Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

2 Answers 2

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.

share|improve this answer
    
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? –  Zsolt Patakfalvi Mar 16 '10 at 15:37
add comment

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.]

share|improve this answer
    
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. –  Zsolt Patakfalvi Mar 16 '10 at 18:29
    
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. –  BCnrd Mar 16 '10 at 21:12
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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