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Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 4, Pages 736–773 (Mi izv2201)  

This article is cited in 43 scientific papers (total in 44 papers)

On the arithmetic of two-dimensional schemes. I. Distributions and residues

A. N. Parshin

Abstract: This paper is devoted to the generalization of the theory of distributions (adeles) on schemes of dimension 2.
Bibliography: 11 titles.

Full text: PDF file (3279 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:4, 695–729

Bibliographic databases:

UDC: 513.6
MSC: Primary 14G99, 14F05; Secondary 14F99
Received: 13.01.1976

Citation: A. N. Parshin, “On the arithmetic of two-dimensional schemes. I. Distributions and residues”, Izv. Akad. Nauk SSSR Ser. Mat., 40:4 (1976), 736–773; Math. USSR-Izv., 10:4 (1976), 695–729

Citation in format AMSBIB
\by A.~N.~Parshin
\paper On the arithmetic of two-dimensional schemes. I.~Distributions and residues
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1976
\vol 40
\issue 4
\pages 736--773
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 4
\pages 695--729

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    This publication is cited in the following articles:
    1. V. G. Lomadze, “On the ramification theory of two-dimensional local fields”, Math. USSR-Sb., 37:3 (1980), 349–365  mathnet  crossref  mathscinet  zmath  isi
    2. A. A. Beilinson, “Residues and adeles”, Funct. Anal. Appl., 14:1 (1980), 34–35  mathnet  crossref  mathscinet  zmath
    3. V. G. Lomadze, “On residues in algebraic geometry”, Math. USSR-Izv., 19:3 (1982), 495–520  mathnet  crossref  mathscinet  zmath
    4. S. V. Vostokov, “Explicit construction of class field theory for a multidimensional local field”, Math. USSR-Izv., 26:2 (1986), 263–287  mathnet  crossref  mathscinet  zmath
    5. Zhongming Tang, “An algebraic approach to the residues in algebraic geometry”, Math Z, 203:1 (1990), 1  crossref  mathscinet  zmath  isi
    6. A. Huber, “On the Parshin-Beilinson Adeles for schemes”, Abh Math Sem Hamburg, 61:1 (1991), 249  crossref
    7. Reinhold Hübl, “Residues of regular and meromorphic differential forms”, Math Ann, 300:1 (1994), 605  crossref  mathscinet  zmath  isi
    8. Fesenko I., “Abelian Local P-Class Field-Theory”, Math. Ann., 301:3 (1995), 561–586  crossref  mathscinet  zmath  isi
    9. J. -L. Brylinski, D. A. Mclaughlin, “The geometry of two-dimensional symbols”, K-Theory, 10:3 (1996), 215  crossref  mathscinet  zmath  isi  elib
    10. Robert A. Bell, “On properties of the n-dimensional norm residue symbol in higher local class field theory”, Journal of Pure and Applied Algebra, 108:1 (1996), 1  crossref
    11. Kazuya MATSUMI, “Arithmetic of three-dimensional complete regular local rings of positive characteristics”, Tohoku Math Publ, 13:13 (1999), 1  crossref  mathscinet
    12. A. B. Zheglov, “On the classification of two-dimensional local skew fields”, Russian Math. Surveys, 54:4 (1999), 858–859  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    13. I-Chiau Huang, “An Explicit Construction of Residual Complexes”, Journal of Algebra, 225:2 (2000), 698  crossref
    14. A. N. Parshin, “The Krichever Correspondence for Algebraic Surfaces”, Funct. Anal. Appl., 35:1 (2001), 74–76  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. A. B. Zheglov, “On the structure of two-dimensional local skew fields”, Izv. Math., 65:1 (2001), 23–55  mathnet  crossref  crossref  mathscinet  zmath
    16. D. V. Osipov, “Krichever correspondence for algebraic varieties”, Izv. Math., 65:5 (2001), 941–975  mathnet  crossref  crossref  mathscinet  zmath
    17. Parshin A.N., “Integrable systems and local fields”, Communications in Algebra, 29:9 (2001), 4157–4181  crossref  isi  elib
    18. Proc. Steklov Inst. Math., 241 (2003), 98–109  mathnet  mathscinet  zmath
    19. D. V. Osipov, “Central extensions and reciprocity laws on algebraic surfaces”, Sb. Math., 196:10 (2005), 1503–1527  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. A. B. Zheglov, D. V. Osipov, “On Some Questions Related to the Krichever Correspondence”, Math. Notes, 81:4 (2007), 467–476  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    21. S. O. Gorchinskiy, “Adele resolvent for bundles of $K$-groups”, Russian Math. Surveys, 62:1 (2007), 184–186  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. D. V. Osipov, A. N. Parshin, “Harmonic analysis on local fields and adelic spaces. I”, Izv. Math., 72:5 (2008), 915–976  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. S. O. Gorchinskiy, “An adelic resolution for homology sheaves”, Izv. Math., 72:6 (2008), 1187–1252  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    24. I. B. Fesenko, “Adelic approach to the zeta function of arithmetic schemes in dimension two”, Mosc. Math. J., 8:2 (2008), 273–317  mathnet  crossref  mathscinet  zmath
    25. Gorchinskiy S., “Notes on the Biextension of Chow Groups”, Motives and Algebraic Cycles: a Celebration in Honour of Spencer J. Bloch, Fields Institute Communications, 56, 2009, 111–148  isi
    26. R. Ya. Budylin, “An adelic construction of Chern classes”, Sb. Math., 202:11 (2011), 1637–1659  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    27. S. V. Vostokov, S. O. Gorchinskiy, A. B. Zheglov, Yu. G. Zarkhin, Yu. V. Nesterenko, D. O. Orlov, D. V. Osipov, V. L. Popov, A. G. Sergeev, I. R. Shafarevich, “Aleksei Nikolaevich Parshin (on his 70th birthday)”, Russian Math. Surveys, 68:1 (2013), 189–197  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    28. S. V. Vostokov, “Shafarevich's paper “A general reciprocity law””, Sb. Math., 204:6 (2013), 781–800  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    29. R. Ya. Budylin, S. O. Gorchinskiy, “Intersections of adelic groups on a surface”, Sb. Math., 204:12 (2013), 1701–1711  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    30. Braunling O., “Geometric Two-Dimensional Ideles With Cycle Module Coefficients”, Math. Nachr., 287:17-18 (2014), 1954–1971  crossref  isi
    31. Camara A., “Locally Convex Structures on Higher Local Fields”, J. Number Theory, 143 (2014), 185–213  crossref  isi
    32. Ivan Fesenko, “Geometric adeles and the Riemann–Roch theorem for $1$-cycles on surfaces”, Mosc. Math. J., 15:3 (2015), 435–453  mathnet  crossref  mathscinet
    33. Gorchinskiy S., Rosly A., “a Polar Complex For Locally Free Sheaves”, Int. Math. Res. Notices, 2015, no. 10, 2784–2829  crossref  isi
    34. St. Petersburg Math. J., 27:6 (2016), 1003–1028  mathnet  crossref  mathscinet  isi  elib
    35. Sergey O. Gorchinskiy, Denis V. Osipov, “Continuous homomorphisms between algebras of iterated Laurent series over a ring”, Proc. Steklov Inst. Math., 294 (2016), 47–66  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    36. Oliver Braunling, Michael Groechenig, Jesse Wolfson, “Tate objects in exact categories (with an appendix by Jan Šťovíček and Jan Trlifaj)”, Mosc. Math. J., 16:3 (2016), 433–504  mathnet  crossref  mathscinet
    37. Braunling O., Groechenig M., Wolfson J., “Operator ideals in Tate objects”, Math. Res. Lett., 23:6 (2016), 1565–1631  crossref  mathscinet  isi
    38. Liu D., “Residues and duality on semi-local two-dimensional adeles”, J. Algebra, 448 (2016), 74–83  crossref  mathscinet  zmath  isi  elib  scopus
    39. Liu D. Zhu Y., “Lca(2), Weil Index, and Product Formula”, Adv. Math., 329 (2018), 1088–1136  crossref  isi
    40. D. V. Osipov, “Adelic quotient group for algebraic surfaces”, St. Petersburg Math. J., 30:1 (2019), 111–122  mathnet  crossref  mathscinet  isi  elib
    41. D. V. Osipov, “Arithmetic surfaces and adelic quotient groups”, Izv. Math., 82:4 (2018), 817–836  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    42. A. N. Parshin, “The Mellin Transform and the Plancherel Theorem for the Discrete Heisenberg Group”, Proc. Steklov Inst. Math., 307 (2019), 174–192  mathnet  crossref  crossref  mathscinet  isi  elib
    43. S. O. Gorchinskiy, D. V. Osipov, “Iterated Laurent series over rings and the Contou-Carrère symbol”, Russian Math. Surveys, 75:6 (2020), 995–1066  mathnet  crossref  crossref  mathscinet  isi  elib
    44. A. I. Madunts, “Mnozhestva skhodimosti mnogomernogo polnogo polya”, Voprosy teorii predstavlenii algebr i grupp. 35, Zap. nauchn. sem. POMI, 492, POMI, SPb., 2020, 125–133  mathnet
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