The higher \(K\)-theory of real curves
Let \(X\) be a smooth real curve, i.e., a smooth one-dimensional reduced quasi-projective scheme over \(\mathbb R\), which is not defined over \(\mathbb C\). It is a classical fact that \(K_{0}(X)\) is closely related to topological invariants of the spaces \(X(\mathbb R)\) and \(X(\mathbb C)\) of real and complex points of \(X\). In this paper, the authors relate the higher \(K\)-groups of \(X\) to such topological invariants. They show that for \(n\neq 0\) the group \(K_n(X)\) is the direct sum of a divisible group and an elementary abelian \(2\)-group. The torsion subgroup of \(K_n(X)\) is a sum of copies of \(\mathbb Q/\mathbb Z\) and of \(\mathbb Z_2\), with the number of summands explicitly described in terms of topological invariants of \(X(\mathbb R)\) and \(X(\mathbb C)\). The results are slightly different depending on whether \(X\) is projective of affine as well as whether \(X(\mathbb R)\) is empty or non-empty. In particular, the torsion subgroups are periodic with period \(8\) if \(X(\mathbb R)\neq\emptyset\) and with period \(4\) is \(X(\mathbb R)=\emptyset\). One important ingredient of the proof is a transfer argument based on Comessatti's theorem. On the other hand, the authors use a spectral sequence converging to \(K\)-theory, whose \(E_2\) term is given by motivic cohomology of \(X\). The relevant motivic cohomology groups are computed using results of Suslin and Voevodsky.
scientific article; zbMATH DE number 1892282