## 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].

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## Questions

>  1. Is a functor of points approach to algebraic cycles and intersection theory "doable"?
>  2. Has someone written about this?
### 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.

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### 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 .