Lemma 59.105.1. Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square
\[ \xymatrix{ E \ar[d]_\pi \ar[r]_ j & X' \ar[d]^ b \\ Z \ar[r]^ i & X } \]
For $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have a distinguished triangle
\[ K \to Ri_*(K|_ Z) \oplus Rb_*(K|_{X'}) \to Rc_*(K|_ E) \to K[1] \]
in $D(X_{\acute{e}tale})$ where $c = i \circ \pi = b \circ j$.
Proof.
The notation $K|_{X'}$ stands for $b_{small}^{-1}K$. Choose a bounded below complex $\mathcal{F}^\bullet $ of abelian sheaves representing $K$. Observe that $i_*(\mathcal{F}^\bullet |_ Z)$ represents $Ri_*(K|_ Z)$ because $i_*$ is exact (Proposition 59.55.2). Choose a quasi-isomorphism $b_{small}^{-1}\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ where $\mathcal{I}^\bullet $ is a bounded below complex of injective abelian sheaves on $X'_{\acute{e}tale}$. This map is adjoint to a map $\mathcal{F}^\bullet \to b_*(\mathcal{I}^\bullet )$ and $b_*(\mathcal{I}^\bullet )$ represents $Rb_*(K|_{X'})$. We have $\pi _*(\mathcal{I}^\bullet |_ E) = (b_*\mathcal{I}^\bullet )|_ Z$ by Lemma 59.91.5 and by Lemma 59.91.12 this complex represents $R\pi _*(K|_ E)$. Hence the map
\[ Ri_*(K|_ Z) \oplus Rb_*(K|_{X'}) \to Rc_*(K|_ E) \]
is represented by the surjective map of bounded below complexes
\[ i_*(\mathcal{F}^\bullet |_ Z) \oplus b_*(\mathcal{I}^\bullet ) \to i_*\left(b_*(\mathcal{I}^\bullet )|_ Z\right) \]
To get our distinguished triangle it suffices to show that the canonical map $\mathcal{F}^\bullet \to i_*(\mathcal{F}^\bullet |_ Z) \oplus b_*(\mathcal{I}^\bullet )$ maps quasi-isomorphically onto the kernel of the map of complexes displayed above (namely a short exact sequence of complexes determines a distinguished triangle in the derived category, see Derived Categories, Section 13.12). We may check this on stalks at a geometric point $\overline{x}$ of $X$. If $\overline{x}$ is not in $Z$, then $X' \to X$ is an isomorphism over an open neighbourhood of $\overline{x}$. Thus, if $\overline{x}'$ denotes the corresponding geometric point of $X'$ in this case, then we have to show that
\[ \mathcal{F}^\bullet _{\overline{x}} \to \mathcal{I}^\bullet _{\overline{x}'} \]
is a quasi-isomorphism. This is true by our choice of $\mathcal{I}^\bullet $. If $\overline{x}$ is in $Z$, then $b_(\mathcal{I}^\bullet )_{\overline{x}} \to i_*\left(b_*(\mathcal{I}^\bullet )|_ Z\right)_{\overline{x}}$ is an isomorphism of complexes of abelian groups. Hence the kernel is equal to $i_*(\mathcal{F}^\bullet |_ Z)_{\overline{x}} = \mathcal{F}^\bullet _{\overline{x}}$ as desired.
$\square$
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