= Tensoring with an invertible sheaf is exact
{title2=$-\otimes L\text{ is exact}$}
For an <invertible sheaf> $L$, tensoring <sheaves of modules> with $L$ is an exact functor: on an <open cover> trivializing $L$, it is naturally the identity functor. Equivalently, its <stalks> are free modules of rank one and hence <flat modules>. Its inverse functor is tensoring with the <dual of a sheaf> $L^\vee$, because $L\otimes L^\vee\cong\mathcal O_X$. This justifies untwisting a surjection onto a positively twisted <coherent sheaf>.
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