The Stacks project

Lemma 59.105.3. 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

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

Suppose given

  1. an object $K'$ of $D^+(X'_{\acute{e}tale})$ with torsion cohomology sheaves,

  2. an object $L$ of $D^+(Z_{\acute{e}tale})$ with torsion cohomology sheaves, and

  3. an isomorphism $\gamma : K'|_ E \to L|_ E$.

Then there exists an object $K$ of $D^+(X_{\acute{e}tale})$ and isomorphisms $f : K|_{X'} \to K'$, $g : K|_ Z \to L$ such that $\gamma = g|_ E \circ f^{-1}|_ E$. Moreover, given

  1. an object $M$ of $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves,

  2. a morphism $\alpha : K' \to M|_{X'}$ of $D(X'_{\acute{e}tale})$,

  3. a morphism $\beta : L \to M|_ Z$ of $D(Z_{\acute{e}tale})$,

such that

\[ \alpha |_ E = \beta |_ E \circ \gamma . \]

Then there exists a morphism $M \to K$ in $D(X_{\acute{e}tale})$ whose restriction to $X'$ is $a \circ f$ and whose restriction to $Z$ is $b \circ g$.

Proof. If $K$ exists, then Lemma 59.105.1 tells us a distinguished triangle that it fits in. Thus we simply choose a distinguished triangle

\[ K \to Ri_*(L) \oplus Rb_*(K') \to Rc_*(L|_ E) \to K[1] \]

where $c = i \circ \pi = b \circ j$. Here the map $Ri_*(L) \to Rc_*(L|_ E)$ is $Ri_*$ applied to the adjunction mapping $E \to R\pi _*(L|_ E)$. The map $Rb_*(K') \to Rc_*(L|_ E)$ is the composition of the canonical map $Rb_*(K') \to Rc_*(K'|_ E)) = R$ and $Rc_*(\gamma )$. The maps $g$ and $f$ of the statement of the lemma are the adjoints of these maps. If we restrict this distinguished triangle to $Z$ then the map $Rb_*(K) \to Rc_*(L|_ E)$ becomes an isomorphism by the base change theorem (Lemma 59.91.12) and hence the map $g : K|_ Z \to L$ is an isomorphism. Looking at the distinguished triangle we see that $f : K|_{X'} \to K'$ is an isomorphism over $X' \setminus E = X \setminus Z$. Moreover, we have $\gamma \circ f|_ E = g|_ E$ by construction. Then since $\gamma $ and $g$ are isomorphisms we conclude that $f$ induces isomorphisms on stalks at geometric points of $E$ as well. Thus $f$ is an isomorphism.

For the final statement, we may replace $K'$ by $K|_{X'}$, $L$ by $K|_ Z$, and $\gamma $ by the canonical identification. Observe that $\alpha $ and $\beta $ induce a commutative square

\[ \xymatrix{ K \ar[r] \ar@{..>}[d] & Ri_*(K|_ Z) \oplus Rb_*(K|_{X'}) \ar[r] \ar[d]_{\beta \oplus \alpha } & Rc_*(K|_ E) \ar[r] \ar[d]_{\alpha |_ E} & K[1] \ar@{..>}[d] \\ M \ar[r] & Ri_*(M|_ Z) \oplus Rb_*(M|_{X'}) \ar[r] & Rc_*(M|_ E) \ar[r] & M[1] } \]

Thus by the axioms of a derived category we get a dotted arrow producing a morphism of distinguished triangles. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EW6. Beware of the difference between the letter 'O' and the digit '0'.