Let \(f: X \to Y\) be a morphism of ringed spaces, let \(\mcf\) be an \(\mco_X\dash\)module, and let \(\mce\) be a locally free \(\mco_Y\dash\)module of finite rank. Prove the projection formula (cf. (II, Ex. 5.1)): \[ R^i f_*(\mcf \tensor f^* \mce) \cong R^i f_*(\mcf) \tensor \mce .\]
Solution
Let \[ 0\longrightarrow\mcf\longrightarrow\mci^0\longrightarrow\mci^1\longrightarrow\cdots \] be an injective resolution of \(\mcf\) as an \(\mco_X\)-module.
1. Tensoring by \(f^*\mce\) is exact and sends injective \(\mco_X\)-modules to injective \(\mco_X\)-modules.
Because \(\mce\) is locally free of finite rank on \(Y\), its pullback \(f^*\mce\) is locally free of finite rank on \(X\). Let \[ (f^*\mce)^\vee=\mathcal Hom_{\mco_X}(f^*\mce,\mco_X). \] There are natural adjunction isomorphisms \[ \operatorname{Hom}_{\mco_X}(\mca,\mcb\tensor f^*\mce) \cong \operatorname{Hom}_{\mco_X}(\mca\tensor(f^*\mce)^\vee,\mcb). \] Both tensor functors with \(f^*\mce\) and with its dual are exact, because these sheaves are locally free. Hence tensoring by \(f^*\mce\) has an exact left adjoint and therefore preserves injective objects. It is itself exact as well.
2. The complex\[ 0\longrightarrow\mcf\tensor f^*\mce \longrightarrow\mci^0\tensor f^*\mce \longrightarrow\mci^1\tensor f^*\mce \longrightarrow\cdots \]is an injective resolution of \(\mcf\tensor f^*\mce\).
1.Each term \(\mci^q\tensor f^*\mce\) is injective by the same step.Thus the displayed complex is an injective resolution.
3. For every \(q\ge0\), the ordinary projection formula gives a natural isomorphism\[ f_*(\mci^q\tensor f^*\mce) \cong f_*\mci^q\tensor\mce. \]
This is the projection formula for ordinary direct images, Hartshorne II, Exercise 5.1, applied to the \(\mco_X\)-module \(\mci^q\) and the finite locally free \(\mco_Y\)-module \(\mce\). The isomorphisms are functorial in \(\mci^q\), so they commute with the differentials of the resolution and identify the two cochain complexes \[ f_*(\mci^\bullet\tensor f^*\mce) \cong f_*\mci^\bullet\tensor\mce. \]
4. Taking cohomology of the right-hand complex gives\[ H^i(f_*\mci^\bullet\tensor\mce) \cong H^i(f_*\mci^\bullet)\tensor\mce. \]
Tensoring by the locally free sheaf \(\mce\) is exact. For an exact functor \(T\) and any cochain complex \(C^\bullet\), kernels, images, and quotients are preserved, so \[ H^i(TC^\bullet)\cong T(H^i(C^\bullet)). \] Apply this to \[ T=(-)\tensor\mce, \qquad C^\bullet=f_*\mci^\bullet. \]
5. For every \(i\ge0\) there is a natural isomorphism\[ \boxed{ R^if_*(\mcf\tensor f^*\mce) \cong R^if_*(\mcf)\tensor\mce}. \]
By step 2, \[ R^if_*(\mcf\tensor f^*\mce) =H^i\bigl(f_*(\mci^\bullet\tensor f^*\mce)\bigr). \] Step 3 identifies this with \[ H^i(f_*\mci^\bullet\tensor\mce), \] which step 4 identifies with \[ H^i(f_*\mci^\bullet)\tensor\mce =R^if_*(\mcf)\tensor\mce. \] Every map used is natural in \(\mcf\) and \(\mce\), so the resulting isomorphism is natural.
6. Q.E.D.
Step 5 is the required projection formula in every degree.