Let \(f: X \to Y\) be a continuous map of topological spaces. Let \(\mcf\) be a sheaf of abelian groups on \(X\), and assume that \(R^i f_*(\mcf) = 0\) for all \(i > 0\). Show that there are natural isomorphisms, for each \(i \geq 0\), \[ H^i(X, \mcf) \cong H^i(Y, f_* \mcf) .\]
This is a degenerate case of the Leray spectral sequence; see Godement [1, II, 4.17.1].
Solution
1. The direct-image functor \(f_*\) sends injective sheaves on \(X\) to injective sheaves on \(Y\).
The inverse-image functor \[ f^{-1}:\operatorname{Ab}(Y)\longrightarrow\operatorname{Ab}(X) \] is exact for sheaves of abelian groups, and it is left adjoint to \(f_*\): \[ \operatorname{Hom}_X(f^{-1}\mcg,\mci) \cong \operatorname{Hom}_Y(\mcg,f_*\mci). \] Let \(\mci\) be injective on \(X\). Given a monomorphism \(\mcg'\hookrightarrow\mcg\) on \(Y\), exactness of \(f^{-1}\) makes \[ f^{-1}\mcg'\hookrightarrow f^{-1}\mcg \] a monomorphism. Injectivity of \(\mci\) therefore makes \[ \operatorname{Hom}_X(f^{-1}\mcg,\mci) \longrightarrow \operatorname{Hom}_X(f^{-1}\mcg',\mci) \] surjective. By adjunction, the same is true for \[ \operatorname{Hom}_Y(\mcg,f_*\mci) \longrightarrow \operatorname{Hom}_Y(\mcg',f_*\mci), \] so \(f_*\mci\) is injective.
2. If\[ 0\longrightarrow\mcf\longrightarrow\mci^0\longrightarrow\mci^1\longrightarrow\cdots \]is an injective resolution on \(X\), then under the hypothesis \(R^if_*\mcf=0\) for \(i>0\),\[ 0\longrightarrow f_*\mcf\longrightarrow f_*\mci^0\longrightarrow f_*\mci^1\longrightarrow\cdots \]is an injective resolution on \(Y\).
By step 1 every \(f_*\mci^q\) is injective. By definition, the cohomology sheaves of the complex \[ f_*\mci^0\longrightarrow f_*\mci^1\longrightarrow\cdots \] are the derived functors \[ R^if_*\mcf. \] The degree-zero cohomology is \(f_*\mcf\), because \(f_*\) is left exact, and all positive-degree cohomology sheaves vanish by hypothesis. Thus adjoining \(f_*\mcf\) in degree zero makes the displayed complex exact. It is therefore an injective resolution of \(f_*\mcf\).
3. Applying global sections to the two resolutions gives the same cochain complex.
For every sheaf \(\mci\) on \(X\), the definition of direct image gives \[ \Gamma(Y,f_*\mci) =(f_*\mci)(Y) =\mci(f^{-1}Y) =\mci(X) =\Gamma(X,\mci). \] Hence term by term, \[ \Gamma(Y,f_*\mci^q)=\Gamma(X,\mci^q), \] and these equalities commute with the differentials. So the global-section complexes calculating the two cohomologies are identical.
4. For every \(i\ge0\) there is a natural isomorphism\[ \boxed{H^i(X,\mcf)\cong H^i(Y,f_*\mcf)}. \]
The injective resolution \(\mci^\bullet\) computes \[ H^i(X,\mcf) =H^i\bigl(\Gamma(X,\mci^\bullet)\bigr). \] By step 2, \(f_*\mci^\bullet\) is an injective resolution of \(f_*\mcf\), so it computes \[ H^i(Y,f_*\mcf) =H^i\bigl(\Gamma(Y,f_*\mci^\bullet)\bigr). \] Step 3 identifies these cochain complexes canonically, hence identifies their cohomology groups. The construction is natural in \(\mcf\) because comparison morphisms between injective resolutions are unique up to homotopy and the identification \(\Gamma(Y,f_*(-))=\Gamma(X,-)\) is functorial.
5. Q.E.D.
Step 4 gives the required natural isomorphisms in every degree.