Thursday, 7 June 2012

ag.algebraic geometry - Flatness of relative canonical bundle

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:XtoB be a flat morphism of finite type of locally Noetherian preschemes. Then, f!(mathcalOB) 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!(mathcalOB) 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!(mathcalOY) 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.]

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