# higher direct image: proof of the proper case.

Hi.

Let $f:X→S$ be a proper, open, surjective morphism of complex reduced spaces with constant fiber dimension n (or universally open morphism with n-fibers between locally noetherian excellent without embedded component schemes on field of characteristic 0). Let $G$ be a coherent sheaf on X.

Claim: $R^{n}f_*G=0$ if and only if $Supp(G)\cap X_{s}$ is nowhere dense in the fiber $X_{s}$.

Outline of the proof:

1) If $Supp(G)\cap X_{s}$ is nowhere dense in the fiber $X_{s}$ and then $dim(Supp(G)\cap X_{s})<n$, it is not hard to see that $R^{n}f_∗G=0$.

For this, we use the fact that $Supp(G)\cap X_{s}$ is a compact subspace of dimension $\leq n-1$ and then admit a fundamental system of $n-1$-complete neighboorhoods (Demailly result on top degre cohomology...) each of them satisfies, in particular, $H^{n}(V, G)=0$; it suffices to write the definition of the germ $R^{n}f_∗G(s)$ as an inductive limit.

2) Suppose now $R^{n}f_∗G=0$. Then we have, by duality, $\pi_*\mathcal{H}om(G, \omega^{n}_{X/S}=0$. The openness and generic smoothness) of $\pi$ imply that $\mathcal{H}om(G, \omega^{n}_{X/S})=0$. Therefore, we can use local $S$-embedding or local $S$ parametrization: $\begin{array}{ccc} X & \overset{\sigma}{\to} & Z \\ & \underset{\pi}{\searrow} & \downarrow q \\ & & S \end{array}$ where $Z$ is an $n+p$-dimensional $S$-smooth complex space, or
$\begin{array}{ccc} X & \overset{f}{\to} & Y \\ & \underset{\pi}{\searrow} & \downarrow q \\ & & S \end{array}$ where $f$ is finite, open and surjective over $Y:=S\times U$ of relative dimension $n$.

Then the isomorphism $$f_{*}{\cal H}om(G,\omega^{n}_{X/S})\simeq{\cal H}om(f_{*}G, {\cal O}_{Y})$$ $$\sigma_{*}{\cal H}om(G,\omega^{n}_{X/S})\simeq {\cal E}xt^{p}(\sigma_{*}G, \Omega^{n+p}_{Z/S})$$ proves the claim. We see that the problem reduce to the case of the projection $q:S\times U\rightarrow S$, where $U$ is either a Stein in ${\Bbb C}^{n+p}$ or in ${\Bbb C}^{n}$, and coherent sheaf $G$ on $S\times U$ satisfiying one of this conditions: $${\cal H}om( G, {\cal O}_{S\times U})=0,\,\,{\rm or }\,\,{\cal E}xt^{p}( G, {\cal O}_{S\times U})=0$$ ` We deduce what we want thanks to this:

Let $k$ be a noetherian local ring with residual field $\mathfrak{k}$, $R$ be a $k$-algebra formally smooth over $k$, $N$ be the Krull dimension of the regular local ring $R{\otimes_{k}}\mathfrak{k}$. Let $q$ be an integer and $M$ an $R$-module of finite type. Then
$${\rm dim}_{(R{\otimes_{k}}\mathfrak{k})} M\otimes_{k}\mathfrak{k}\leq q\,\Longleftrightarrow {\rm E}xt_{R}^{j}(M, R)= 0 \quad \text{for} \quad j<N-q$$

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Sorry. I dont understand anything!!!! –  kaddar Sep 15 '10 at 19:34
Check you TeX code, you left some dollar signs out. In case, the "only if" part is wrong: take $S$ to be a point, $X = \mathbb P^1$, and $G = \mathcal O_X$. And what is the question? –  Angelo Sep 15 '10 at 20:00
I fixed your LaTeX code, but I still don't see a question. –  S. Carnahan Sep 16 '10 at 7:47
The question was (see the "Higher direct image..") If $f:X\rightarrow S$ is a proper, open and surjective morphism of reduced complex spaces countable at infinity and $G$ is a coherent sheaf on $X$. Then $R^{n}f_{*}G=0$ if and only if $Supp(G)\cap X_{s}$ is nowhere dense in the fiber $X_{s}$. –  kaddar Sep 16 '10 at 8:34
But this is not true, consider the map from projective space (of poistive dimension) over a field $k$ to the spectrum of the field and the tautological sheaf $\mathcal O(1)$ on projective space. In the other direction it is OK. –  Torsten Ekedahl Sep 16 '10 at 8:53