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.



divisors in non-compact toric varieties
As Piotr mentioned in the comments, you don't need completeness of the fan to be sure that the sequence
$$0 \to M \to \oplus \mathbb Z D\_i \to Cl(X\_{\Sigma}) \to 0$$
is exact. The only condition that you actually need is that fan $\Sigma \subset N\_{\mathbb R}$ is not contained inside (real) linear subspace of codimension 1. See Cox Little Schenck Theorem 4.1.3.


I believe you wanted to say "since $X$ is **Calabi-Yau** (necessary non-compact) the fan *is* contained in hyperplane" (which is true). In this case you can always write
$$0 \to M/(\operatorname{ker}(M \to \bigoplus \mathbb Z D\_i)) \to \bigoplus \mathbb Z D\_i \to Cl(X\_{\Sigma}) \to 0$$
This sequence is exact for any toric variety $X\_{\Sigma}$ associated with fan $\Sigma$. And $\operatorname{ker}(M \to \bigoplus \mathbb Z D\_i)$ is exactly the covectors which are ortohonal to the fan generators, so you can write
$$0 \to M/L^{\bot} \to \bigoplus \mathbb Z D\_i \to Cl(X\_{\Sigma}) \to 0$$
where $L = \operatorname{span}\_{\mathbb Z} \Sigma(1)^{gen} \subset N$ is $\mathbb Z$-submodule inside $N$ generated by ray generators of fan $\Sigma$ and $L^{\bot} = \{\phi \in M | \forall l \in L : \phi(l) = 0\}$ the annihilator.


