Let $\mathcal{C}\rightarrow S$ a flat projective family of locally complete intersection projective curves over a integral noetherian scheme (say a spectrum of a local ring). I was wondering whether there was a simple (without derived category) way to construct the "relative dualizing sheaf" $\omega_{\mathcal{C}/S}$; specially whether it was an invertible sheaf and whether its restriction to (geometric) fibers (generic or special)was the usual dualizing sheaf (of locally complete intersection).
See the paper
Kleiman, Steven L. Relative duality for quasicoherent sheaves. Compositio Math. 41 (1980), no. 1, 39–60.
You'll find a detailed nonderived construction and a verification of the main properties of $\omega_{X/S}$.

Thank you very much. I think the section on duality theory of Qing Liu's Algebraic geometry and arithmetic curves is also a good reference. – user052715 Jul 22 '15 at 14:29