Local-Global
Example: Hartshorne, Example \(\mathrm{III}.10.0.2\)¶
Let \(f : X \to Y\) be a morphism of schemes with \(X\) integral and Noetherian, and let \(\Omega_{X/Y}\) denote the sheaf of relative Kähler differentials. The claim is that \(\Omega_{X/Y}\) is locally free of rank \(n\) if and only if the fiber dimension is constant:
The forward direction. If \(\Omega_{X/Y} \cong \mathcal{O}_X^{\oplus n}\) locally, then at every point \(x \in X\) the fiber \(\Omega_{X/Y} \otimes k(x) \cong k(x)^n\) has dimension \(n\). This is immediate.
The converse. Since \(X\) is Noetherian and \(f\) is a morphism of schemes, \(\Omega_{X/Y}\) is a coherent \(\mathcal{O}_X\)-module. At each point \(x \in X\), the hypothesis \(\dim_{k(x)} \Omega_{X/Y} \otimes k(x) = n\) means that \(\Omega_{X/Y,x}\) requires exactly \(n\) generators as an \(\mathcal{O}_{X,x}\)-module, by Nakayama's lemma. By choosing lifts of a basis for the fiber, we obtain a surjection \(\mathcal{O}_{X,x}^n \twoheadrightarrow \Omega_{X/Y,x}\) in a neighborhood of \(x\).
The key input is that \(X\) is integral. The generic stalk \(\Omega_{X/Y,\eta}\) is a vector space over the function field \(K(X)\), and the constant fiber dimension forces \(\operatorname{rank}_{K(X)} \Omega_{X/Y,\eta} = n\). At every point \(x\), the surjection \(\mathcal{O}_{X,x}^n \twoheadrightarrow \Omega_{X/Y,x}\) must then be an isomorphism: its kernel \(K_x\) satisfies \(K_x \otimes K(X) = 0\) (since the generic rank is already \(n\)), and on an integral Noetherian scheme a coherent subsheaf of a free module with zero generic stalk is itself zero. This establishes local freeness of rank \(n\).
This is a standard and useful fact: a coherent sheaf on an integral Noetherian scheme with constant fiber dimension is locally free. (See, e.g., Hartshorne II, Exercise 5.8 or Vakil, Theorem 13.7.2.)
Reinterpretation via cohomology and base change. There is an instructive (if circular) way to recover the same conclusion from the base change machinery. Consider the identity morphism \(\operatorname{id} : X \to X\) and the coherent sheaf \(\mathscr{F} = \Omega_{X/Y}\). If \(\Omega_{X/Y}\) is flat over \(\mathcal{O}_X\) (via \(\operatorname{id}\), this simply means flat as an \(\mathcal{O}_X\)-module), then the base change theorem (Hartshorne III.12.11, or Grauert's theorem in the proper case) guarantees that the natural map
is an isomorphism for all \(x\). Since \(R^0(\operatorname{id})_* \Omega_{X/Y} = \Omega_{X/Y}\) and the fiber of \(\operatorname{id}\) over \(x\) is \(\operatorname{Spec} k(x)\), the right-hand side is \(\Omega_{X/Y} \otimes k(x)\). By the semicontinuity theorem, the constancy of \(\dim_{k(x)} \Omega_{X/Y} \otimes k(x)\) then implies that \(R^0(\operatorname{id})_*(\Omega_{X/Y}) = \Omega_{X/Y}\) is locally free.
The circularity is worth noting explicitly: on an integral Noetherian scheme, a coherent sheaf is flat if and only if it is locally free. So assuming flatness to deduce local freeness invokes the conclusion as a hypothesis. Nevertheless, this reformulation illustrates how the cohomology and base change theorem subsumes pointwise-to-global upgrade results, and in more general settings (e.g., families over a non-integral base where flatness is a genuine additional hypothesis), the base change approach becomes the essential tool.