Bivariant Chow groups and the topology of envelopes (Q1928434)

Let \(K\) be an algebraically closed field and let \(Sch_K\) denote the category of schemes over \(K\). The main objective of this paper is to extend the bivariant Chow theory, defined and studied by \textit{W. Fulton} and \textit{R. MacPherson} in [Categorical framework for the study of singular spaces. Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)], to the category of presheaves \(Pre_K\) on \(Sch_K\). Let \(S\) be a scheme in \(Sch_K\). A morphism \(f:\mathcal{F}\rightarrow\mathcal{G}\) of presheaves of sets on \(Sch_K/S\) is said to be representable, if for any object \(Y/S\in Sch_K/S\) and a morphism \(h_{Y/S}\rightarrow\mathcal{G}\) of presheaves on \(Sch_K/S\), the fibre product presheaf \(\mathcal{F}\times_{\mathcal{G}}h_{Y/S}\) is a representable presheaf on \(Sch_K/S\). The author associates to each representable morphism \(f:\mathcal{F}\rightarrow\mathcal{G}\), a bivariant Chow group \(CH_S^p(f:\mathcal{F}\rightarrow\mathcal{G})\) for each \(p\in\mathbb{Z}\), and show that, when the presheaves \(\mathcal{F}, \mathcal{G}\) are representable respectively by \(W, Z\in Sch_K/S'\) with \(S'\rightarrow S=\mathrm{Spec}{K}\) a smooth morphism, there exists a natural isomorphism from \(\mathrm{CH}_{S'}^p(f_{S'}:h_{W/S'}\rightarrow h_{Z/S'})\) to the Fulton-MacPherson bivariant Chow group \(\mathrm{CH}_{FM}^p(f:W\rightarrow Z)\).