The aim of this note is to remark that the injectivity theorems of Kollar and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening (by dropping the positivity hypothesis) of the Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves (proved in [EL]) and to extend this result to higher cohomology classes as well (cf. Theorem 4.1). For global sections, this is a slightly more general statement of the algebraic version of an analytic result of Siu [S] based on the original Skoda theorem. In Section 4 we list a few consequences of this type of result, such as the surjectiv-ity of various multiplication or cup product maps and the corresponding version of the geometric effective Nullstellensatz.
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