The Stacks project

Lemma 30.17.3. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma (X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $X$ is quasi-compact and quasi-separated, the canonical map

\[ H^ p_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^ p(X_ s, \mathcal{F}) \]

which maps $\xi /s^ n$ to $s^{-n}\xi $ is an isomorphism.

Proof. Note that for $p = 0$ this is Properties, Lemma 28.17.2. We will prove the statement using the induction principle (Lemma 30.4.1) where for $U \subset X$ quasi-compact open we let $P(U)$ be the property: for all $p \geq 0$ the map

\[ H^ p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^ p(U_ s, \mathcal{F}) \]

is an isomorphism.

If $U$ is affine, then both sides of the arrow displayed above are zero for $p > 0$ by Lemma 30.2.2 and Properties, Lemma 28.26.4 and the statement is true. If $P$ is true for $U$, $V$, and $U \cap V$, then we can use the Mayer-Vietoris sequences (Cohomology, Lemma 20.8.2) to obtain a map of long exact sequences

\[ \xymatrix{ H^{p - 1}_*(U \cap V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^ p_*(U \cup V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^ p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \oplus H^ p_*(V, \mathcal{L}, \mathcal{F})_{(s)} \ar[d] \\ H^{p - 1}(U_ s \cap V_ s, \mathcal{F}) \ar[r]& H^ p(U_ s \cup V_ s, \mathcal{F}) \ar[r] & H^ p(U_ s, \mathcal{F}) \oplus H^ p(V_ s, \mathcal{F}) } \]

(only a snippet shown). Observe that $U_ s \cap V_ s = (U \cap V)_ s$ and that $U_ s \cup V_ s = (U \cup V)_ s$. Thus the left and right vertical maps are isomorphisms (as well as one more to the right and one more to the left which are not shown in the diagram). We conclude that $P(U \cup V)$ holds by the 5-lemma (Homology, Lemma 12.5.20). This finishes the proof. $\square$

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