Let f:X→Y be a proper morphism. In section 6.4. of Liu’s book he introduces the r-dualizing sheaf ωf for f which satisfies
for all quasi-coherent sheaves F on X.
In the special case of f being finite he proves that the 0-dualizing sheaf is given by f!OY=HomOY(f∗OX,OY) where this is considered an OX-module via multiplication into the argument.
Let X be a proper, one-dimensional scheme over the field k and let Y=P1k. Let ωf denote the 0-dualizing sheaf for f. Let Z denote an irreducible component of X. Let j:Z→X denote the corresponding closed immersion. Then f∘j is again finite and we denote its dualizing sheaf by ωf∘j.
My question is: How is the restriction of ωf to Z (via the pullback j∗) related to the 0-dualizing sheaf ωf∘j for the morphism f∘j?
Are they isomorphic? In general, I don’t think so. But maybe in specific cases they are. Do there exist canonical maps between them?
I am grateful for any kind of help.