A Weil divisor "D" is said to be Cartier if and only if the sheaf "O"("D") is invertible. When this happens, "O"("D") (with its embedding in "MX") is the line bundle associated to a Cartier divisor. More precisely, if "O"("D") is invertible, then there exists an open cover {"Ui"} such that "O"("D") restricts to a trivial bundle on each open set. For each "Ui", choose an isomorphism formula_66 The image of formula_67 under this map is a section of "O"("D") on "Ui". Because "O"("D") is defined to be a subsheaf of the sheaf of rational functions, the image of 1 may be identified with some rational function "fi". The collection formula_68 is then a Cartier divisor. This is well-defined because the only choices involved were of the covering and of the isomorphism, neither of which change the Cartier divisor. This Cartier divisor may be used to produce a sheaf, which for distinction we will notate "L"("D"). There is an isomorphism of "O"("D") with "L"("D") defined by working on the open cover {"Ui"}. The key fact to check here is that the transition functions of "O"("D") and "L"("D") are compatible, and this amounts to the fact that these functions all have the form formula_69