$K$-theory of cones of smooth varieties

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 K"ahler differentials on the variety.

💡 Research Summary

The paper investigates the algebraic K‑theory of the homogeneous coordinate ring R of a smooth projective variety X over a field k of characteristic zero. The authors develop a comprehensive method to compute the low‑degree K‑groups (K₀, K₁, and negative K‑groups) and to describe higher K‑theory in terms of geometric invariants of the embedding X⊂ℙⁿₖ.

The central observation is that the affine cone Spec R is A¹‑contractible, which allows the reduced K‑theory eKₙ(R)=Kₙ(R)/Kₙ(k) to be expressed via cdh‑cohomology and the homotopy fibre of cyclic homology. By exploiting the standard “A¹‑trick” for graded algebras, the authors show that eKₙ(R) can be identified with the homotopy groups of the fibre FHC(R) of the map HC(R)→HC_cdh(R). This yields a long exact sequence linking cyclic homology, cdh‑cohomology of differential forms, and the reduced K‑groups.

A key technical tool is the decomposition of K‑theory under Adams operations ψᵏ. The authors prove that K₀(R) splits as \