ag.algebraic geometry - Comparison of Local and Global Duality

In the case of varieties over a perfect field this question is explained with great detail in the book by J. Lipman:

Dualizing sheaves, differentials and residues on algebraic varieties. (French summary)
Astérisque No. 117 (1984)

known by some people as "Lipman's blue book". If you want to get rid of the base field then you should look at Greenlees May duality for schemes and Grothedieck duality for formal schemes, see joint work by Alonso, Jeremías & Lipman.