Zero cycles on a threefold with isolated singularities. (English) Zbl 1096.14004
Let \(X\) be a quasi-projective threefold over a field \(k\) of characteristic zero with Cohen-Macaulay isolated singularities. Let \(\text{CH}^3(X)\) denote the Chow group of zero cycles on \(X\) as is defined in [M. Levine and C. A. Weibel, J. Reine Angew. Math. 359, 106–120 (1985; Zbl 0555.14004)]. Let \(F^3K_0(X)\) denote the subgroup of the Grothendieck group of vector bundles \(K_0(X)\), generated by the classes of smooth codimension three closed points on \(X\). Furthermore for any closed subscheme \(Z\) of \(X\), let \(K_0(X, Z)\) be the relative \(K\)-group as defined in [K. R. Coombes and V. Srinivas, Invent. Math. 70, 1–12 (1982; Zbl 0507.14014)], and let \(F^3K_0(X,Z)\) be the subgroup of \(K_0(X,Z)\) generated by the classes of smooth points of \(X- Z\). Let \(p: \widetilde X\to X\) be a resolution of singularities such that \(p\) is a blow up of \(X\) along a closed subscheme \(Z\) whose support is the set of singular points of \(X\), and let \(E\) be the reduced exceptional divisor on \(\widetilde X\). Under these circumstances the main theorem of this paper shows that for all sufficiently large \(n\), the natural surective maps \(F^3K_0(\widetilde X,nE)\to F^3K_0(\widetilde X,(n-1)E)\) and \(F^3K_0(X)\to F^3K_0(\widetilde X,nE)\) are isomorphisms.
Since the map CH\(^3(X)\to F^3K_0(X)\) is shown by M. Levine [Proc. Symp. Pure Math. 46, 451–462 (1987; Zbl 0635.14007)] to be an isomorphism when \(k\) is algebraically closed and \(X\) is affine or projective, the theorem implies in this case that \(\text{CH}^3(X)\cong\varprojlim_n\,F^3K_0(\widetilde X,nE)\). A crucial idea for the proof is to employ the Northcott-Rees theory of reduction of ideals in [C. Weibel, Duke Math. J. 108, No. 1, 1–35 (2001; Zbl 1092.14014)], which assures the existence of a ideal subsheaf \({\mathcal J}\) of the ideal sheaf \({\mathcal I}\) of the center of \(p\) such that \({\mathcal J}{\mathcal I}^n={\mathcal I}^{n+1}\) for sufficiently large \(n\), and the fact that one can choose \({\mathcal J}\) to be a local complete intersection ideal sheaf by the Cohen-Macaulayness of \(X\). If we denote the blow up of \(X\) along \({\mathcal J}\) by \(p': X'\to X\), then the map \(F^3K_0(X)\to F^3K_0(X')\) is known to be injective by [A. Krishna and V. Srinivas, Ann. Math. (2) 156, 155–195 (2002; Zbl 1060.14015)]. This fact together with a careful analysis of the functoriality of various maps between relative \(K\)-groups concludes the proof of the theorem.
Since the map CH\(^3(X)\to F^3K_0(X)\) is shown by M. Levine [Proc. Symp. Pure Math. 46, 451–462 (1987; Zbl 0635.14007)] to be an isomorphism when \(k\) is algebraically closed and \(X\) is affine or projective, the theorem implies in this case that \(\text{CH}^3(X)\cong\varprojlim_n\,F^3K_0(\widetilde X,nE)\). A crucial idea for the proof is to employ the Northcott-Rees theory of reduction of ideals in [C. Weibel, Duke Math. J. 108, No. 1, 1–35 (2001; Zbl 1092.14014)], which assures the existence of a ideal subsheaf \({\mathcal J}\) of the ideal sheaf \({\mathcal I}\) of the center of \(p\) such that \({\mathcal J}{\mathcal I}^n={\mathcal I}^{n+1}\) for sufficiently large \(n\), and the fact that one can choose \({\mathcal J}\) to be a local complete intersection ideal sheaf by the Cohen-Macaulayness of \(X\). If we denote the blow up of \(X\) along \({\mathcal J}\) by \(p': X'\to X\), then the map \(F^3K_0(X)\to F^3K_0(X')\) is known to be injective by [A. Krishna and V. Srinivas, Ann. Math. (2) 156, 155–195 (2002; Zbl 1060.14015)]. This fact together with a careful analysis of the functoriality of various maps between relative \(K\)-groups concludes the proof of the theorem.
Reviewer: Fumio Hazama (Hatoyama)
MSC:
| 14C25 | Algebraic cycles |
| 14J30 | \(3\)-folds |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
References:
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