\(K\)-theory of cones of smooth varieties. (English) Zbl 1267.14013
Let \(X\) be a smooth projective variety \(X\) over a field \(k\) of characteristic zero and \(R\) its homogeneous coordinate ring. The authors compute the lower \(K\)-theory: \(K_{i}(R), i\leq 1\) in terms of the Zariski cohomology groups \(H^{*}(X, {\mathcal O}(t))\) and \(H^{*}(X, {\Omega}^{*}_{X}(t))\). Here \({\mathcal O}(1)\) is the ample line bundle of the embedding and \({\Omega}_{X}^{*}\) denotes the Kähler differentials of \(X\) relative to \(\mathbb Q\). For \(X\) equal to a curve \(C\) the authors calculate \(K_{0}(R)\), \(K_{1}(R)\) and show that \(K_{-1}(R)=\bigoplus H^{1}(C, {\mathcal O}(n))\). As far as higher \(K\)-theory is concerned, the authors compute \(K_{n}(R)/K_{n}(k)\) for the conic \(xy=z^{2}\).
Reviewer: Piotr Krasoń (Szczecin)
MSC:
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 19E20 | Relations of \(K\)-theory with cohomology theories |
| 19E08 | \(K\)-theory of schemes |
| 19D50 | Computations of higher \(K\)-theory of rings |
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