Remark 28.28.4. With assumptions and notation of Lemma 28.28.3. Denote the displayed map of the lemma by $\theta _\mathcal {F}$. Note that the isomorphism $f^*\mathcal{O}_ Y(n) \to \mathcal{L}^{\otimes n}$ of Lemma 28.28.2 is just $\theta _{\mathcal{L}^{\otimes n}}$. Consider the multiplication maps

$\widetilde{M} \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(n) \longrightarrow \widetilde{M(n)}$

see Constructions, Equation (27.10.1.5). Pull this back to $X$ and consider

$\xymatrix{ f^*\widetilde{M} \otimes _{\mathcal{O}_ X} f^*\mathcal{O}_ Y(n) \ar[r] \ar[d]_{\theta _\mathcal {F} \otimes \theta _{\mathcal{L}^{\otimes n}}} & f^*\widetilde{M(n)} \ar[d]^{\theta _{\mathcal{F} \otimes \mathcal{L}^{\otimes n}}} \\ \mathcal{F} \otimes \mathcal{L}^{\otimes n} \ar[r]^{\text{id}} & \mathcal{F} \otimes \mathcal{L}^{\otimes n} }$

Here we have used the obvious identification $M(n) = \Gamma _*(X, \mathcal{L}, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. This diagram commutes. Proof omitted.

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