divisors in non-compact toric varieties

Question

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.

Answer 1

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.