A cohomological characterization of Alexander schemes
For a smooth algebraic variety \(X\) its Chow group \(A*X\) has a ring structure given by the intersection product. However, when \(X\) has singularities the situation is much more complicated. In some cases it is impossible to define a natural intersection product on the group \(A*X\) [see \textit{A. Zobel}, Mathematika 8, 39-44 (1961; Zbl 0099.15902)]. When the singularities are mild, various attempts have been made to define such products. Danilov constructed intersection products for simplicial toric varieties [\textit{V. I. Danilov}, Russ. Math. Surv. 33, No. 2, 97-154 (1978); translation from Usp. Mat. Nauk 33, No. 2(200), 85-134 (1978; Zbl 0425.14013)], \textit{D. Mumford} did this for the moduli space of stable curves [in: Arithmetic and geometry, Pap. dedic I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271-328 (1983; Zbl 0554.14008)]. \textit{S. Kleiman} and \textit{A. Thorup} defined the category of \(C_\mathbb{Q}\) orthocyclic schemes in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick 1985, part II, Proc. Symp. Pure Math. 46, 321-370 (1987; Zbl 0664.14031). These schemes have been exensively studied by \textit{A. Vistoli} [Invent. Math. 97, No. 3, 613-670 (1989; Zbl 0694.14001) and Compos. Math. 70, No. 3, 199-225 (1989; Zbl 0702.14002)] who called them Alexander schemes. The category of Alexander schemes is convenient from the point of view of the intersection theory with \(\mathbb{Q}\)-coefficients. In the paper the author answers affirmatively the conjecture of A. Vistoli that the Alexander property is Zariski local. The author constructs a category of bivariant sheaves and shows (theorem 4.6) that the scheme is Alexander if and only if all the first cohomolgoy groups of the bivariant sheaves vanish. Then the author proves that a union of two Alexander subschemes is again Alexander.
scientific article; zbMATH DE number 1336940