Remark 53.4.3. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 53.4.1. If $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee $ then we have canonical isomorphisms

This follows from the lemma and Cohomology, Lemma 20.48.5. If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then we have canonical isomorphisms

by Lemma 53.4.2. In particular if $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then we have

and

## Comments (2)

Comment #7163 by Xuande Liu on

Comment #7305 by Johan on