2
$\begingroup$

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!

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

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,

Relative duality for quasicoherent sheaves Kleiman, Steven L. Compositio Math. 41 (1980), no. 1, 39–60. (NUMDAM)

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":

Dualizing sheaves, differentials and residues on algebraic varieties Lipman, Joseph Astérisque(1984), no. 117

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)

Bivariance, Grothendieck duality and Hochschild homology I: Construction of a bivariant theory
Alonso-Tarrío, Leovigildo; Jeremías-López, Ana; Lipman, Joseph; Asian J. Math. 15 (2011), no. 3, 451–497.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you very much, this is exactly what I was looking for! $\endgroup$
    – Emiliano Ambrosi
    Commented Nov 29, 2023 at 17:41
  • $\begingroup$ I have just one further question: do you know how does the map H^1(X,\Omega^1_{X/k})---> k looks in some concrete situation? For example, I can image what is it if X is the union of two P^1, but I can't guess what the map does if X is a double line for example (i.e. x^2=0 inside P^2). Is it the composition of the map to the reduced part and the trace of this one? $\endgroup$
    – Emiliano Ambrosi
    Commented Nov 29, 2023 at 17:44
  • $\begingroup$ Basically you take a Noether normalization $X \to P^1 \to \operatorname{Spec}(k)$ and then they integral on $X$ is the composition of the trace from $X$ to $P^1$ and the integral on $P^1$ that you can compute with Czech cocycles. $\endgroup$
    – Leo Alonso
    Commented Nov 29, 2023 at 19:13

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .