Throughout, consider a Cartesian square \[ \begin{array}{ccc} X' & \xrightarrow{g'} & X \\ \downarrow{\scriptstyle f'} & & \downarrow{\scriptstyle f} \\ Y' & \xrightarrow{g} & Y \end{array} \] and the base change map \(g^* R^i f_* \mathcal{F} \to R^i f'_* g'^* \mathcal{F}\).
(Flat base change)
If \(f\) is quasicompact and separated, \(g\) is flat, and \(\mathcal{F}\) is quasicoherent on \(X\), the base change map is an isomorphism for all \(i\). [@Har10a, Proposition III.9.3]
(Extension of the base field)
Let \(X\) be a quasicompact separated scheme over a field \(k\), \(\mathcal{F}\) a quasicoherent sheaf on \(X\), and \(K/k\) a field extension, with \(X_K = X \times_k \Spec K\) and \(\mathcal{F}_K\) the pullback of \(\mathcal{F}\). Then \(H^i(X_K, \mathcal{F}_K) \cong H^i(X, \mathcal{F}) \otimes_k K\) for every \(i\); in particular \(\dim_k H^i(X, \mathcal{F}) = \dim_K H^i(X_K, \mathcal{F}_K)\) [@Har10a, Proposition III.9.3].
(Proper base change)
If \(f\) is proper, \(g\) is arbitrary, and \(\mathcal{F}\) is a torsion étale sheaf on \(X\), the base change map \(g^* R^i f_* \mathcal{F} \to R^i f'_* g'^* \mathcal{F}\) is an isomorphism for all \(i\). The same holds for sheaves of abelian groups on locally compact Hausdorff spaces when \(f\) is proper. In particular the stalk of \(R^i f_* \mathcal{F}\) at a geometric point \(\bar{y}\) is \(H^i(X_{\bar{y}}, \mathcal{F})\).
(Smooth base change)
If \(g\) is smooth, \(f\) is quasicompact and quasiseparated, and \(\mathcal{F}\) is a torsion étale sheaf with torsion orders invertible on \(Y\), the base change map is an isomorphism for all \(i\).
Without flatness of \(g\), quasicoherent base change fails. Let \(E\) be an elliptic curve over an algebraically closed field, \(p \in E\), \(f \colon E \times E \to E\) the second projection, and \(\mathcal{L} = \OO(\Delta - \{p\} \times E)\). The restriction of \(\mathcal{L}\) to the fibre \(E \times \{t\}\) is \(\OO_E(t - p)\), which has no nonzero sections for \(t \neq p\), so the torsion-free sheaf \(f_* \mathcal{L}\) is \(0\). For \(g\) the inclusion of the point \(p\), \(g^* f_* \mathcal{L} = 0\), while the fibre gives \(H^0(E, \OO_E) = k\). The coherent statement comparing \(R^i f_* \mathcal{F} \otimes \kappa(y)\) with \(H^i(X_y, \mathcal{F}_y)\) at arbitrary points is the theorem on cohomology and base change.
Dependencies
D-SCHBCBase change, and properties stable under ituses
Backlinks
T-LADICPROPERProper pushforward preserves constructible $\ell$-adic sheavesuses