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).
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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 $\omega_{X/S}$.

$\begingroup$ 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. $\endgroup$ – user052715 Jul 22 '15 at 14:29