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We note that the vanishing and injectivity theorems of Koll\'ar andEsnault-Viehweg can be used to give a quick algebraic proof of a strengtheningof the Ein-Lazarsfeld Skoda-type division theorem for global sections ofadjoint line bundles vanishing along suitable multiplier ideal sheaves, and toextend it to higher cohomology classes as well. For global sections, this isalso a slightly more general statement of the algebraic translation of ananalytic result of Siu. Along the way we write down an injectivity statementfor multiplier ideals, and its standard consequences.

Global division of cohomology classes via injectivity