Let $k$ be a field, $X$ and $Y$ two connected $k$-varieties, and $f:X\rightarrow Y$ a dominant projective morphism of relative dimension $d$.
I would like to know under which condition there is a natural "trace" morphism
$$R^df_*\Omega_{X/Y}^d\rightarrow \mathcal O_Y,$$
which is not necessarily an isomorphism, but that it is an isomorphism on the smooth fibers (if they exist).
I think that if $f$ is l.c.i. morphism (which I might be okay to assume), then there is a dualizing sheaf $\omega_{X/Y}$ and a trace isomorphism
$$R^df_*\omega_{X/Y}\rightarrow \mathcal O_Y.$$
In this situation, what is the relation between $\omega_{X/Y}$ and $\Omega_{X/Y}^d$? Is there a morphism between them?
Thank you in advance!
Best Answer
For reasonable schemes, there is always a sheaf $\omega_{X/Y}$ and a canonical isomorphism $$ \int_f \colon R^df_*\omega_{X/Y}\rightarrow \mathcal O_Y $$ See the paper by Kleiman,
The morphism $$ c_{X/Y} \colon \Omega_{X/Y}^d\rightarrow \omega_{X/Y} $$
is called the fundamental class. It is discussed in detail in the case that $X$ is an algebraic variety over $Y = \operatorname{Spec}(k)$ where $k$ is a perfect field in Lipman's "blue book":
In general, one uses the so called fundamental class in Hochschild homology. This is discussed in the paper (that uses the full machinery of Grothendieck duality and derived categories of sheaves)