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Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 2, Pages 227–261 (Mi izv2107)  

This article is cited in 23 scientific papers (total in 24 papers)

The Tannaka–Artin problem and reduced $K$-theory

V. P. Platonov


Abstract: We solve the old Tannaka–Artin problem, which states the following in modern terminology: is the reduced Whitehead group $SK_1(A)$ of a finite-dimensional division algebra $A$ trivial?
We work out a method for computing the group $SK_1(A)$ based on a reduction to the computation of a group of special protective conorms – a new object in field theory – and we discover unexpected connections with number theory.
Bibliography: 23 titles.

Full text: PDF file (3534 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:2, 211–243

Bibliographic databases:

UDC: 513.6
MSC: Primary 16A40, 12A65, 12A80; Secondary 16A18, 16A54, 12A60, 12F25
Received: 27.06.1975

Citation: V. P. Platonov, “The Tannaka–Artin problem and reduced $K$-theory”, Izv. Akad. Nauk SSSR Ser. Mat., 40:2 (1976), 227–261; Math. USSR-Izv., 10:2 (1976), 211–243

Citation in format AMSBIB
\Bibitem{Pla76}
\by V.~P.~Platonov
\paper The Tannaka--Artin problem and reduced $K$-theory
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1976
\vol 40
\issue 2
\pages 227--261
\mathnet{http://mi.mathnet.ru/izv2107}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=407082}
\zmath{https://zbmath.org/?q=an:0338.16005}
\transl
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 2
\pages 211--243
\crossref{https://doi.org/10.1070/IM1976v010n02ABEH001686}


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    Remarks

    This publication is cited in the following articles:
    1. V. P. Platonov, “The infinitude of the reduced Whitehead group in the Tannaka–Artin problem”, Math. USSR-Sb., 29:2 (1976), 167–176  mathnet  crossref  mathscinet  zmath  isi
    2. V. I. Yanchevskii, “Reduced unitary $K$-theory and division rings over discretely valued Hensel fields”, Math. USSR-Izv., 13:1 (1979), 175–213  mathnet  crossref  mathscinet  zmath  isi
    3. V. I. Yanchevskii, “Reduced unitary $K$-theory. Aplications to algebraic groups”, Math. USSR-Sb., 38:4 (1981), 533–548  mathnet  crossref  mathscinet  zmath  isi
    4. A. E. Zalesskii, “Linear groups”, Russian Math. Surveys, 36:5 (1981), 63–128  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. A. S. Merkur'ev, A. A. Suslin, “Cohomology of Severi–Brauer varieties and the norm residue homomorphism”, Math. USSR-Izv., 21:2 (1983), 307–340  mathnet  crossref  mathscinet  zmath
    6. V. P. Platonov, “The arithmetic theory of algebraic groups”, Russian Math. Surveys, 37:3 (1982), 1–62  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. Yu. L. Ershov, “Henselian valuations of division rings and the group $SK_1$”, Math. USSR-Sb., 45:1 (1983), 63–71  mathnet  crossref  mathscinet  zmath
    8. V. P. Platonov, V. I. Yanchevskii, “Dieudonné's conjecture on the structure of unitary groups over a division ring, and Hermitian $K$-theory”, Math. USSR-Izv., 25:3 (1985), 573–599  mathnet  crossref  mathscinet  zmath
    9. A. S. Merkur'ev, “The group $SK_2$ for quaternion algebras”, Math. USSR-Izv., 32:2 (1989), 313–337  mathnet  crossref  mathscinet  zmath
    10. Pat Morandi, “The Henselization of a valued division algebra”, Journal of Algebra, 122:1 (1989), 232  crossref
    11. A. P. Monastyrnyi, V. I. Yanchevskii, “Whitehead groups of spinor groups”, Math. USSR-Izv., 36:1 (1991), 61–100  mathnet  crossref  mathscinet  zmath  adsnasa
    12. Bill Jacob, Adrian Wadsworth, “Division algebras over Henselian fields”, Journal of Algebra, 128:1 (1990), 126  crossref
    13. S. I. Adian, E. I. Zel'manov, G. A. Margulis, S. P. Novikov, A. S. Rapinchuk, L. D. Faddeev, V. I. Yanchevskii, “Vladimir Petrovich Platonov (on his 60th birthday)”, Russian Math. Surveys, 55:3 (2000), 601–610  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    14. R Hazrat, “SK1-like Functors for Division Algebras”, Journal of Algebra, 239:2 (2001), 573  crossref
    15. Jürgen Ritter, Alfred Weiss, “Toward equivariant Iwasawa theory, II”, Indagationes Mathematicae, 15:4 (2004), 549  crossref
    16. R. Hazrat, “Reduced K-theory of Azumaya algebras”, Journal of Algebra, 305:2 (2006), 687  crossref
    17. Roozbeh Hazrat, “SK1 of Azumaya algebras over Hensel Pairs”, Math Z, 2008  crossref  zmath  isi
    18. Tim Wouters, “L'invariant de Suslin en caractéristique positive”, J K-Theory, 2010, 1  crossref
    19. V. I. Yanchevskiǐ, “Homogeneous skew-fields of non-commutative rational functions and their reduced Whitehead groups”, J. Math. Sci. (N. Y.), 183:5 (2012), 727–747  mathnet  crossref
    20. Roozbeh Hazrat, “Stability of unitary SK1 of Azumaya algebras”, Arch. Math, 2011  crossref
    21. Tim Wouters, “Comparing Invariants of SK1”, Algebr Represent Theor, 2011  crossref
    22. A. R. Wadsworth, “Unitary SK1 of semiramified graded and valued division algebras”, manuscripta math, 139:3-4 (2012), 343  crossref
    23. V. I. Yanchevskii, “Reduced Whitehead groups and conjugacy problem for special unitary groups of anisotropic hermitian forms”, J. Math. Sci. (N. Y.), 192:2 (2013), 250–262  mathnet  crossref  mathscinet
    24. N. M. Glazunov, “Quadratic forms, algebraic groups and number theory”, Chebyshevskii sb., 16:4 (2015), 77–89  mathnet  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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