Base change theorems

Builds on Noetherian scheme Projective morphism Flat morphism Quasi-coherent sheaf Coherent sheaf Fiber of a morphism Residue field of a point Sheaf cohomology

Statement

Let $Y$ be a Noetherian scheme and let $f \colon X \to Y$ be a morphism. For a sheaf $\mathcal{F}$ on $X$ , $R^i f_* \mathcal{F}$ is the higher direct image, the sheaf on $Y$ associated to $V \mapsto H^i(f^{-1}(V), \mathcal{F})$ , where $H^i$ is sheaf cohomology. For $y \in Y$ with residue field $k(y)$ , let $X_y$ be the fiber and $\mathcal{F}_y$ the restriction of $\mathcal{F}$ to $X_y$ .

(Flat base change.) Let $f$ be separated and of finite type, let $\mathcal{F}$ be quasi-coherent, and let $u \colon Y' \to Y$ be a flat morphism with $Y'$ Noetherian. Let $v \colon X' = X \times_Y Y' \to X$ and $g \colon X' \to Y'$ be the projections. Then the natural maps $u^* R^i f_* \mathcal{F} \to R^i g_* (v^* \mathcal{F})$ are isomorphisms for all $i \ge 0$ .

In the following, let $f$ be projective and let $\mathcal{F}$ be a coherent sheaf on $X$ that is flat over $Y$ .

(Semicontinuity.) For each $i \ge 0$ , the function $y \mapsto \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is upper semicontinuous on $Y$ , and the Euler characteristic $y \mapsto \sum_i (-1)^i \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is locally constant.
(Grauert.) If $Y$ is integral and $y \mapsto \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is constant, then $R^i f_* \mathcal{F}$ is locally free and the natural map $R^i f_* \mathcal{F} \otimes k(y) \to H^i(X_y, \mathcal{F}_y)$ is an isomorphism for every $y \in Y$ .
(Cohomology and base change.) Let $y \in Y$ and suppose that the natural map $\varphi^i(y) \colon R^i f_* \mathcal{F} \otimes k(y) \to H^i(X_y, \mathcal{F}_y)$ is surjective. Then $\varphi^i(y)$ is an isomorphism, and $\varphi^{i-1}(y)$ is surjective if and only if $R^i f_* \mathcal{F}$ is locally free in a neighbourhood of $y$ .

Remarks

The idea of the proof of parts 2 to 4: locally on an affine $Y = \operatorname{Spec} A$ , there is a bounded complex $K^\bullet$ of finitely generated free $A$ -modules with a special property. For every $A$ -algebra $B$ , it computes the cohomology after base change: $H^i(X \times_Y \operatorname{Spec} B, \mathcal{F}_B) \cong H^i(K^\bullet \otimes_A B)$ . All statements then become facts about ranks of matrices of functions on $Y$ . Part 1 follows from the Čech complex of an affine cover, since flat base change commutes with taking cohomology of a complex.

Example: let $f \colon X \to Y$ be a flat projective family of curves over an integral base, with all fibers smooth of genus $g$ . Then $h^0(X_y, \mathcal{O}) = 1$ and $h^1(X_y, \mathcal{O}) = g$ for all $y$ , so by part 3 $f_* \mathcal{O}_X$ and $R^1 f_* \mathcal{O}_X$ are locally free of ranks $1$ and $g$ .

Flatness is needed in part 2. Let $Y = \mathbb{A}^1_k$ , and let $X$ be the disjoint union of $Y$ and the closed point $0$ . Let $f$ be the identity on $Y$ and the inclusion on the point, and let $\mathcal{F} = \mathcal{O}_X$ . Then $f$ is finite, hence projective, but $\mathcal{F}$ is not flat over $Y$ . The Euler characteristic of the fibers is $2$ at $0$ and $1$ elsewhere, so it is not locally constant.

Going deeper

For a student

The base change theorems are used to construct vector bundles on parameter spaces, such as the Hodge bundle $f_* \omega_{X/Y}$ on a moduli space of curves. Semicontinuity says that a property like $H^1(X_y, \mathcal{F}_y) = 0$ is open in families, which is the starting point of deformation arguments. Flat base change lets one compute higher direct images after a flat change of base, for example after passing to a completion or to an étale neighbourhood.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`1NC789DATR`
URI	`m2://1NC789DATR`
Kind	statement, theorem
Writing	W0 stub
MSC	14F06, 14D06
Status	This copy of the site has no build data. Run `m2 status 1NC789DATR`.
About	Sheaf cohomology
Uses	Noetherian scheme, Projective morphism, Flat morphism, Quasi-coherent sheaf, Coherent sheaf, Fiber of a morphism, Residue field of a point, Sheaf cohomology
Same as	Wikidata Q4866385 Wikipedia Base change theorems
Changed	2 November 2026