Let $X$ be a smooth quasi-projective toric variety of dimension $n$ over $\mathbb C$. Take it to be non-compact, so its fan is not complete. (A good example to keep in mind is a toric Calabi-Yau.)

If we denote by $D_i$ for $i=1,\ldots,d$ the irreducible subvarieties of codimension one that are stable under the torus action (corresponding to edges of the fan), what can be said about the kernel $K$ in the exact sequence $$ 0 \to K \to \oplus_{i=1}^d \mathbb Z D_i \to A_{n-1}(X) \to 0 $$ in this case?

I know that if $D_i$ is also compact, then its Poincare dual $PD[D_i]$ defines a class in compact support cohomology $H^2_c (X)$, but I'm confused how this is related with usual $H^2(X)$, hence my question above.

iscontained in a hyperplane, right? if so, it is meaningful to ask for a characterization of the kernel, or more generally how to best deal with divisors in the non-compact setting. $\endgroup$