory has a discussion of the Chow group of a blowup and that, which is important, should certainly appear in the article. -- Taku (talk) 23:37, 8 August 2017 (UTC)

I noticed the article only considers the situation only over a field; I think, as already suggested above, we should mention the case over Z (or some other regular ring). Most stuff should work over a regular ring, right? -- Taku (talk) 23:47, 8 August 2017 (UTC)

Intersection Product

This page should discuss the construction of the intersection product for the $\mathbb {Q}$ chow ring using the derived tensor product from $D^{b}(X)$. This is how grothendieck originally constructed intersection theory in SGA 6. — Preceding unsigned comment added by 71.212.185.82 (talk) 15:31, 24 August 2017 (UTC)

Wrong definition of CH or of the intersection?

I think that there is a mistake on the definition of the Chow group! Indeed the i-cycles are not linear combination of i-dimensional subvarieties, but i-co-dimensionale sub-varieties.

The fact that the intersection send an i-cycle and a j-cycle to an i+j-cycle should show that the definition with dimension (and not with co-dimension) is wrong. (If I intersect two points in general position I do not get a point, but the empty "variety": the 0 cycle, if I intersect two curve I do not get a 1+1=2-dimensional space (a plane): I get a point...).

Please correct what's wrong (it may be the definition of CH^* or maybe the definition of the product? I think that the wrong definition is of the Chow Group CH) Marco Baracchin (talk) 15:31, 30 September 2023 (UTC)