(Semicontinuity)
Let \(f: X \to T\) be projective with \(T\) Noetherian and \(\mcf\) coherent on \(X\), flat over \(T\). Then \(t \mapsto h^i(X_t, \mcf_t)\) is upper semicontinuous on \(T\), and \(t \mapsto \chi(\mcf_t)\) is locally constant.
(Grauert and base change)
If \(T\) is reduced and \(t \mapsto h^i(X_t,\mcf_t)\) is constant, then \(R^i f_* \mcf\) is locally free and \[ R^i f_* \mcf \tensor k(t) \iso H^i(X_t, \mcf_t) \] for all \(t\). Without reducedness there is still Grothendieck’s criterion: if the base change map in degree \(i\) is surjective at \(t\), it is an isomorphism near \(t\), and \(R^i f_* \mcf\) is locally free at \(t\) exactly when the map in degree \(i-1\) is surjective as well.
The shape to hold: individual \(h^i\) can only jump up on closed subsets, the alternating sum cannot jump at all, and cohomology commutes with restriction to a fibre exactly when nothing jumps.
Flatness is the load-bearing hypothesis throughout, and dropping it destroys even the constancy of \(\chi\). Reducedness of the base is what Grauert needs, and it is where non-reduced bases genuinely fail.
The standard specimen is a family of line bundles of degree \(0\) on a curve: \(h^0\) is \(0\) at a general point and \(1\) at the trivial bundle, a jump on a closed subset, while \(\chi = 1 - g\) throughout. That single example answers “give an example where cohomology does not commute with base change”.
Dependencies
D-COHEULERThe Euler characteristic and the Hilbert polynomialusesT-COHFFThe theorem on formal functionsuses
Backlinks
T-BASECHANGEFlat, proper and smooth base changerelated-toP-AGXMISCTSENPONEBUNDLEA family of genus-zero curves over a curve is a $\PP^1$-bundleuses