On the finite generation of a family of Ext modules

Let $Q$ be a Noetherian ring with finite Krull dimension and let $mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $projdim_Q M$ finite. Set $R = bigoplus_{ngeq 0}I^n$, the Rees-Algebra of $I$. Let $N = bigoplus_{j geq 0}N_j$ be a finitely generated graded $R$-module. We show that [bigoplus_{jgeq 0}bigoplus_{igeq 0} Ext^{i}_{A}(M,N_j) ] is a finitely generated bi-graded module over $Sc = R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings.