It is hard for me not to mention the following splendid result by Auslander-Lichtenbaum:

If $R$ is regular local with Krull dimension $d$, then for any finitely generated module $M$, $M^{\otimes n}$ is torsion-free for some $n\geq d$ if and only if $M$ is flat.

Auslander only proved it for unfamified regular local rings (see the references here) but the key ingredient for the proof, namely certain Tor-rigidity property for finitely generated modules was later completed by Lichtenbaum.

Very recently, this was generalized to some extend to non-finitely generated case in this paper.

I believe Auslander's theorem extends to the case when $R$ is an isolated hypersurface singularity over a char. $0$ field and $d =\dim R$ is even (because the key ingredients are now available for that case).

It is hard for me not to mention the following splendid result by Auslander-Lichtenbaum:

If $R$ is regular local with Krull dimension $d$, then for any finitely generated module $M$, $M^{\otimes n}$ is torsion-free for some $n\geq d$ if and only if $M$ is flat.

Auslander only proved it for unfamified regular local rings (see the references here) but the key ingredient for the proof, namely certain Tor-rigidity property for finitely generated modules was later completed by Lichtenbaum.

Very recently, this was generalized to some extend to non-finitely generated case in this paper.

I believe Auslander's theorem extends to the case when $R$ is an isolated hypersurface singularity over a char. $0$ field and $d =\dim R$ is even (because the key ingredients are now available for that case).

It is hard for me not to mention the following splendid result by Auslander-Lichtenbaum:

If $R$ is regular local with Krull dimension $d$, then for any finitely generated module $M$, $M^{\otimes n}$ is torsion-free for some $n\geq d$ if and only if $M$ is flat.

Auslander only proved it for unfamified regular local rings (see the references here) but the key ingredient for the proof, namely certain Tor-rigidity property for finitely generated modules was later completed by Lichtenbaum.

Very recently, this was generalized to some extend to non-finitely generated case in this paper.

I believe Auslander's theorem extends to the case when $R$ is an isolated hypersurface singularity over a char. $0$ field and $d =\dim R$ is even (because the key ingredients are now available for that case).

It is hard for me not to mention the following splendid result by Auslander-Lichtenbaum:

If $R$ is regular local with Krull dimension $d$, then for any finitely generated module $M$, $M^{\otimes n}$ is torsion-free for some $n\geq d$ if and only if $M$ is flat.

Auslander only proved it for unfamified regular local rings (see the references here) but the key ingredient for the proof, namely certain Tor-rigidity property for finitely generated modules was later completed by Lichtenbaum.

Very recently, this was generalized to some extend to non-finitely generated case in this paper.

I believe Auslander's theorem extends to the case when $R$ is an isolated hypersurface singularity over a char. $0$ field and $d =\dim R$ is even (because the key ingredients are now available for that case).