Lemma 50.10.3. The “boundary” map $\delta : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2]$ in $D(X, f^{-1}\mathcal{O}_ S)$ coming from the short exact sequence in Lemma 50.10.2 is the map of Remark 50.4.3 for $\xi = c_1^{dR}(\mathcal{L})$.
Proof. To be precise we consider the shift
\[ 0 \to \Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{L^\star /S, 0}[1] \to \Omega ^\bullet _{X/S} \to 0 \]
of the short exact sequence of Lemma 50.10.2. As the degree zero part of a grading on $(L^\star \to X)_*\Omega ^\bullet _{L^\star /S}$ we see that $\Omega ^\bullet _{L^\star /S, 0}$ is a differential graded $\mathcal{O}_ X$-algebra and that the map $\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{L^\star /S, 0}$ is a homomorphism of differential graded $\mathcal{O}_ X$-algebras. Hence we may view $\Omega ^\bullet _{X/S}[1] \to \Omega ^\bullet _{L^\star /S, 0}[1]$ as a map of right differential graded $\Omega ^\bullet _{X/S}$-modules on $X$. The map $\text{Res} : \Omega ^\bullet _{L^\star /S, 0}[1] \to \Omega ^\bullet _{X/S}$ is a map of right differential graded $\Omega ^\bullet _{X/S}$-modules since it is locally defined by the rule $\text{Res}(\omega ' + \text{d}\log (s) \wedge \omega ) = \omega $, see proof of Lemma 50.10.2. Thus by the discussion in Differential Graded Sheaves, Section 24.32 we see that $\delta $ comes from a map $\delta ' : \Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}[2]$ in the derived category $D(\Omega ^\bullet _{X/S}, \text{d})$ of right differential graded modules over the de Rham complex. The uniqueness averted in Remark 50.4.3 shows it suffices to prove that $\delta (1) = c_1^{dR}(\mathcal{L})$.
We claim that there is a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \mathcal{O}_ X^* \ar[r] \ar[d]_{\text{d}\log } & E \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \Omega ^\bullet _{X/S}[1] \ar[r] & \Omega ^\bullet _{L^\star /S, 0}[1] \ar[r] & \Omega ^\bullet _{X/S} \ar[r] & 0 } \]
where the top row is a short exact sequence of abelian sheaves whose boundary map sends $1$ to the class of $\mathcal{L}$ in $H^1(X, \mathcal{O}_ X^*)$. It suffices to prove the claim by the compatibility of boundary maps with maps between short exact sequences. We define $E$ as the sheafification of the rule
\[ U \longmapsto \{ (s, n) \mid n \in \mathbf{Z},\ s \in \mathcal{L}^{\otimes n}(U)\text{ generator}\} \]
with group structure given by $(s, n) \cdot (t, m) = (s \otimes t, n + m)$. The middle vertical map sends $(s, n)$ to $\text{d}\log (s)$. This produces a map of short exact sequences because the map $Res : \Omega ^1_{L^\star /S, 0} \to \mathcal{O}_ X$ constructed in the proof of Lemma 50.10.2 sends $\text{d}\log (s)$ to $1$ if $s$ is a local generator of $\mathcal{L}$. To calculate the boundary of $1$ in the top row, choose local trivializations $s_ i$ of $\mathcal{L}$ over opens $U_ i$ as in Section 50.9. On the overlaps $U_{i_0i_1} = U_{i_0} \cap U_{i_1}$ we have an invertible function $f_{i_0i_1}$ such that $f_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} s_{i_0}|_{U_{i_0i_1}}^{-1}$ and the cohomology class of $\mathcal{L}$ is given by the Čech cocycle $\{ f_{i_0i_1}\} $. Then of course we have
\[ (f_{i_0i_1}, 0) = (s_{i_1}, 1)|_{U_{i_0i_1}} \cdot (s_{i_0}, 1)|_{U_{i_0i_1}}^{-1} \]
as sections of $E$ which finishes the proof. $\square$
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