# Relative dualizing sheaf (reference, behavior)

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).

You'll find a detailed non-derived construction and a verification of the main properties of $\omega_{X/S}$.