There was a conjecture that the natural homomorphism \(K^0(X)\to K_0(X)\) between the Grothendieck groups of vector bundles and coherent sheaves on a simplicial toric variety or even on a quasi-projective orbifold \(X\) becomes an isomorphism after tensoring with \(\mathbb Q\) [see M. Brion and M. Vergne, J. Reine Angew. Math. 482, 67–92 (1997; Zbl 0862.14006); D. A. Cox, in: Geometry of toric varieties, Sémin. Congr. 6, 1–41 (2002; Zbl 1050.14001)].
The paper gives a negative answer even for projective simplicial toric varieties of dimension \(\geq3\). More precisely, \(K^0(X)_{\mathbb Q}\to K_0(X)_{\mathbb Q}\) is a surjection for a simplicial toric variety \(X\) due to Brion and Vergne [loc. cit.]. If \(X\) is quasiprojective, then \(K_0(X)_{\mathbb Q}\simeq A_{*}(X)_{\mathbb Q}\) [see P. Baum, W. Fulton and R. MacPherson, Publ. Math., Inst. Hautes Étud. Sci. 45, 101–145 (1975; Zbl 0332.14003)], the latter group being finitely generated [W. Fulton, R. MacPherson, F. Sottile and B. Sturmfels, J. Algebr. Geom. 4, No. 1, 181–193 (1995; Zbl 0819.14019)].
However, the author constructs a series of examples where \(K^0(X)\) has infinite rank provided that the base field is an infinite extension of \(\mathbb Q\). The main building block is a so-called basic configuration, which is a triple of monoids in \(\mathbb Z^n\) satisfying certain saturation and separation properties. Each basic configuration determines a non-complete toric variety \(X\) covered by two affine charts. If the above monoids are simplicial and non-free, then \(X\) is simplicial, but singular, and \(K^0(X)\) has infinite rank. This is proved using the higher \(K\)-theory of semigroup algebras. If \(n\geq3\), then there exists a basic configuration of simplicial non-free monoids. Finally, it is possible to attach several smooth affine charts to \(X\) so that \(X\) becomes projective and still \(K^0(X)\) has infinite rank by Mayer-Vietoris.