The Stacks project

Proof. Let $U \subset X$ be an affine open mapping into an affine open of $S$. Then there is an isomorphism $K|_ U \to L|_ U$ by Lemma 48.28.2 and Dualizing Complexes, Lemma 47.27.2. The reader can reuse the argument of that lemma in the schemes case to obtain a proof in this case. We will instead use a glueing argument.

Suppose we have an isomorphism $\alpha : K \to L$. Then $\alpha (\xi ) = u \eta $ for some invertible section $u \in H^0(W, \Delta _*\mathcal{O}_ X) = H^0(X, \mathcal{O}_ X)$. (Because both $\eta $ and $\alpha (\xi )$ are generators of an invertible $\Delta _*\mathcal{O}_ X$-module by assumption.) Hence after replacing $\alpha $ by $u^{-1}\alpha $ we see that $\alpha (\xi ) = \eta $. Since the automorphism group of $K$ is $H^0(X, \mathcal{O}_ X^*)$ by Lemma 48.28.3 there is at most one such $\alpha $.

Let $\mathcal{B}$ be the collection of affine opens of $X$ which map into an affine open of $S$. For each $U \in \mathcal{B}$ we have a unique isomorphism $\alpha _ U : K|_ U \to L|_ U$ mapping $\xi $ to $\eta $ by the discussion in the previous two paragraphs. Observe that $\text{Ext}^ i(K|_ U, K|_ U) = 0$ for $i < 0$ and any open $U$ of $X$ by Lemma 48.28.3. By Cohomology, Lemma 20.45.2 applied to $\text{id} : X \to X$ we get a unique morphism $\alpha : K \to L$ agreeing with $\alpha _ U$ for all $U \in \mathcal{B}$. Then $\alpha $ sends $\xi $ to $\eta $ as this is true locally. $\square$