The Stacks project

Lemma 50.23.2. The gysin map (50.23.1.1) is compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

Proof. This follows from an almost trivial calculation once we correctly interpret this. First, we recall that the functor $\mathcal{H}^ c_ Z$ computed on the category of $\mathcal{O}_ X$-modules agrees with the similarly defined functor on the category of abelian sheaves on $X$, see Cohomology, Lemma 20.34.8. Hence, the differential $\text{d} : \Omega ^ p_{X/S} \to \Omega ^{p + 1}_{X/S}$ induces a map $\mathcal{H}^ c_ Z(\Omega ^ p_{X/S}) \to \mathcal{H}^ c_ Z(\Omega ^{p + 1}_{X/S})$. Moreover, the formation of the extended alternating Čech complex in Derived Categories of Schemes, Remark 36.6.4 works on the category of abelian sheaves. The map

\[ \mathop{\mathrm{Coker}}\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^ c_ Z(\mathcal{F}) \]

used in the construction of $c_{f_1, \ldots , f_ c}$ in Derived Categories of Schemes, Remark 36.6.10 is well defined and functorial on the category of all abelian sheaves on $X$. Hence we see that the lemma follows from the equality

\[ \text{d}\left( \frac{\tilde\omega \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}\right) = \frac{\text{d}(\tilde\omega ) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c} \]

which is clear. $\square$


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Your email address will not be published.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 0G85. Beware of the difference between the letter 'O' and the digit '0'.