= Direct-image tensor comparison
{title2=$f_*\mathcal F\otimes_{\mathcal O_Y}f_*\mathcal G\to f_*(\mathcal F\otimes_{\mathcal O_X}\mathcal G)$}
Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two <direct image sheaves> on $V$. The <universal property of the tensor product of modules> and <sheafification> produce the displayed morphism. Composing with the <pullback-direct-image adjunction unit> gives the <projection formula for sheaves>. The comparison itself need not be an <isomorphism>.
Back to article page
