Relative dualizing sheaf (References)

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do you know any reference for the relative dualizing sheaf of a family of curves? I´ve been looking through the literature and I can not find anything. Thank you for your time.

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Hartshorne's Residues and Duality is the classic original reference for Grothendieck duality. It has a number of technical issues, which are fixed in Brian Conrad's Grothendieck Duality and Base Change, available on the web for free from the author's website. These texts are very general - if you're looking for something more specific, please clarify your question.