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 ?

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 $S$.

Question: Is the coherent sheaf $f^{*}G\otimes \omega^{n}_{X/S}$ torsion free fiber wise or on all of $X$?

We have a similar result in EGA3, \$6. but only if $\omega^{n}_{X/S}$ is flat sheaf over $S$...

Thank you.

  • $\begingroup$ Why is your earlier involving flatness and torsion-freeness now deleted? That was a very nice question, especially in view of Angelo's affirmative answer. Anyway, taking $N = A$ and $A$ a domain, you're asking if every finite $B$-module with $A$-depth at least 2 is torsion-free over $B$; that is clearly false with examples in which $B$ is a domain and $M$ a quotient of $B$. You need more hypotheses. $\endgroup$ – Boyarsky Jun 26 '10 at 12:45

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