The Stacks project

Remark 50.4.3. Let $f : X \to S$ be a morphism of schemes. Let $\xi \in H_{dR}^ n(X/S)$. According to the discussion Differential Graded Sheaves, Section 24.32 there exists a canonical morphism

\[ \xi ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[n] \]

in $D(f^{-1}\mathcal{O}_ S)$ uniquely characterized by (1) and (2) of the following list of properties:

  1. $\xi '$ can be lifted to a map in the derived category of right differential graded $\Omega ^\bullet _{X/S}$-modules, and

  2. $\xi '(1) = \xi $ in $H^0(X, \Omega ^\bullet _{X/S}[n]) = H^ n_{dR}(X/S)$,

  3. the map $\xi '$ sends $\eta \in H^ m_{dR}(X/S)$ to $\xi \cup \eta $ in $H^{n + m}_{dR}(X/S)$,

  4. the construction of $\xi '$ commutes with restrictions to opens: for $U \subset X$ open the restriction $\xi '|_ U$ is the map corresponding to the image $\xi |_ U \in H^ n_{dR}(U/S)$,

  5. for any diagram as in Remark 50.4.2 we obtain a commutative diagram

    \[ \xymatrix{ Rf_*\Omega ^\bullet _{X/S} \otimes _{q^{-1}\mathcal{O}_ T}^\mathbf {L} Rf_*\Omega ^\bullet _{X/S} \ar[d]_{\xi ' \otimes \text{id}} \ar[r]_-\mu & Rf_*\Omega ^\bullet _{X/S} \ar[d]^{\xi '} \\ Rf_*\Omega ^\bullet _{X/S}[n] \otimes _{q^{-1}\mathcal{O}_ T}^\mathbf {L} Rf_*\Omega ^\bullet _{X/S} \ar[r]^-\mu & Rf_*\Omega ^\bullet _{X/S}[n] } \]

    in $D(Y, q^{-1}\mathcal{O}_ T)$.

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