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Algebraic K-Theory of Rings

Quillen により , algebraic K -theory するためには , exact category すればよい

なのは , algebraic K -theory であるが , Carlsson Goldfarb [ CG16 ] かれているように , R から exact category 方法 はいくつか ある

  • finitely generated projective R -module exact category
  • R Noetherian のとき , finitely generate R -module exact category
  • Carlsson Goldfarb [ CG16 ] による infinite group group ring する

もちろん , これらに Quillen てできた spectrum , には , るものである のものが K * ( R ) かれ , R algebraic K -theory ばこれを

なのは , algebraic K -theory であるが , これについては , Weibel [ Wei13 ] 最後 section ていることがまとめられている そこに いてい ないことで , chromatic homotopy theory から なこととして , Mitchell [ Mit90 ] がある

  • n 2 K ( ) K ( n )-homology である つまり , K ( ) K (1)-local である

について 調 べた , についたものを 以下 げる :

  • algebraic closure of finite field (Quillen [ Qui72 ] )
  • algebraically closed field local field など (Suslin [ Sus83 Sus84 ] )
  • Azumaya algebra (Hazrat and Millar [ HM10 ] )
  • truncated polynomial algebra (Angeltveit Gerhardt Hill Lindenstrauss [ AGHL14 ] )
  • planar cuspidal curve k [ x,y ] ( x b - y a ) (Hesselholt [ Hes14 ] )

, , Farrell-Jones などとの , について 調 られている

References

[AFR00]     C. S. Aravinda, F. T. Farrell, and S. K. Roushon. Algebraic K -theory of pure braid groups. Asian J. Math. , 4(2):337–343, 2000, http://dx.doi.org/10.4310/AJM.2000.v4.n2.a4 .

[AGHL14]     Vigleik Angeltveit, Teena Gerhardt, Michael A. Hill, and Ayelet Lindenstrauss. On the algebraic K -theory of truncated polynomial algebras in several variables. J. K-Theory , 13(1):57–81, 2014, arXiv:1206.0247 .

[BFJPP00]     E. Berkove, F. T. Farrell, D. Juan-Pineda, and K. Pearson. The Farrell-Jones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic K -theory of Bianchi groups. Trans. Amer. Math. Soc. , 352(12):5689–5702, 2000, http://dx.doi.org/10.1090/S0002-9947-00-02529-0 .

[BJPL04]     Ethan Berkove, Daniel Juan-Pineda, and Qin Lu. Algebraic K -theory of mapping class groups. K -Theory , 32(1):83–100, 2004, arXiv:math/0305425 .

[BJPP01]     Ethan Berkove, Daniel Juan-Pineda, and Kimberly Pearson. The lower algebraic K -theory of Fuchsian groups. Comment. Math. Helv. , 76(2):339–352, 2001, http://dx.doi.org/10.1007/PL00000382 .

[BJPP02]     E. Berkove, D. Juan-Pineda, and K. Pearson. A geometric approach to the lower algebraic K -theory of Fuchsian groups. Topology Appl. , 119(3):269–277, 2002, http://dx.doi.org/10.1016/S0166-8641(01)00068-2 .

[CG16]     Gunnar Carlsson and Boris Goldfarb. On modules over infinite group rings. Internat. J. Algebra Comput. , 26(3):451–466, 2016, arXiv:1509.02402 .

[FR00]     F. T. Farrell and Sayed K. Roushon. The Whitehead groups of braid groups vanish. Internat. Math. Res. Notices , (10):515–526, 2000, http://dx.doi.org/10.1155/S1073792800000283 .

[GJP15]     John Guaschi and Daniel Juan-Pineda. A survey of surface braid groups and the lower algebraic K -theory of their group rings. In Handbook of group actions. Vol. II , volume 32 of Adv. Lect. Math. (ALM) , pages 23–75. Int. Press, Somerville, MA, 2015, arXiv:1302.6536 .

[GJPML]     John Guaschi, Daniel Juan-Pineda, and Silvia Millán-López. The lower algebraic K -theory of the braid groups of the sphere, arXiv:1209.4791 .

[Hes14]     Lars Hesselholt. On the K -theory of planar cuspical curves and a new family of polytopes. In Algebraic topology: applications and new directions , volume 620 of Contemp. Math. , pages 145–182. Amer. Math. Soc., Providence, RI, 2014, arXiv:1303.6060 .

[HM10]     Roozbeh Hazrat and Judith R. Millar. On graded simple algebras. J. Homotopy Relat. Struct. , 5(1):113–124, 2010, arXiv:1003.4538 .

[Mit90]     S. A. Mitchell. The Morava K -theory of algebraic K -theory spectra. K -Theory , 3(6):607–626, 1990, http://dx.doi.org/10.1007/BF01054453 .

[Qui72]     Daniel Quillen. On the cohomology and K -theory of the general linear groups over a finite field. Ann. of Math. (2) , 96:552–586, 1972, http://dx.doi.org/10.2307/1970825 .

[Sus83]     A. Suslin. On the K -theory of algebraically closed fields. Invent. Math. , 73(2):241–245, 1983, http://dx.doi.org/10.1007/BF01394024 .

[Sus84]     Andrei A. Suslin. On the K -theory of local fields. In Proceedings of the Luminy conference on algebraic K -theory (Luminy, 1983) , volume 34, pages 301–318, 1984, http://dx.doi.org/10.1016/0022-4049(84)90043-4 .

[Wei13]     Charles A. Weibel. The K -book , volume 145 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2013. An introduction to algebraic K -theory.