Let R be a local noetherian ring and let M, N be two finitely generated modules.
Is it true that if $M \otimes_R N$ has finite length, then $Tor_i^R(M,N)$ also has finite length for all i?
I know a reference for a special case of this in Serre's book on local algebra. But I really don't want to assume R to be regular. My guess is that this is true as $supp(M \otimes^L N) = supp(M) \cap supp(N)$ and $supp(M \otimes^L N) = \bigcup_i supp Tor_i(M,N)$, but I can't make it precise.