Fiberwise torsion free and generically null sheaf for flat morphism - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:54:20Z http://mathoverflow.net/feeds/question/33641 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33641/fiberwise-torsion-free-and-generically-null-sheaf-for-flat-morphism Fiberwise torsion free and generically null sheaf for flat morphism kaddar 2010-07-28T10:42:39Z 2011-06-30T07:22:12Z <p>Hi.</p> <p>Has some one an example of sheaf $A$ on flat morphism $f:X\rightarrow S$ of reduced complex spaces with fibers of constant positive dimension (or locally noetherian excellent schemes without embedded components) which satisfies the properties :</p> <p>1) there is a dense open subset $V$ of $X$ (smooth or Cohen-Macaulay locus) on which $A$ is canonically null,</p> <p>2) For every subset $F$ s.t $F\cap f^{-1}(s)$ has empty interior in $f^{-1}(s)$, we have ${\cal H}^{0}_{F}(A) = 0$,</p> <p>3) There is some fibers on which the restriction of $A$ is not trivialy null.</p> <p>Thank you.</p> http://mathoverflow.net/questions/33641/fiberwise-torsion-free-and-generically-null-sheaf-for-flat-morphism/33653#33653 Answer by kaddar for Fiberwise torsion free and generically null sheaf for flat morphism kaddar 2010-07-28T13:55:04Z 2010-07-28T13:55:04Z <p>Dear Brian,</p> <p>I dont know how to add comments!</p> <p>A is a coherent sheaf on $X$ which is null on dense open subset containing the smooth locus of $f$ and $F\cap X_{s}$ is nowhere dense in $X_{s}$.</p> <p>The motivation is giving by the question:</p> <p>for $f:X\rightarrow S$ flat morphism of reduced complex spaces with purely $n$-dimensional fibers and $S$-flat relative canonical sheaf $\omega^{n}_{X/S}$, is the canonical morphism $$\Theta:f{*}G\otimes \omega^{n}_{X/S}\rightarrow {\cal H}^{-n}(f^{!}G)$$</p> <p>injectiv (or $S$-injective) for all torsion free coherent sheaf $G$ on $S$ ?</p>