Duke Mathematical Journal

Roitman’s theorem for singular complex projective surfaces

Article information

Source
Duke Math. J., Volume 84, Number 1 (1996), 155-190.

Dates
First available in Project Euclid: 19 February 2004

https://projecteuclid.org/euclid.dmj/1077243631

Digital Object Identifier
doi:10.1215/S0012-7094-96-08405-7

Mathematical Reviews number (MathSciNet)
MR1394751

Zentralblatt MATH identifier
0864.14026

Citation

Barbieri-Viale, L.; Pedrini, C.; Weibel, C. Roitman’s theorem for singular complex projective surfaces. Duke Math. J. 84 (1996), no. 1, 155--190. doi:10.1215/S0012-7094-96-08405-7. https://projecteuclid.org/euclid.dmj/1077243631

References

• [1] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, vol. 305, Springer-Verlag, Berlin, 1973.
• [2] L. Barbieri-Viale, Zero-cycles on singular varieties: torsion and Bloch's formula, J. Pure Appl. Algebra 78 (1992), no. 1, 1–13.
• [3] L. Barbieri-Viale and V. Srinivas, On the Néron-Severi group of a singular variety, J. Reine Angew. Math. 435 (1993), 65–82.
• [4] L. Barbieri-Viale and V. Srinivas, The Néron-Severi group and the mixed Hodge structure on $H\sp 2$. Appendix to: “On the Néron-Severi group of a singular variety”, J. Reine Angew. Math. 450 (1994), 37–42.
• [5] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), Springer, Berlin, 1973, 349–446. Lecture Notes in Math., Vol. 342.
• [6] A. Beilinson, Higher regulators and values of $L$-functions, J. Soviet Math. 30 (1985), 2036–2070.
• [7] K. Brown and S. Gersten, Algebraic $K$-theory as generalized sheaf cohomology, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, 266–292. Lecture Notes in Math., Vol. 341.
• [8] A. Collino, Quillen's $K$-theory and algebraic cycles on singular varieties, Geometry today (Rome, 1984), Progr. Math., vol. 60, Birkhäuser Boston, Boston, MA, 1985, pp. 75–85.
• [9] A. Collino, Torsion in the Chow group of codimension two: the case of varieties with isolated singularities, J. Pure Appl. Algebra 34 (1984), no. 2-3, 147–153.
• [10] J.-L. Colliot-Thélène, Cycles algébriques de torsion et $K$-théorie algébrique, Arithmetic algebraic geometry (Trento, 1991) ed. E. Ballico, Lecture Notes in Math., vol. 1553, Springer-Verlag, Berlin, 1993, pp. 1–49.
• [11] J.-L. Colliot-Thélène and W. Raskind, $\scr K\sb 2$-cohomology and the second Chow group, Math. Ann. 270 (1985), no. 2, 165–199.
• [12]1 P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57.
• [12]2 P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77.
• [13] H. Esnault, Une remarque sur la cohomologie du faisceau de Zariski de la $K$-théorie de Milnor sur une variété lisse complexe, Math. Z. 205 (1990), no. 3, 373–378.
• [14] H. Esnault and E. Viehweg, Deligne-Beĭlinson cohomology, Beĭlinson's conjectures on special values of $L$-functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 43–91.
• [15] O. Gabber, $K$-theory of Henselian local rings and Henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70.
• [16] S. Geller and L. Roberts, Kahler differentials and excision for curves, J. Pure Appl. Algebra 17 (1980), no. 1, 85–112.
• [17] S. Geller and C. Weibel, $K\sb1(A,\,B,\,I)$, J. Reine Angew. Math. 342 (1983), 12–34.
• [18] H. Gillet, Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math. 40 (1981), no. 3, 203–289.
• [19] H. Gillet, On the $K$-theory of surfaces with multiple curves and a conjecture of Bloch, Duke Math. J. 51 (1984), no. 1, 195–233.
• [20] H. Gillet, Deligne homology and Abel-Jacobi maps, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 285–288.
• [21] J. Graham, Continuous symbols on fields of formal power series, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, 474–486. Lecture Notes in Math., Vol. 342.
• [22] R. Hoobler, Étale cohomology of geometric local rings is Galois cohomology, to appear.
• [23] J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87.
• [24] M. Levine, Bloch's formula for singular surfaces, Topology 24 (1985), no. 2, 165–174.
• [25] M. Levine, Torsion zero-cycles on singular varieties, Amer. J. Math. 107 (1985), no. 3, 737–757.
• [26] M. Levine, Zero-cycles and $K$-theory on singular varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 451–462.
• [27] M. Levine and C. Weibel, Zero cycles and complete intersections on singular varieties, J. Reine Angew. Math. 359 (1985), 106–120.
• [28] S. Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), no. 1, 183–215.
• [29] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 195–279.
• [30] J. P. May, Simplicial objects in algebraic topology, Van Nostrand Math. Stud., No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967.
• [31] J. Milnor and J. Stasheff, Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, Princeton, N. J., 1974.
• [32] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961), no. 9, 5–22.
• [33] C. Pedrini and C. Weibel, $K$-theory and Chow groups on singular varieties, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 339–370.
• [34] C. Pedrini and C. Weibel, Divisibility in the Chow group of zero-cycles on a singular surface, Astérisque (1994), no. 226, 10–11, 371–409.
• [35] W. Raskind, On $K\sb 1$ of curves over global fields, Math. Ann. 288 (1990), no. 2, 179–193.
• [36] A. A. Rojtman, The torsion of the group of $0$-cycles modulo rational equivalence, Ann. of Math. (2) 111 (1980), no. 3, 553–569.
• [37] A. Suslin, On the $K$-theory of local fields, J. Pure Appl. Algebra 34 (1984), no. 2-3, 301–318.
• [38] A. Suslin, Torsion in $K\sb 2$ of fields, $K$-Theory 1 (1987), no. 1, 5–29.
• [39] R. Thomason and T. Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435.
• [40] C. Weibel, Mayer-Vietoris sequences and mod $p$ $K$-theory, Algebraic $K$-theory, Part I (Oberwolfach, 1980), Lecture Notes in Math., vol. 966, Springer, Berlin, 1982, pp. 390–407.
• [41] C. Weibel, Étale Chern classes at the prime $2$, Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 249–286.
• [42] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.