×

Intersection theory using Adams operations. (English) Zbl 0632.14009

The \(\lambda\)-structure of algebraic K-theory with supports is used to prove three results on intersection theory. The first result is a vanishing theorem for intersection multiplicities which was conjectured by Serre who proved it in several cases; another proof was obtained by Roberts. The second result describes, in positive characteristic, the action of the Frobenius endomorphism on the Euler characteristic of a complex; it is a special case of a conjecture by Szpiro. Finally, a multiplicative structure on the Chow groups of any noetherian regular scheme, after tensoring these by \({\mathbb{Q}}\) is introduced.
Reviewer: L.Vaserstein

MSC:

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Atiyah, M., Tall, D.O.: Group representations, ?-rings and theJ-homomorphism. Topology8, 253-297 (1969) · Zbl 0176.52701
[2] Bass, H.: AlgebraicK-theory. New York: Benjamin 1968 · Zbl 0174.30302
[3] Baum, P., Fulton, W., MacPherson, R.: Riemann-Roch for singular varieties. Publ. Math., Inst. Hautes Etud. Sci.45, 101-145 (1975) · Zbl 0332.14003
[4] Beilinson, A.: Higher regulators and values ofL-functions. J. Sov. Math.30, 2036-2070 (1985) · Zbl 0588.14013
[5] Brown, K.S., Gersten, S.M.: AlgebraicK-theory as generalized cohomology. Lect. Notes in Math., vol. 341, pp. 266-292. Berlin-Heidelberg-New York: Springer 1973
[6] Deligne, P.: Letter to A. Grothendieck, 23/10/1967
[7] Dold, A.: Homology of symmetric products and other functors of complexes. Ann. Math.68, 54-80 (1958) · Zbl 0082.37701
[8] Dold, A., Puppe, D.: Homologie nicht-additiver Funktoren. Anwendungen. Ann. Inst. Fourier11, 201-312 (1961) · Zbl 0098.36005
[9] Eilenberg, S., Maclane, S.: On the groupsH(?,n), II. Ann. Math.60, 49-139 (1954) · Zbl 0055.41704
[10] Fulton, W., Lang, S.: Riemann-Roch algebra. Grundlehren der math. Wissenschaften, vol. 277. Berlin-Heidelberg-New York-Tokyo: Springer 1985 · Zbl 0579.14011
[11] Fulton, W.: Intersection theory. Ergebnisse der Math. Berlin-Heidelberg-New York-Tokyo: Springer 1984 · Zbl 0541.14005
[12] Gillet, H.: Riemann-Roch theorems for higher algebraicK-theory. Adv. Math.40, (n0 3) 203-289 (1981) · Zbl 0478.14010
[13] Gillet, H.: An introduction to higher dimensional Arakelov theory, to appear in the proceedings of the AMS, Summer Research Conference on Arithmetical Geometry, Arcata (1985)
[14] Gillet, H., Messing, W.: Cycle classes and Riemann-Roch for crystalline cohomology. Duke Math. J. (in press) (1987) · Zbl 0651.14014
[15] Gillet, H., Soul?, C.:K-th?orie et nullit? des multiplicit?s d’intersection. C. R. Acad. Sci. Paris, S?r. I,300, (n0 3) 71-74 (1985) · Zbl 0587.13007
[16] Gillet, H., Soul?, C.: Intersection sur les vari?t?s d’Arakelov. C. R. Acad. Sci., Paris, S?r. I.299, (n0 12) 563-566 (1984) · Zbl 0607.14003
[17] Grothendieck, A., Berthelot, P., Illusie, L.: SGA 6, Th?orie des intersections et th?or?me de Riemann-Roch. Lect. Notes in Math., vol. 225. Berlin-Heidelberg-New York: Springer 1971 · Zbl 0218.14001
[18] Grothendieck, A.: SGA2, Cohomologie locale des faisceaux coh?rents et Th?or?mes de Lefschetz locaux et globaux. North-Holland Pub. Comp. 1968
[19] Hartshorne, R.: Algebraic geometry. Graduate Texts in Maths., vol. 52. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001
[20] Illusie, L.: Complexe cotangent et d?formations, I. Lect. Notes in Math., vol. 239. Berlin-Heidelberg-New York: Springer 1971 · Zbl 0224.13014
[21] Kratzer, C.: Op?rations d’Adams et repr?sentations de groupes. Ann. Sci. Ec. Norm. Super., IV. Ser.9, 309-377 (1976)
[22] Matsumura, H.: Commutative algebra. New York: Benjamin 1970 · Zbl 0211.06501
[23] Quillen, D.: AlgebraicK-theory, I. Lect. Notes in Math., vol. 341, pp. 85-147. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0292.18004
[24] Roberts, P.: On the vanishing of intersection multiplicities of perfect complexes. Bull. Am. Math. Soc., New Ser.13, 127-130 (1985) · Zbl 0585.13004
[25] Serre, J-P.: Alg?bre locale, multiplicit?s. Lect. Notes in Math., vol. 11 (3?me ?d. 1975). Berlin-Heidelberg-New York: Springer 1975
[26] Serre, J-P.: Groupes de Grothendieck des sch?mas en groupes d?ploy?s. Publ. Math., Inst. Hautes Etud. Sci.34, 37-52 (1968) · Zbl 0195.50802
[27] Soul?, C.: Op?rations enK-th?orie alg?brique. Can. J. Math.37 (n0 3) 488-550 (1985) · Zbl 0575.14015
[28] Szpiro, L.: Sur la th?orie des complexes parfaits. In: Sharp, R.Y. (ed.), Commutative Algebra: Durham 1981. London Math. Soc. Lecture Note Series, vol. 72, pp. 83-90. Cambridge University Press 1982
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.