𝐾-theory of cones of smooth varieties

Cortiñas, G.; Haesemeyer, C.; Walker, M.; Weibel, C.

doi:10.1090/S1056-3911-2011-00583-3

$K$-theory of cones of smooth varieties

Authors: G. Cortiñas, C. Haesemeyer, M. E. Walker and C. Weibel
Journal: J. Algebraic Geom. 22 (2013), 13-34
DOI: https://doi.org/10.1090/S1056-3911-2011-00583-3
Published electronically: December 27, 2011
MathSciNet review: 2993045
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Abstract | References | Additional Information

Abstract: Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $k$ of characteristic 0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve, then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=\bigoplus H^1(C,\mathcal {O}(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kähler differentials on the variety.

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Additional Information

G. Cortiñas
Affiliation: Departamento Matemática, FCEyN-UBA, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina
MR Author ID: 18832
ORCID: 0000-0002-8103-1831
Email: gcorti@dm.uba.ar

C. Haesemeyer
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095
MR Author ID: 773007
Email: chh@math.ucla.edu

M. E. Walker
Affiliation: Department of Mathematics, University of Nebraska–Lincoln, Lincoln, Nebraska 68588
Email: mwalker5@math.unl.edu

C. Weibel
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901
MR Author ID: 181325
Email: weibel@math.rutgers.edu

Received by editor(s): March 2, 2010
Received by editor(s) in revised form: November 23, 2010
Published electronically: December 27, 2011
Additional Notes: The first author’s research was supported by CONICET and partially supported by grants PICT 2006-00836, UBACyT-X051, and MTM2007-64704.
The second and third authors were partially supported by NSF grants
The fourth author was supported by NSA and NSF grants