The Stacks project

Lemma 50.17.2. With notation as in More on Morphisms, Lemma 37.17.3 and denoting $f : X \to S$ the structure morphism there is a canonical distinguished triangle

\[ \Omega ^\bullet _{X/S} \to Rb_*(\Omega ^\bullet _{X'/S}) \oplus i_*\Omega ^\bullet _{Z/S} \to i_*Rp_*(\Omega ^\bullet _{E/S}) \to \Omega ^\bullet _{X/S}[1] \]

in $D(X, f^{-1}\mathcal{O}_ S)$ where the four maps

\[ \begin{matrix} \Omega ^\bullet _{X/S} & \to & Rb_*(\Omega ^\bullet _{X'/S}), \\ \Omega ^\bullet _{X/S} & \to & i_*\Omega ^\bullet _{Z/S}, \\ Rb_*(\Omega ^\bullet _{X'/S}) & \to & i_*Rp_*(\Omega ^\bullet _{E/S}), \\ i_*\Omega ^\bullet _{Z/S} & \to & i_*Rp_*(\Omega ^\bullet _{E/S}) \end{matrix} \]

are the canonical ones (Section 50.2), except with sign reversed for one of them.

Proof. Choose a distinguished triangle

\[ C \to Rb_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \to i_*Rp_*\Omega ^\bullet _{E/S} \to C[1] \]

in $D(X, f^{-1}\mathcal{O}_ S)$. It suffices to show that $\Omega ^\bullet _{X/S}$ is isomorphic to $C$ in a manner compatible with the canonical maps. By the axioms of triangulated categories there exists a map of distinguished triangles

\[ \xymatrix{ C' \ar[r] \ar[d] & b_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \ar[r] \ar[d] & i_*p_*\Omega ^\bullet _{E/S} \ar[r] \ar[d] & C'[1] \ar[d] \\ C \ar[r] & Rb_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \ar[r] & i_*Rp_*\Omega ^\bullet _{E/S} \ar[r] & C[1] } \]

By Lemma 50.17.1 part (3) and Derived Categories, Proposition 13.4.23 we conclude that $C' \to C$ is an isomorphism. By Lemma 50.17.1 part (2) the map $i_*\Omega ^\bullet _{Z/S} \to i_*p_*\Omega ^\bullet _{E/S}$ is an isomorphism. Thus $C' = b_*\Omega ^\bullet _{X'/S}$ in the derived category. Finally we use Lemma 50.17.1 part (1) tells us this is equal to $\Omega ^\bullet _{X/S}$. We omit the verification this is compatible with the canonical maps. $\square$


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Your email address will not be published.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 0FUV. Beware of the difference between the letter 'O' and the digit '0'.