Advertisement

Selecta Mathematica

, Volume 23, Issue 2, pp 1235–1247 | Cite as

Analogues of Gersten’s conjecture for singular schemes

  • Amalendu Krishna
  • Matthew Morrow
Article

Abstract

We formulate analogues, for Noetherian local \({\mathbb {Q}}\)-algebras which are not necessarily regular, of the injectivity part of Gersten’s conjecture in algebraic K-theory and prove them in various cases. Our results suggest that the algebraic K-theory of such a ring should be detected by combining the algebraic K-theory of both its regular locus and the infinitesimal thickenings of its singular locus.

Keywords

Gersten’s conjecture Algebraic K-theory Singular schemes 

Mathematics Subject Classification

Primary 19E08 Secondary 14B05 

Notes

Acknowledgments

The second author would like to thank the Tata Institute of Fundamental Research for its hospitality during a visit in November 2012. The authors would like to thank the anonymous referee for carefully reading the paper and suggesting many improvements.

References

  1. 1.
    Cortiñas, G., Guccione, J., Guccione, J.: Decomposition of the Hochschild and cyclic homology of commutative differential graded algebras. J. Pure Appl. Algebra 83(3), 219–235 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Cortiñas, G., Haesemeyer, C., Schlichting, M., Weibel, C.: Cyclic homology, cdh-descent and negative \(K\). Ann. Math. 167, 549–573 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hartshorne, R.: Algebraic geometry. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Math, vol. 52. Springer, New York (1977)Google Scholar
  4. 4.
    Feĭgin, B.L., Tsygan, B.L.: Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen 19(2), 52–62, 96 (1985) (Russian) Google Scholar
  5. 5.
    Kassel, A., Sletsjøe, A.: Base change, transitivity and Künenth formulas for Quillen decomposition of Hochschild homology. Math. Scand. 70(2), 186–192 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Krishna, A.: On \(K_{2}\) of one-dimensional local rings. K-Theory 35(1–2), 139–158 (2005)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Loday, J.-L.: Cyclic homology. In: Berger, M., Eckmann, B., Varadhan, S.R.S. (eds.) Grundlehren der Mathematischen Wissenschaften, vol. 301. Springer, Berlin (1992). Appendix E by María O. RoncoGoogle Scholar
  8. 8.
    Morrow, M.: Pro cdh-descent for cyclic homology and \(K\)-theory. J. Inst. Math. Jussieu 15(3), 539–567 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Morrow, M.: Pro unitality and pro excision in algebraic \(K\)-theory and cyclic homology. J. Reine Angew. Math. (2015)Google Scholar
  10. 10.
    Morrow, M.: \(K\)-theory of one-dimensional rings via pro-excision. J. Inst. Math. Jussieu 13(2), 225–272 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Morrow, M.: A singular analogue of Gersten’s conjecture and applications to \(K\)-theoretic adèles. Comm. Algebra 43(11), 4951–4983 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Nesterenko, Y., Suslin, A.: Homology of general linear group over a local ring and Milnor \(K\)-theory. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 121–146 (1989)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Soulé, C.: Opérations en \(K\)-théorie algébrique. Canad. J. Math. 37(3), 488–550 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Temkin, M.: Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. 219(2), 488–522 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Weibel, C.: Cyclic homology for schemes. Proc. AMS 124(6), 1655–1662 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Weibel, C.: An introduction to homological algebra. In: Garling, D.J.H., tom Dieck, T., Walters, P. (eds.) Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.MumbaiIndia
  2. 2.BonnGermany

Personalised recommendations