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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)].

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