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Register a new user$K$-theory of cones of smooth varieties
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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 Kahler differentials on the variety.
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