An asymptotic bound for Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings

Set $A := Q/({\bf z})$, where $Q$ is a polynomial ring over a field, and ${\bf z} = z_1,\ldots,z_c$ is a homogeneous $Q$-regular sequence. Let $M$ and $N$ be finitely generated graded $A$-modules, and $I$ be a homogeneous ideal of $A$. We show that (1) $\mathrm{reg}\left( \mathrm{Ext}_A^{i}(M, I^nN) \right) \le \rho_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b \mbox{ for all } i, n \ge 0$, (2) $\mathrm{reg}\left( \mathrm{Ext}_A^{i}(M,N/I^nN) \right) \le \rho_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all } i, n \ge 0$, where $b$ and $b'$ are some constants, $f := \mathrm{min}\{ \mathrm{deg}(z_j) : 1 \le j \le c \}$, and $\rho_N(I)$ is an invariant defined in terms of reduction ideals of $I$ with respect to $N$. There are explicit examples which show that these inequalities are sharp.