Skip to main content

Cyclic homology, Serre's local factors and λ-operations

  • Alain Connes (a1) and Caterina Consani (a2)

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring , provides the right theory to obtain, using λ-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

Hide All
1.Beilinson A., Higher regulators and values of L-functions. (Russian) Current problems in mathematics 24, 181238. Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
2.Bloch S., Kato K., L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift I, 333400, Progr. Math. 86, Birkhauser Boston, Boston, MA, 1990.
3.Connes A., Spectral sequence and homology of currents for operator algebras, Mathematisches Forschunginstitut oberwolfach, Tagungsbericht 42/81.
4.Connes A., Cohomologie cyclique etfoncteurs Extn. C. R. Acad. Sci. Paris Sér. I Math. 296(23) (1983), 953958.
5.Connes A., Noncommutative differential geometry. Inst.Hautes Études Sci. Publ.Math. No. 62 (1985), 257360.
6.Connes A., Noncommutative geometry, Academic Press (1994).
7.Consani C., Double complexes and Euler L-factors. Compositio Math. 111(3) (1998), 323358.
8.Deligne P., Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math. 33 (1979) part II, 313346.
9.Demazure M., Gabriel P., Groupes algébriques, Tome I. (French) Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970.
10.Deninger C., On the Γ-factors attached to motives, Invent. Math. 104 (1991), 245261.
11.Deninger C., Motivic L-functions and regularized determinants. Motives (Seattle, WA, 1991), 707743, Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
12.Feigin B. and Tsygan B., Additive K-theory, Lecture notes in Math. 1289, Springer-Verlag, 1987, 97209.
13.Gillman L., Jerison M., Rings of continuous functions. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York 1960.
14.Grothendieck A., Produits tensoriels topologiques et espaces nuclaires. Mem. Am. Math. Soc. 16 (1955), 140 pp.
15.Grothendieck A., On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95103.
16.Görtz U., Wedhorn T., Algebraic Geometry I, Vieweg + Teubner (2010).
17.Hartshorne R., Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York Heidelberg Berlin 1977.
18.Hood C., Jones J., Some properties of cyclic homology groups, K-Theory 1 (1987), 361384.
19.Karoubi M., Homologie cyclique et K-théorie. (French) [Cyclic homology and K-theory] Astérisque 149 (1987), 147 pp.
20.Kontsevich M., Solomon Lefschetz Memorial Lecture series: Hodge structures in non-commutative geometry. Notes by Ernesto Lupercio. Contemp. Math. 462, Noncommutative geometry in mathematics and physics 121, Amer. Math. Soc., Providence, RI, 2008.
21.Liu Q., Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics 6. Oxford Science Publications. Oxford University Press, Oxford, 2002.
22.Loday J.L., Quillen D, Homologie cyclique et homologie de l'algèbre de Lie des matrices. (French) [Cyclic homology and homology of the Lie algebra of matrices] C. R. Acad. Sci. Paris Sr. I Math. 296(6) (1983), 295297.
23.Loday J.L., Cyclic homology. Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin, 1998.
24.Manin Yu. I., Lectures on zeta functions and motives (according to Deninger and Kurokawa). Columbia University Number Theory Seminar (New York, 1992). Astérisque 228(4) (1995), 121-163.
25.Oesterlé J., Nombres de Tamagawa et groupes unipotents. Invent. Math. 78 (1984), 1388.
26.Rinehart G., Differential forms on general commutative algebras. Trans. AMS 108 (1963), 195222.
27.Ray D.B., Singer I.M., Analytic torsion for complex manifolds. Ann. Math. 98 (1973), 154177.
28.Schneider P., Introduction to the Beilinson conjectures. Beilinson's conjectures on special values of L-functions. 135, Perspect. Math. 4, Academic Press, Boston, MA, 1988.
29.Serre J. P., Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures). Sém. Delange-Pisot-Poitou, exp. 19, 1969/1970.
30.Serre J. P., Géométrie algébrique et géométrie analytique. (French) Ann. Inst. Fourier, Grenoble 6 (19551956), 142.
31.Tsygan B. L., Homology of matrix Lie algebras over rings and the Hochschild homology. (Russian) Uspekhi Mat. Nauk 38 (1983), no. 2(230), 217218.
32.Weibel C., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge University Press, Cambridge, 1994. xiv + 450 pp.
33.Weibel C., Cyclic Homology for schemes. Proc. Amer. Math. Soc. 124(6) (1996), 16551662.
34.Weibel C., The Hodge filtration and cyclic homology. K-Theory 12(2) (1997), 145164.
35.Weibel C., Geller S., Etale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991), 368388.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 13 *
Loading metrics...

Abstract views

Total abstract views: 100 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th January 2018. This data will be updated every 24 hours.