OO D
For a Weil divisor $D$ on a normal variety $X$ the associated sheaf ${\cal O}_X(D)$ is defined by $H^0(U, {\cal O}_X(D)) = \{ f \in {\mathbb C}(X)^* | (div(f)+D)|_U \geq 0 \} \cup \{0\}$. The sheaf associated to a Weil divisor is reflexive; it is equal to its bidual. A divisor is Cartier if and only if the associated sheaf is a line bundle
The first examples show that the associated sheaves are reflexive.
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Two Weil divisors $D$ and $E$ are linearly equivalent if $D = E + div(f)$, for some $f \in {\mathbb C}(X)^*$. Linearly equivalent divisors produce isomorphic sheaves.
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