What is the Theorem on Coherent Cohomology and Base Change good for?

One version of the theorem is:

Suppse $f \colon X \to Y$ is a proper morphism of noetherian schemes and $F$ is a $Y$-flat coherent sheaf on $Y$. If $y_0 \in X$ and $q \in \mathbb{Z}$ have the property that the natural map

$$\alpha^{q}(y_0) \colon k(y_0) \otimes R^{q}f_{*}(F) \to H^{q}(X_{y_0}, F/\mathfrak{m}_{y_0}F)$$

is surjective, then there exists a neighborhood $U \subset Y$ with the property that $\alpha^{q}(y)$ is an isomorphism for all $y \in Y$. Furthermore, $\alpha^{q-1}(y_0)$ is surjective if and only if $R^{q}f_{*}(F)$ is locally free of finite type in a neighbohood of $y_0$.

What are some results that make use of this theorem?

Here are the applications I have seen:

1) Top-dimensional cohomology commutes with base change:

If the fibers $X_{y}$ all have dimension at most $n$, then trivially $\alpha^{n+1}(y_0)$ is surjective and $R^{n+1}f_{*}(F)$ is locally free of finite type. Thus, $\alpha^{n}(y_0)$ is always an isomorphism. For example, if $f: X \to Y$ is a family of curves, then the fibers of $R^{1}f_{*} (\mathcal{O}_{X})$ are the 1st cohomology groups of the fibers of $f$.

This fact is used to construction the Hodge bundle on the Deligne--Mumford moduli stack of stable curves.

2) The direct image of the structure sheaf of a family of varieties is the sheaf of regular functions on the base:

Trivially $\alpha^{-1}(y_0)$ is surjective. Thus, if $\alpha^{0}(y_0)$ is surjective, then $f_{*}(F)$ is locally free of finite type on a neighborhood of $y_0$. When $f$ is surjective with geometrically reduced and connected fibers and $F=\mathcal{O}_{X}$, then it is not hard to show that $\alpha^{0}(y_0)$ is surjective (every global function on $X_y$ is constant). In particular, the direct image of the structure sheaf is locally free of finite type, and one can show further that it is $\mathcal{O}_{Y}$, the trivial locally free module of rank $1$.

When the condition

$$\mathcal{O} = f_{*}(\mathcal{O}_{X}) $$

is satisfied, it is a theorem of Artin that the relative Picard space (parameterizing line bundles on the fibers of $X_{y}$) exists.

3) The Picard bundle is a vector bundle.

Suppose $C$ is a smooth, projective curve of genus $g$ (over a field) and $Y = J^{d}_{C}$ is its degree $d$ Jacobian (parameterizing line bundles on $C$ of degree $d$). If $d \ge 2g-1$, then the Picard bundle $E$ is a vector bundle with the property the fiber over $y_0 \in Y$ is $H^{0}(C, L)$, where $L$ is the line bundle corresponding to $y_0$. If $F$ is a universal family of line bundles on $X = C \times J^{d}_{C}$, then the Theorem on Base Change implies that $E$ can be defined to be the direct image of $F$ under the projection map $f \colon X \to Y$. Indeed, both $\alpha^{-1}(y_0)$ and $\alpha^{1}(y_0)$ are always surjective as they map to the zero module.

The projectivization of the Picard bundle is the $d$-th symmetric power of the curve, and this description provides one approach to studying the symmetric power.

What are some other applications?

  • 3
    $\begingroup$ Of course, all of the applications you use (particularly (2)) have tons of secondary applications. If you're willing to broaden what you consider to be a "cohomology and base change" theorem, then proper and smooth base change for torsion sheaves on the etale site, for example, are important for defining etale cohomology with compact support, which is a crucial ingredient in the proof of the Weil conjectures. $\endgroup$ – Daniel Litt Jul 23 '11 at 7:51
  • 5
    $\begingroup$ Grothendieck used this theorem to embed the Quot functor into a Grassmannian and thus prove its representability. See FGA for details. $\endgroup$ – AFK Jul 23 '11 at 12:02
  • $\begingroup$ Community Wiki? It's a good question which has no definite answer. $\endgroup$ – Joël Mar 31 '12 at 0:55

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