# 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 [1].

## 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

[1] 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 [1] 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]

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