Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism $$CDiv_{T_{N}}(X_{\Sigma}) \cong ker(\oplus_{i}M/M(\sigma_{i}) \to \oplus_{i<j}M/M(\sigma_{i} \cap \sigma_{j}))$$
(cf Exercise 4.2.4 Cox, Little, and Schenck).
where $M(\sigma) = \sigma^{\perp}\cap M.$ I am having trouble understanding what exactly the kernel is here where the mapping is defined via $$(m_{i})_{i} \to (m_{i} - m_{j})_{i<j}$$
On that note, is there a more intuitive way to think of divisors in general? The only way I understand them so far is that prime divisors are codimension 1 irreducible subvarieties that give rise to discrete valuation rings and we can talk about orders of vanishing along divisors. Is there a more geometric way to think of them?
Thanks