9
$\begingroup$

I was wondering if there is any general theorem, which guarantees the flatness of $\omega_{X/B}$ over $B$ for a flat morphism $f : X \to B$ of schemes of finite type over $\mathbb{C}$ with equidimensional fibers. I am specially interested in statements which apply to the case of non-reduced $B$. Here by $\omega_{X/B}$ I mean $h^{-n}(f^! \mathcal{O}_B)$ where $n$ is the relative dimension of $f$ and $h^{-n}$ means the cohomology of the complex at the $-n$-th position.

$\endgroup$

2 Answers 2

8
$\begingroup$

A sufficient condition is that $f:X \rightarrow B$ is flat and locally finite type map between locally noetherian schemes such that its fibers are local complete intersection morphism (see EGA IV$\_4$, 19.2 for relative lci maps), in which case $\omega_{X/B}$ is even a line bundle on $X$ whose formation commutes with any base change on $B$. This covers the case of flat families of semistable curves, for example, and this lci condition on fibers is open on the base.

$\endgroup$
1
  • $\begingroup$ Thanks for the nice answer and for the precise reference. Is it, by any chance, known for the little more general case of Gorenstein morphisms? $\endgroup$ Mar 16, 2010 at 15:37
8
$\begingroup$

It sounds like you may want Exercise 9.7 in Hartshorne's "Residues and Duality". I paraphrase the statement:


Exercise 9.7 (RD): Let $f: X \to B$ be a flat morphism of finite type of locally Noetherian preschemes. Then, $f^!(\mathcal{O}_B)$ has a unique non-zero cohomology sheaf, which is flat over $B$, iff all the fibers of $f$ are Cohen-Macaulay schemes of the right dimension. Moreover $f^!(\mathcal{O}_B)$ is isomorphic to (a shift of) an invertible sheaf (on $X$) iff all the fibers of $f$ are Gorenstein schemes of the right dimension.


In particular, this addresses the case you mention in your comment ($f$ Gorenstein), since then $f^!(\mathcal{O}_Y)$ is locally free on $X$ and, since $f$ is flat, certainly flat over $B$.

[Aside: I believe I learned this reference from Brian's book "Grothendieck Duality and Base Change", which I think also contains a proof of this.]

$\endgroup$
2
  • $\begingroup$ Thanks again, so it seems everything works in the CM case. Now the new question is can we weaken it even more. What about $S_r$. I am specially interested in the $S_2$ case. $\endgroup$ Mar 16, 2010 at 18:29
  • 1
    $\begingroup$ Theorem 3.5.1 in my duality book is the reference for a solution to the Exercise 9.7. (When I wrote my original response I wanted to say the CM case is sufficient, but I didn't remember offhand if the flatness of that sheaf over the base also held. So I was lazy and just went with lci, which is certainly somewhat restrictive.) I would guess that assuming just $S_2$ is insufficient, but have no idea on a counterexample. $\endgroup$
    – BCnrd
    Mar 16, 2010 at 21:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.