K-theory of cones of smooth varieties.

GUILLERMO CORTIÑAS; CHRISTIAN HAESEMEYER; MARK E. WALKER; CHARLES A. WEIBEL

JOURNAL OF ALGEBRAIC GEOMETRY

UNIV PRESS INC

Año: 2013 vol. 22 p. 13 - 34

1056-3911

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 termsof the geometry of the projective embedding of $X$.In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, andprove that $K_{-1}(R)=oplus H^1(C,cO(n))$.The formula for $K_0(R)$ involves the Zariski cohomology of twistedK"ahler differentials on the variety.