I read about for any separable morphism of non-singular varieties $f:X'\to X$, one can define a homomorphism $\text{Tr}:f\_\*(\Omega\_{X'}^q) \to \Omega\_{X}^q$,so that the map $\Omega\_{X}^q \to f\_\*(\Omega\_{X'}^q) $splits. But I didn't find a reference about how is it done? 



Trace map for separated morphism of non-singular varieties
If you do not mind a reference that contains a serious mistake, you can use the following.


MR1716049 (2000h:13016) Reviewed   

Zannier, Umberto   

A note on traces of differential forms. (English summary)   

J. Pure Appl. Algebra 142 (1999), no. 1, 91–97.   

13N05 (14F10)   

[article](http://www.sciencedirect.com/science/article/pii/S0022404998000450)


Part of the mistake is corrected in the following erratum.


MR2507247 (2010h:14007) Reviewed   

de Jong, A. J.(1-CLMB); Starr, Jason(1-SUNYS)   

Erratum: Cubic fourfolds and spaces of rational curves [MR2085418].   
 
Illinois J. Math. 52 (2008), no. 1, 345–346.   

14C05 (14E08) 


