# Is there a functor of points approach to algebraic cycles and intersection theory?

## Motivation

Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is particularly powerful. As a result I am more familiar with a scheme viewed as its functor of points than viewed as locally ringed space.

Recently I started looking into algebraic cycles and intersection theory. Most of this makes use of the underlying set of the scheme/variety one is working with. I am convinced that the "locally ringed spaces POV" is more natural here, and want to enhance my familiarity with it.

However, in training my intuition, sometimes I find that I would like to check some things via the "functor of points" approach. To give a concrete example, I find it pretty hard to compute the push-forward of a given algebraic cycle .

## Questions

1. Is a functor of points approach to algebraic cycles and intersection theory "doable"?

### Remarks

Ad 1. I can see a definition of a prime cycle as a morphism of schemes that is a closed immersion, with some other properties probably. However I am not sure how to define all the adequate equivalence relations that one usually encounters. Also I would not know how to define the intersection multiplicity.

Ad 2. Something introductory, preferably. That would be great.

Ad replies. I do not intend to launch a debate "Functor of points POV vs. locally ringed spaces POV". I am convinced that both POVs have their advantages.

### Note

 In particular (and this is very particular, as in localized) I really do not get the computation of $i^{*} \Delta_{\xi}$ in the proof of Lemma 5.1.5 of Zhang's paper "Gross–Schoen Cycles and Dualising Sheaves", available at http://www.arxiv.org/abs/0812.0371 .

 By now the motivation for this question (as explained in Note  above) is no longer there. Moreover, I am not so afraid anymore of the usual computations in intersection theory. (I probably got more used to it à la the von Neumann quote.) Further, I learned about Fulton's book “Intersection theory”, and though I have not read much of it, it seems to be a very good treatment of the subject.

Nevertheless, I am still very interested in a functor of points POV to intersection theory. I do have my vague doubts whether it is realistic to wish for such a thing, though. [/Edit]

• I'm not sure how much this satisfies you, but I did a little bit of a literature search and I found that several people have at least tried to do intersection theory on stacks. It's not exactly what you asked for but at least stacks are done F.O.P. style. In Gillet's "Intersection theory on algebraic stacks and $Q$-varieties" ('84), he says this goes back to Mumford looking at $\mathfrak{M}_g$ in "Towards an enumerative geometry of the moduli space of curves" ('83). I think after that was Gillet paper I mentioned, followed by Vistoli('89), Joshua('99), and more recent work by Gillet('09). Jun 22, 2014 at 19:32
• @JoeB — Thanks for the references! As I wrote in the edit, by now I'm pretty satisfied with the usual approach. Nevertheless, it sounds interesting, and I will have a look at the papers you mentioned.
– jmc
Jun 23, 2014 at 8:21
• You might look at Triangulated categories of motives in positive characteristic by Shane Kelly where there is a definition of presheaves of relative cycles. Aug 27, 2016 at 6:48
• Do you mean effective cycles, or arbitrary, up to rational equivalence or by themselves? There is an example of classes of zero cycles on a K3 surface, which - if I remember it right - cannot be represented by a scheme (or even a stack). Oct 6, 2016 at 2:55
• One thing that comes to mind is bivariant intersection theory (see e.g. chapter 17 of Fulton), which is used for Riemann–Roch on singular varieties. But this assumes quite a bit about Chow groups already, so it's very far from a ground-up approach to functor of points intersection theory. Dec 15, 2020 at 0:14