I have a disjoint union of open sets $U_1,..., U_k$ on a variety $X$. In Fultons "Introduction to toric varieties", he used $X=X_\Sigma$ a toric variety and $U_i=\mathcal{O}(\sigma_i)$ orbits. He then used $$ A_k\left(\bigsqcup_i U_i \right) = \bigoplus_i A_k(U_i) $$ as a fact. Is this true in general, that the chow group of a (finite) disjoint union is the direct sum (for finite this equals to the product) of all chow groups?
I think the argument could be something like this: If we take a subvariety on $\sqcup_i U_i$, it must be a subvariety in each $U_i$ as they are distinct. But then we can look at the morphism $$ V \mapsto (V\cap U_1, ..., V\cap U_k) \\ V_1\cup ... \cup V_k\leftarrow\hspace{-5pt}\small| (V_1,...,V_k) $$ which should be an isomorphism. Is this true and why? I appreciate any suggestions.