torsion freeness of tensor product - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:52:13Z http://mathoverflow.net/feeds/question/29588 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29588/torsion-freeness-of-tensor-product torsion freeness of tensor product kaddar 2010-06-26T08:10:02Z 2010-06-26T08:10:02Z <p>Hi.</p> <p>Let $f:A\rightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)\geq 2$ and $N$ is torsion free. Then it is true that $N\otimes _{A}M$ is torsion free ?</p> <p>Motivation: Let $f:X\rightarrow S$ be a proper and flat morphism of reduced finite dimensional complex spaces with n-dimensional fibers. Let $\omega^{n}_{X/S}$ be the canonical relative sheaf (which is fiber wise of prof >1) and $G$ torsion free coherent sheaf on $S$.</p> <p>Question: Is the coherent sheaf $f^{*}G\otimes \omega^{n}_{X/S}$ torsion free fiber wise or on all of $X$?</p> <p>We have a similar result in EGA3, \$6. but only if$\omega^{n}_{X/S}$is flat sheaf over$S\$...</p> <p>Thank you.</p>