The Stacks project

Lemma 109.19.3. Let $X \to S$ be a family of curves with Gorenstein fibres equidimensional of dimension $1$ (Lemma 109.12.2). Then the relative dualizing sheaf $\omega _{X/S}$ is an invertible $\mathcal{O}_ X$-module whose formation commutes with arbitrary base change.

Proof. This is true because the pullback of the relative dualizing module to a fibre is invertible by the discussion above. Alternatively, you can argue exactly as in the proof of Lemma 109.19.1 and deduce the result from Duality for Schemes, Lemma 48.25.10. $\square$

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