G₀ of affine, simplicial toric varieties

Let X be an affine, simplicial toric variety over a field. Let G₀ denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let F¹G₀ denote the first step of the filtration on G₀ by codimension of support. Then G₀(X)⊕ F¹G₀(X) and F¹G₀(X) is a finite abelian group. In dimension 2, we show that F¹G₀(X) is a finite cyclic group and determine its order. In dimension 3, F¹G₀(X) is determined up to a group extension of the Chow group A¹(X) by the Chow group A²(X). We determine the order of the Chow group A¹(X) in this case. A conjecture on the orders of A¹(X) and A²(X) is formulated for all dimensions.