# ag.algebraic geometry – Integral isomorphism between \$K_0(X)\$ and \$A(X)\$ for toric varieties

Let $$X$$ be a smooth projective toric variety. The Chern character gives an isomorphism of rings:
$$operatorname{Ch}:K_{0}(X)otimesmathbb{Q} to A(X)otimes mathbb{Q}$$
where $$K_{0}(X)$$ is the Grothendieck group of vector bundles on $$X$$ and $$A(X)$$ is the Chow ring of $$X$$. This map seems only well-defined over $$mathbb{Q}$$, but I was wondering (likely naively) if there is possibly an integral isomorphism (i.e. without tensor with $$mathbb{Q}$$)?

Why might we hope for such a map? Fulton and Strumfels showed that there exists an isomorphism $$mathcal{D}_{X}:A^{k}(X)to operatorname{Hom}(A_{K}(X),mathbb{Z})$$ where $$A^{k}(X)$$ and $$A_{k}(X)$$ are the Chow cohomology and homology groups respectively. In particular, this means that the Chow ring $$A(X)$$ of a smooth toric variety is torsion free. In the couple very (very) simple examples I’ve done $$K_{0}(X)$$ also seems torsion free, although I am unsure whether this is true generally.

Of course, even if both $$K_{0}(X)$$ and $$A(X)$$ are torsion free there need not be an isomorphism between them, but one can hope.