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Zero cycles and the Euler class groups of smooth real affine varieties. (English) Zbl 0949.14005
The main result of the paper under review is that, given a real smooth algebraic variety $$X=\text{Spec } A$$ of dimension $$n\geq 2$$ and a projective $$A$$-module $$P$$ of rank $$n$$ such that the corresponding vector bundle over $$X({\mathbb R})$$ has a nowhere vanishing section, then $$P$$ has a unimodular element (i.e., splits off a free summand of rank one), provided the top Chern class $$C_n(P)\in \text{CH}_0(X)$$ is zero and $$P$$ has trivial determinant. As a consequence, a unimodular element in $$P$$ does exist if $$C_n(P)=0$$, $$\det(P)=0$$, and $$X({\mathbb R})$$ has no compact connected component. Without any geometric assumption on $$X({\mathbb R})$$, if either $$n$$ is odd or the canonical module $$K_{{\mathbb R}(X)}$$ of the ring $${\mathbb R}(X)$$ of real regular functions on $$X({\mathbb R})$$ is trivial, then $$P$$ has a unimodular element iff $$P\otimes{\mathbb R}(X)$$ has a unimodular element. If $$X ({\mathbb R})$$ is compact, it is well known that $$P\otimes{\mathbb R}(X)$$ has a unimodular element iff the associated vector bundle on $$X({\mathbb R})$$ has a nowhere vanishing section. It is well known also that the unimodular element in $$P$$ exists if $$\text{ rk}(P)>n$$, or if $$X$$, $$P$$ are considered over an algebraically closed field, $$\text{ rk}(P)=n$$ and $$C_n(P)=0$$. The main tool is Nori’s Euler class group $$E(A)$$ of $$A$$, and its unoriented version $$E_0(A)$$. $$E_0(A)$$ is, by definition, the quotient of the group $$Z_0(X)$$ of zero-cycles on $$X$$ by the subgroup generated by those cycles $$M_1+\dots{}+M_r$$ for which $$M_i\neq M_j$$ for $$i\neq j$$ and the associated closed subscheme of $$X$$ is a complete intersection. The authors show that, over the field $${\mathbb R}$$, the natural epimorphism $$E_0(A)\to \text{ CH}_0 (X)$$ is an isomorphism. Another important ingredient is an earlier result by the authors [S. M. Bhatwadekar and R. Sridharan, Invent. Math. 133, No. 1, 161-192 (1998; Zbl 0936.13005)].

##### MSC:
 14C25 Algebraic cycles 13C10 Projective and free modules and ideals in commutative rings 14P05 Real algebraic sets
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