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arXiv · 0905.4642

$K$-theory of cones of smooth varieties

Guillermo Cortiñas·Christian Haesemeyer·Mark E. Walker·Charles A. Weibel
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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 $K_{-1}(R)=\oplus H^1(C,\cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kähler differentials on the variety.

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Guillermo Cortiñas, Christian Haesemeyer, Mark E. Walker, Charles A. Weibel. 2010-02-22. $K$-theory of cones of smooth varieties. https://arxiv.org/abs/0905.4642

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