Abstract
The problem of defining intersection products on the Chow groups of schemes has a long history. Perhaps the first example of a theorem in intersection theory is Bézout’s theorem, which tells us that two projective plane curves C and D, of degrees c and d and which have no components in common, meet in at most cd points. Furthermore if one counts the points of C ∩ D with multiplicity, there are exactly cd points. Bezout’s theorem can be extended to closed subvarieties Y and Z of projective space over a field k, ℙ n k , with dim(Y) + dim(Z) = n and for which Y ∩ Z consists of a finite number of points.
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References
Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958.
Théorie des intersections et théorème de Riemann–Roch. Springer-Verlag, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre, Lecture Notes in Mathematics, Vol. 225.
Paul Baum, William Fulton, and Robert MacPherson. Riemann–Roch for singular varieties. Inst. Hautes Études Sci. Publ. Math., (45):101–145, 1975.
Pierre Berthelot. Altérations de variétés algébriques (d’après A. J. de Jong). Astérisque, (241):Exp. No. 815, 5, 273–311, 1997. Séminaire Bourbaki, Vol. 1995/96.
A.A. Be?linson. Higher regulators and values of L-functions. In Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, pages 181–238. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
J. G. Biswas and V. Srinivas. The Chow ring of a singular surface. Proc. Indian Acad. Sci. Math. Sci., 108(3):227–249, 1998.
S. Bloch. A note on Gersten’s conjecture in the mixed characteristic case. In Applications of algebraic K -theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), volume 55 of Contemp. Math., pages 75–78. Amer. Math. Soc., Providence, RI, 1986.
Spencer Bloch. K2 and algebraic cycles. Ann. of Math. (2), 99:349–379, 1974.
Spencer Bloch and Arthur Ogus. Gersten’s conjecture and the homology of schemes. Ann. Sci. École Norm. Sup. (4), 7:181–201 (1975), 1974.
Kenneth S. Brown. Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc., 186:419–458, 1974.
Kenneth S. Brown and Stephen M. Gersten. Algebraic K-theory as generalized sheaf cohomology. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 266–292. Lecture Notes in Math., Vol. 341. Springer-Verlag, Berlin, 1973.
Luther Claborn and Robert Fossum. Generalizations of the notion of class group. Illinois J. Math., 12:228–253, 1968.
Jean-Louis Colliot-Thélène, Raymond T. Hoobler, and Bruno Kahn. The Bloch-Ogus-Gabber theorem. In Algebraic K -theory (Toronto, ON, 1996), volume 16 of Fields Inst. Commun., pages 31–94. Amer. Math. Soc., Providence, RI, 1997.
A.J. de Jong. Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math., (83):51–93, 1996.
Samuel Eilenberg and Norman Steenrod. Foundations of algebraic topology. Princeton University Press, Princeton, New Jersey, 1952.
A. D. Elmendorf, I. Kříž, M. A. Mandell, and J. P. May. Modern foundations for stable homotopy theory. In Handbook of algebraic topology, pages 213–253. North-Holland, Amsterdam, 1995.
William Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, 1998.
Ofer Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K -theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59–70. Amer. Math. Soc., Providence, RI, 1992.
Ofer Gabber. Gersten’s conjecture for some complexes of vanishing cycles. Manuscripta Math., 85(3–4):323–343, 1994.
H. Gillet. Some new Gysin homomorphisms for the Chow homology of varieties. Proc. London Math. Soc. (3), 50(1):57–68, 1985.
H. Gillet and C. Soulé. Intersection theory using Adams operations. Invent. Math., 90(2):243–277, 1987.
H. Gillet and C. Soulé. Descent, motives and K-theory. J. Reine Angew. Math., 478:127–176, 1996.
H. Gillet and C. Soulé. Filtrations on higher algebraic K-theory. In Algebraic K -theory (Seattle, WA, 1997), volume 67 of Proc. Sympos. Pure Math., pages 89–148. Amer. Math. Soc., Providence, RI, 1999.
Henri Gillet. Riemann–Roch theorems for higher algebraic K-theory. Adv. in Math., 40(3):203–289, 1981.
Henri Gillet. Universal cycle classes. Compositio Math., 49(1):3–49, 1983.
Henri Gillet. Deligne homology and Abel-Jacobi maps. Bull. Amer. Math. Soc. (N.S.), 10(2):285–288, 1984.
Henri Gillet. Homological descent for the K-theory of coherent sheaves. In Algebraic K -theory, number theory, geometry and analysis (Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 80–103. Springer-Verlag, Berlin, 1984.
Henri Gillet. Intersection theory on algebraic stacks and Q-varieties. J. Pure Appl. Algebra, 34(2–3):193–240, 1984.
Henri Gillet. Gersten’s conjecture for the K-theory with torsion coefficients of a discrete valuation ring. J. Algebra, 103(1):377–380, 1986.
Henri Gillet. K-theory and intersection theory revisited. K -Theory, 1(4):405–415, 1987.
Henri Gillet and Marc Levine. The relative form of Gersten’s conjecture over a discrete valuation ring: the smooth case. J. Pure Appl. Algebra, 46(1):59–71, 1987.
Henri Gillet and William Messing. Cycle classes and Riemann–Roch for crystalline cohomology. Duke Math. J., 55(3):501–538, 1987.
Henri A. Gillet and Robert W. Thomason. The K-theory of strict Hensel local rings and a theorem of Suslin. J. Pure Appl. Algebra, 34(2–3):241–254, 1984.
Daniel R. Grayson. Products in K-theory and intersecting algebraic cycles. Invent. Math., 47(1):71–83, 1978.
Daniel R. Grayson. Weight filtrations via commuting automorphisms. K -Theory, 9(2):139–172, 1995.
A.Grothendieck.Élémentsdegéométriealgébrique.IV.Étudelocaledessché-mas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math., (24):231, 1965.
A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV. Inst. Hautes Études Sci. Publ. Math., (32):356, 1967.
Alexander Grothendieck. La théorie des classes de Chern. Bull. Soc. Math. France, 86:137–154, 1958.
Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J. Amer. Math. Soc., 13(1):149–208, 2000.
J. F. Jardine. Presheaves of symmetric spectra. J. Pure Appl. Algebra, 150(2):137–154, 2000.
Shun-ichi Kimura. Fractional intersection and bivariant theory. Comm. Algebra, 20(1):285–302, 1992.
Steven L. Kleiman. Misconceptions about Kx. Enseign. Math. (2), 25(3–4):203–206 (1980), 1979.
Steven L. Kleiman. Cartier divisors versus invertible sheaves. Comm. Algebra, 28(12):5677–5678, 2000. Special issue in honor of Robin Hartshorne.
Ch. Kratzer. λ-structure en K-théorie algébrique. Comment. Math. Helv., 55(2):233–254, 1980.
Andrew Kresch. Canonical rational equivalence of intersections of divisors. Invent. Math., 136(3):483–496, 1999.
Andrew Kresch. Cycle groups for Artin stacks. Invent. Math., 138(3):495–536, 1999.
Marc Levine. Bloch’s formula for singular surfaces. Topology, 24(2):165–174, 1985.
Marc Levine and Chuck Weibel. Zero cycles and complete intersections on singular varieties. J. Reine Angew. Math., 359:106–120, 1985.
C.R.F. Maunder. The spectral sequence of an extraordinary cohomology theory. Proc. Cambridge Philos. Soc., 59:567–574, 1963.
K.H. Paranjape. Some spectral sequences for filtered complexes and applications. J. Algebra, 186(3):793–806, 1996.
Claudio Pedrini and Charles A. Weibel. K-theory and Chow groups on singular varieties. In Applications of algebraic K -theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), volume 55 of Contemp. Math., pages 339–370. Amer. Math. Soc., Providence, RI, 1986.
Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85–147. Lecture Notes in Math., Vol. 341. Springer-Verlag, Berlin, 1973.
Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967.
Joel Roberts. Chow’s moving lemma. In Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), pages 89–96. Wolters-Noordhoff, Groningen, 1972. Appendix 2 to: “Motives” (Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), pp. 53–82, Wolters-Noordhoff, Groningen, 1972) by Steven L. Kleiman.
Paul Roberts. The vanishing of intersection multiplicities of perfect complexes. Bull. Amer. Math. Soc. (N.S.), 13(2):127–130, 1985.
Paul C. Roberts. Recent developments on Serre’s multiplicity conjectures: Gabber’s proof of the nonnegativity conjecture. Enseign. Math. (2), 44(3–4):305–324, 1998.
Markus Rost. Chow groups with coefficients. Doc. Math., 1:No. 16, 319–393 (electronic), 1996.
Pierre Samuel. Algèbre locale. Mem. Sci. Math., no. 123. Gauthier-Villars, Paris, 1953.
Stefan Schwede. S-modules and symmetric spectra. Math. Ann., 319(3):517–532, 2001.
Jean-Pierre Serre. Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965., volume 11 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1965.
Clayton Sherman. Some theorems on the K-theory of coherent sheaves. Comm. Algebra, 7(14):1489–1508, 1979.
Clayton Sherman. Group representations and algebraic K-theory. In Algebraic K -theory, Part I (Oberwolfach, 1980), volume 966 of Lecture Notes in Math., pages 208–243. Springer-Verlag, Berlin, 1982.
Christophe Soulé. Opérations en K-théorie algébrique. Canad. J. Math., 37(3):488–550, 1985.
A.A. Suslin. Homology of GLn, characteristic classes and Milnor K-theory. In Algebraic K -theory, number theory, geometry and analysis (Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 357–375. Springer-Verlag, Berlin, 1984.
R.W. Thomason and Thomas Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 247–435. Birkhäuser Boston, Boston, MA, 1990.
Jean-Louis Verdier. Le théorème de Riemann–Roch pour les intersections complètes. In Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974–75), pages 189–228. Astérisque, No. 36–37. Soc. Math. France, Paris, 1976.
Angelo Vistoli. Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math., 97(3):613–670, 1989.
Charles A. Weibel. Pic is a contracted functor. Invent. Math., 103(2):351–377, 1991.
Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.
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Gillet, H. (2005). K-Theory and Intersection Theory. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_7
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