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VANISHING OF THE NEGATIVE HOMOTOPY $K$ -THEORY OF QUOTIENT SINGULARITIES

Abstract

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy $K$ -theory groups of a Noetherian scheme $X$ of Krull dimension $d$ vanish below $-d$ . In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy $K$ -theory groups vanish below $-1$ . Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy $K$ -theory group.

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O. Iyama and R. Takahashi , Tilting and cluster tilting for quotient singularities, Math. Ann. 356(3) (2013), 10651105.

B. Keller , D. Murfet and M. Van den Bergh , On two examples by Iyama and Yoshino, Compos. Math. 147(2) (2011), 591612.

V. Lunts and D. Orlov , Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853908.

C. Weibel , Mayer–Vietoris sequences and mod-p K-theory, Lecture Notes in Maths 966 (1983), 390407.

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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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