Let $R$ be a commutative ring with unit. Vanishing of $\operatorname{Tor}_0(M,N)$ (see here) for two finitely generated $R$ modules $M$ and $N$ implies $\operatorname{Ann}M+ \operatorname{Ann}N=R$. Can we get something similar for vanishing of $\operatorname{Tor}_i(M,N)$ for $i\geq 1$?
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I suggest you clarify what you mean by `something similar'. For example, given any ideal $I$, $\mathrm{Tor}^1(R,R/I)=0$, but $\mathrm{Ann}, R+\mathrm{Ann}, R/I=I$. – Mohan Aug 12 '15 at 14:46
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@Mohan, I mean a ring theoretic condition like in the case of $Tor_0$ will be a good start. – Aug 12 '15 at 15:15
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I do not understand. In $\mathrm{Tor}^0$case there is no ring theoretic condition. So, what do you mean? – Mohan Aug 12 '15 at 15:38
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$Tor^{0}(M,N)$ vanishes implies $Ann M +Ann N=R$, which i call a ring theoretic condition. – Aug 12 '15 at 16:21
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If either $M$ or $N$ is flat (or, stronger condition, projective, or, even stronger condition, free, as in Mohan's example), then Tor$_i(M,N)=0$ for all $i\geq 1$. This seems like bad news, because it says that if one of $M$ and $N$ is nice, the other can be whatever you like. But on second thought, I guess that doesn't rule out the possibility of a nice answer. – Hugh Thomas Aug 12 '15 at 20:54
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@Hugh, you are right, i saw this and thought this is really bad news, but there can still be a "satisfying" answer. – Aug 13 '15 at 09:42