Grauert's direct image theorem

Remarks

When $Y$ is a single point, $R^q f_* \mathcal{F}$ is the vector space $H^q(X, \mathcal{F})$ and $X$ is compact, so the theorem contains the Cartan–Serre finiteness theorem. The case where $f$ is finite and $q = 0$ is easier and is proved with the Weierstrass division theorem. The general case is much harder. Grauert's original proof was long; a later proof by Kiehl and Verdier uses Čech cochains of Stein coverings and compact operators between Fréchet modules.

Example: if $A \subseteq X$ is an analytic subset with its reduced structure, then $f_* \mathcal{O}_A$ is coherent on $Y$, and its support is $f(A)$. Since supports of coherent sheaves are analytic, this gives Remmert's proper mapping theorem.

Properness is necessary. Consider the inclusion $f \colon \C \setminus \set{0} \to \C$. The stalk of $f_* \mathcal{O}$ at $0$ is the ring of holomorphic functions on punctured discs around $0$, modulo agreement near $0$. It is not finitely generated over $\mathcal{O}_{\C,0}$. Indeed it is nonzero and multiplication by $z$ is bijective on it, so Nakayama's lemma rules out finite generation.

Going deeper

For a student

The theorem is the basic finiteness result of relative complex analytic geometry. It is used to study families of compact complex spaces, for example to prove semicontinuity of the dimensions of cohomology groups of fibres. It is the analytic analogue of the coherence of higher direct images of coherent sheaves under proper morphisms of schemes.