Cartier divisors also have a sheaf-theoretic description. A fractional ideal sheaf is a sub-formula_57-module of formula_46 A fractional ideal sheaf "J" is invertible if, for each "x" in "X", there exists an open neighborhood "U" of "x" on which the restriction of "J" to "U" is equal to formula_59 where formula_60 and the product is taken in formula_46 Each Cartier divisor defines an invertible fractional ideal sheaf using the description of the Cartier divisor as a collection formula_49 and conversely, invertible fractional ideal sheaves define Cartier divisors. If the Cartier divisor is denoted "D", then the corresponding fractional ideal sheaf is denoted "O"("D") or "L"("D").