Tensor by a pulled-back finite-rank bundle upstairs, or tensor every higher direct image downstairs: the result is the same.Suggest a slogan
(Projection formula)
Let \(f \colon X \to Y\) be a morphism of ringed spaces, \(\mathcal{F}\) an \(\OO_X\)-module, and \(\mathcal{E}\) a locally free \(\OO_Y\)-module of finite rank. Then there are natural isomorphisms \[ R^i f_* (\mathcal{F} \otimes_{\OO_X} f^* \mathcal{E}) \cong R^i f_* \mathcal{F} \otimes_{\OO_Y} \mathcal{E} \] for all \(i \geq 0\). [@Har10a, Exercise III.8.3]
For \(\pi \colon \PP(\mathcal{V}) \to Y\) a projective bundle and \(\mathcal{L}\) a line bundle on \(Y\), \(\pi_*(\OO(m) \otimes \pi^* \mathcal{L}) \cong \Sym^m \mathcal{V} \otimes \mathcal{L}\) for \(m \geq 0\).
What is Serre's projection formula?
Dependencies
D-MODPULLPullback and pushforward of sheaves of modulesuses