Let $A:=\mathbb{C}[t], M$ a finitely generated $A$ module. Denote by $m_\alpha$ the maximal ideal generated by $t-\alpha$ for $\alpha \in \mathbb{C}$, $S_\alpha$ the multiplicative set which is the complement of $m_\alpha$, $A_\alpha$ the local ring $S_\alpha^{-1}A$ and $M_\alpha$ the localized module $S_\alpha^{-1}M$. Suppose for some $\alpha \in \mathbb{C}$, $\mathrm{Tor}^1_{A_\alpha}(M_\alpha,\mathbb{C})=0$. Does it imply that $\mathrm{Tor}^1_{A_\beta}(M_\beta,\mathbb{C})=0$ for any $\beta \in \mathbb{C}$?
EDIT Assume further that the coherent sheaf associated to $M$ is supported on the whole of the affine line $\mathrm{Spec}(A)$ (by support I mean the definition mentioned in Hartshorne Ex. II.$5.6$)