I'm struggling with the proof of Leray's theorem. Here's the statement:
Theorem Let $X$ be a topological space, $\mathscr{F}$ a sheaf on $X$ and $\mathscr{U} = \{ U_i \}_{i \in I}$ an open cover of $X$. If $\mathrm{H}^p(V, \mathscr{F}) = 0$ for all $p$ and all $V = U_{i_0}\cap \dots U_{i_n}, i_j \in I$ then $\check{\mathrm{H}}^p(\mathscr{U}, \mathscr{F}) \simeq \mathrm{H}^p(X, \mathscr{F})$ for all $p$.
I'm stack at the end of the proof. Let's embed $\mathscr{F}$ into a flasque sheaf $\mathscr{G}$ and then let us take the quotient $\mathscr{H}$. Then we have a short exact sequence of sheaves:
$$0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0$$
From $\mathrm{H}^1(V, \mathscr{F}) = 0$ for all intersections we get that the short sequence of Cech complexes
$$0 \rightarrow C^{\ast}(\mathscr{U}, \mathscr{F}) \rightarrow C^{\ast}(\mathscr{U}, \mathscr{G}) \rightarrow C^{\ast}(\mathscr{U}, \mathscr{F}) \rightarrow 0$$
is exact. Comparing the long exact sequence we then get in Cech cohomology with the one we already had from sheaf cohomology, we obtain that:
- $\check{\mathrm{H}}^1(\mathscr{U}, \mathscr{F}) \simeq \mathrm{H}^1(X, \mathscr{F})$
- $\check{\mathrm{H}}^{p-1}(\mathscr{U}, \mathscr{H}) \simeq \check{\mathrm{H}}^p(X, \mathscr{F})$
The professor then concludes by induction. What I don't understand is why he is able to use the induction hypothesis. We have the diagram:
$$ \require{AMScd} \begin{CD} 0 @>>> \check {\mathrm{H}}^{p-1}(\mathscr{U}, \mathscr{H}) @>\simeq>> \check {\mathrm{H}}^p(X, \mathscr{F}) @>>> 0 \\ @VVV @VVV @VVV @VVV\\ 0 @>>>\mathrm{H}^{p-1}(\mathscr{U}, \mathscr{H}) @>\simeq>> \mathrm{H}^p(X, \mathscr{F}) @>>> 0 \end{CD} $$
He says that on the $p-1$ Cech cohomology group of $\mathscr{H}$ we can use the induction hypothesis. I don't succeed in understanding why: we don't know that the cohomology group of $\mathscr{H}$ are zero on the intersection of open sets took from $\mathscr{U}$, right? I would be very grateful if someone could give me an hint to conclude the proof.
I think it's important to point out that in class we did't talk about neither quasi-coherent sheaves or schemes.
From the long exact sequence $$ \cdots \to H^k(V,{\cal G}) \to H^k(V, {\cal H}) \to H^{k+1} (V, {\cal F}) \to \cdots, $$ one sees that the vanishing of the first and last groups (between the $\cdots$) implies the vanishing of the the middle.