If $R$ is a commutative noetherian ring and $I$ is an ideal of $R$, $M$ is an $R$module. Does $Tor_i^R(R/I, M)$ is finitely generated for $i\ge 0$ imply $Ext^i_R(R/I, M)$ is finitely generated for $i\ge 0$?

In fact, the two are equivalent. My apologies for the length of this argument  if someone else has a shorter one, I'd be happy to hear it. Let $(a_1,\ldots,a_k) = I$, and let $K_\bullet$ be the Koszul complex associated to this set of generators. Note that its zero'th homology group is $R/I$, and all the homology groups are finitely generated $R/I$modules because $R$ is Noetherian. Let C be a Serre class of Rmodules (i.e. one such that for any $0 \to A' \to A \to A'' \to 0$ exact, $A$ is in $C$ if and only if $A'$ and $A''$ are both in C). The result you ask is obtained by letting C be the class of finitely generated $R$modules (which is only a Serre class because $R$ is Noetherian). We have that the following are equivalent for an Rmodule M:
The implication 1 => 2 follows inductively by writing $N$ in a short exact sequence $0 \to J \to \oplus R/I \to N \to 0$ and applying the long exact sequence of Tor. The implication 2 => 3 follows from a hyperhomology spectral sequence. The implication 3 => 4 is immediate from the definition of the Koszul complex. The implication 4 => 1 is proved inductively. The hyperhomology spectral sequence $$ E^2_{p,q} = Tor_p(H_q(K), M) \Rightarrow H_{p+q}(K \otimes_R M) $$ first shows $E^2_{0,0} = Tor_0(R/I,M)$ is in C. If $Tor_i(R/I,M)$ is in C for $0 \leq i \leq m$, the above argument implies that $Tor_i(N,M)$ is in C for all finitely generated $N$, which forces $E^2_{p,q}$ to be in C for all $p \leq m$. As the abutment is in C, this forces $E^2_{m+1,0} = Tor_{m+1}(R/I,M)$ to be in C. Now, there is an exactly analogous string of implications in Ext. The following are equivalent:
However, the Koszul complex has selfduality; the tensor product complex $K \otimes_R M$ visibly has the same homology groups as a shift of the Homcomplex $\underline{Hom}_R(K,M)$. Therefore, the two versions of statement (4) are equivalent. 

