Hi.
Question 1: If $f:A\rightarrow B$ be a morphism of local noetherian rings with $B$ is $A$-flat. Let $M$ (resp. $N$) be a $B$ (resp. $A$-)-module of finite type (fin. generated). We assume that $depth_{A}(M)\geq 2$, $M$ is $B$-torsion free and N is $A$-torsion free. Then it is true that $N\otimes_{A}M$ is torsion free ?
I remind the 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 $depth>1$ and torsion free on $X$, $G$ torsion free coherent sheaf on $S$.
Question: Is the coherent sheaf $f^{*}G\otimes \omega^{n}_{X/S}$ (which is generically torsion free) torsion free fiber wise or on all of X?
Remark: We can reduce this question to smooth $f$ and especially to the projection $S\times U\rightarrow S$ and replace $\omega^{n}_{X/S}$ by torsion free coherent sheaf on $S\times U$ which is of $depth>1$ fiber wise…
Thank you.
P.S: Thanks to Boyarski for his remark. The last question on flatness and torsion freeness is not deleted but in another count of kaddar with the same name "kaddar".
Best Answer
Tensor products of non-free modules typically will not be torsion-free, even if you assume good depth conditions on the modules.
In discussing the counter-example I will assume $B=A$ and depth $A$ at least $2$. Let $(m,k)$ be the maximal ideal and residue field respectively. Let $M$ be the first syzygy of $m$ (second syzygy of $k$). Then $\text{depth}_A(M)=2$ and $M$ is certainly torsion-free (being a submodule of a free module). Let $N$ be a torsion-free f.g $A$-module.
Claim: $M\otimes_AN$ is torsion free iff $\text{pd}_AN\leq 1$.
Proof: Tensor the exact sequence $0\to M\to F \to m\to 0$ with $N$ we get an exact sequence: $$0 \to \text{Tor}_1^A(m,N)\to M\otimes N \to F\otimes N$$
$\text{Tor}_1^A(m,N) = \text{Tor}_2^A(k,N)$ is killed by $m$, so it is the torsion part of $M\otimes N$ (note the quotient embeds into the torsion -free $F\otimes N$). Thus, the tensor product is torsion-free iff $\text{Tor}_2^A(k,N)=0$, which forces $\text{pd}_AN\leq 1$.
This also shows that regarding the second question, one can not improve much from the EGA result, since $\text{pd}_A\omega_A <\infty$ forces the canonical module to be free.
PS: you should merge your accounts and link to your other questions, for the benefits of the readers.