Zero cycles and projective modules.

*(English)*Zbl 0839.13007This 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 \(s_0 (M) = 0\).

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 \(s_0 (M) = 0\).

Reviewer: J.Rosenknop (Berlin)