Let $f: X \to Z$ and $g: Y \to Z$ be smooth morphisms of smooth projective varieties. Consider the fibered product \begin{array}{ccc} X \times_Z Y &\stackrel{\tilde{f}}{\longrightarrow}& Y\\ \downarrow^{\tilde{g}}&&\downarrow^{g} \\ X& \stackrel{f}{\longrightarrow}& Z. \end{array} How one can compute canonical bundle of fibered product $X'=X \times_Z Y$?
My hunch is that $\Omega^1_{X'}$ should be fibered product in the category of coherent sheaves on $X'$ : \begin{array}{ccc} \Omega^1_{X'} &\to&{\tilde{f}^*}\Omega^1_Y\\ \downarrow &&\downarrow \\ \tilde{g}^* \Omega^1_X &\to&{(g\tilde{f})^*}\Omega^1_Z \end{array}
Is it true? Even if this is true how one can compute determinant of a vector bundle given as a fibered product?
I don't know if this is good enough, but you could get some different expressions using the following facts:
(1) One has $\Omega_{X \times_Z Y/Z} = \Omega_{X/Z} \boxplus \Omega_{Y/Z}$ (pull back to the product and take the direct sum).
(2) In general, if $X \xrightarrow{f} Y \xrightarrow{g} Z$ are morphism with $f$ smooth (to get an injection on the left) then there is an exact sequence $$ 0 \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0. $$
(3) If $\mathscr{F}' \to \mathscr{F} \to \mathscr{F}''$ is a short exact sequence of finite rank locally free sheaves then $\det \mathscr{F} \simeq \det \mathscr{F}' \otimes \det \mathscr{F}''$.
It seems to me that using these you can express the differentials on the product using the differentials on the original three schemes together with some knowledge of the pullbacks.