Bloch's formula for singular surfaces
This is one paper in a series of articles devoted to extensions of the Bloch-Quillen formula \(CH^ p(X)\approx H^ p_{Zar}(X,{\mathcal K}_ p)\) to singular varieties. This subject was started by \textit{A. Collino} [Ill. J. Math. 25, 654-666 (1981; Zbl 0496.14005)] who considered the case when X has only one singular point. It has been pursued by \textit{C. Pedrini} and \textit{C. A. Weibel} [''K-theory and Chow groups on singular varieties'', in Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 339-370 (1986), a work which contains an extensive bibliography] and by the author [''A geometric theory of the Chow ring of a singular variety'' (preprint)]. In the present paper, the author recalls one quite simple definition of the relative Chow group \(CH^ 2(X,S)\) when X is a surface [N.B. There is another one due to C. Pedrini and C. Weibel (loc. cit.); when S is not finite, these definitions lead to different groups]. The main result of this sophisticated paper, which builds upon Collino's paper as well as on Bloch's original paper [\textit{S. Bloch}, Ann. Math., II. Ser. 99, 349-379 (1974; Zbl 0298.14005)] is then that for X a surface over an infinite field k, the groups \(CH^ 2(X,S)\) and \(H^ 2_{Zar}(X,{\mathcal K}_ 2)\) are isomorphic. This seems to indicate that the chosen definition of \(CH^ 2(X,S)\) (a definition which does not impose any restriction on S) is the appropriate one.
scientific article; zbMATH DE number 3964008
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