# zbMATH — the first resource for mathematics

Zero cycles and projective modules. (English) Zbl 0839.13007
This work contains proofs and extensions of results announced earlier [the author, Bull. Am. Math. Soc., New Ser. 19, No. 1, 315-317 (1988; Zbl 0656.13008)]. – Let $$A$$ be a commutative Noetherian ring of Krull dimension $$\dim A = n$$, $$K_0 (A)$$ be the Grothendieck group of finitely generated $$A$$-modules, identified with the Grothendieck group of the category of finite $$A$$-modules of finite projective dimension. For such a module $$M$$, let $$(M)$$ be its image in $$K_0 (A)$$. Let $$F^nK_0 (A)$$ be the subgroup of $$K_0 (A)$$ generated by all $$(A/{\mathfrak M})$$, where $${\mathfrak M}$$ is a maximal ideal of $$A$$ such that $$A_{\mathfrak M}$$ is a regular local ring of dimension $$n$$.
Let us assume further that $$A$$ is a reduced affine $$k$$-algebra over a field $$k$$, $$P$$ is a projective $$A$$-module of rank $$n$$, $$C_n (P) = \sum_i (-1)^i (\bigwedge^i P^*)$$, where $$P^*$$ is the dual of $$P$$. Suppose that $$F^n K_0 (A)$$ has no $$(n - 1)$$! torsion. It is proved that if $$k$$ is algebraically closed, then there exists a $$P'$$ such that $$(P) = (P' \oplus A) + {(-1)^{n - 1} \over (n - 1)!} C_n(P)$$ and (in some other cases, as well) $C_n (P) = 0 \Leftrightarrow (\exists P') P \approx P' \oplus A. \tag{*}$ Some applications to generation of $$A$$-modules are considered. For $${\mathfrak p} \in \text{Spec} A$$, let us the denoted by $$\mu_{ \mathfrak p} (M)$$ the number of elements in a minimal set of generators for $$M_{\mathfrak p}$$ and $\delta (M) = \text{supp} \bigl \{\mu_{\mathfrak p} (M) + \dim A/{\mathfrak p} \mid {\mathfrak p} \in \text{supp} (M), \dim A/{\mathfrak p} < n \bigr\}.$ Some conditions for every $$A$$-module $$M$$ to be generated by $$\delta (M)$$ elements are established, in particular, $$F^n K_0 (A) = 0$$ or the condition (*) for every projective $$A$$-module $$P$$.
Finally, the case is studied, where $$A$$ is a regular domain and $$X = \text{Spec} A$$ is a smooth affine $$n$$-dimensional variety. The group $$F^n K_0 (A)$$ can be identified with the Chow group $$\text{CH}^n (X)$$ of zero cycles on $$X$$ here. A class $$s_0 (M)$$ in $$\text{CH}^n (X)$$ is defined (the “Segre class”) and some properties related to it are investigated, in particular: $$M$$ is generated by $$\delta (M)$$ elements if and only if $$s_0 (M) = 0$$.

##### MSC:
 13C10 Projective and free modules and ideals in commutative rings 14C05 Parametrization (Chow and Hilbert schemes) 13D15 Grothendieck groups, $$K$$-theory and commutative rings
##### Keywords:
Segre class; Grothendieck group; Chow group
Full Text: