Here’s the projection formula:
Lemma
Let \(f:X\to Y\) be a morphism of smooth projective schemes, \(\mathcal F\) a sheaf on \(X\) and \(\mathcal E\) a sheaf on \(Y\). Then \[f_*(\mathcal F\otimes f^*\mathcal E) = f_*\mathcal F\otimes \mathcal E.\]
Here’s the derived version:
Lemma
Let \(f:X\to Y\) be a morphism of smooth projective schemes, \(\mathcal F^\bullet\in D^b(X)\) \(X\) and \(\mathcal E^\bullet \in D^B(Y)\). Then \[Rf_*(\mathcal F^\bullet \otimes^{L}Lf^*\mathcal E^\bullet) \simeq Rf_*\mathcal F^\bullet \otimes^L\mathcal E^\bullet\]
See also flat base change.