The Stacks project

Lemma 59.105.2. Let $X$ be a scheme and let $K \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Let $Z \subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square

\[ \xymatrix{ E \ar[d] \ar[r] & X' \ar[d]^ b \\ Z \ar[r] & X } \]

Then there is a canonical long exact sequence

\[ H^ p_{\acute{e}tale}(X, K) \to H^ p_{\acute{e}tale}(X', K|_{X'}) \oplus H^ p_{\acute{e}tale}(Z, K|_ Z) \to H^ p_{\acute{e}tale}(E, K|_ E) \to H^{p + 1}_{\acute{e}tale}(X, K) \]

First proof. This follows immediately from Lemma 59.105.1 and the fact that

\[ R\Gamma (X, Rb_*(K|_{X'})) = R\Gamma (X', K|_{X'}) \]

(see Cohomology on Sites, Section 21.14) and similarly for the others. $\square$

Second proof. By Lemma 59.102.7 these cohomology groups are the cohomology of $X, X', E, Z$ with values in some complex of abelian sheaves on the site $(\mathit{Sch}/X)_{ph}$. (Namely, the object $a_ X^{-1}K$ of the derived category, see Lemma 59.102.1 above and recall that $K|_{X'} = b_{small}^{-1}K$.) By More on Flatness, Lemma 38.36.1 the ph sheafification of the diagram of representable presheaves is cocartesian. Thus the lemma follows from the very general Cohomology on Sites, Lemma 21.26.3 applied to the site $(\mathit{Sch}/X)_{ph}$ and the commutative diagram of the lemma. $\square$

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