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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