Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



$ 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
Published electronically: December 27, 2011
MathSciNet review: 2993045
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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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G. Cortiñas
Affiliation: Departamento Matemática, FCEyN-UBA, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina

C. Haesemeyer
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095

M. E. Walker
Affiliation: Department of Mathematics, University of Nebraska–Lincoln, Lincoln, Nebraska 68588

C. Weibel
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901

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

American Mathematical Society