Asymptotic depth of ext modules over complete intersection rings (Q2236789)

Let \(Q\) be a Noetherian ring of finite Krull dimension, and let \(f = f_1,\ldots, f_c\) be a \(Q\)-regular sequence. Set \(A=Q/(f)\). Suppose \(M\) is a finitely generated \(A\)-module with \(\mathrm{projdim}_Q(M)<\infty\) and \(I\) is an ideal of \(A\). Let \(N=\bigoplus_{n\geq0}N_n\) be a finitely generated \(\mathbb{R}(I)\)-module, where \(\mathbb{R}(I)=\bigoplus_{n\geq0}I^nt^n\) is the Rees ring of \(A\) with respect to \(I\). Let \(J\) be an ideal of \(A\). In the present paper, the authors prove that \(\mathrm{grade}(J,\mathrm{Ext}^{2i+t}_A(M, N_n))\) is constant for every fixed \(t=0,1\) and all sufficiently large \(n,i\). They also prove that if \(I\) is an ideal of a local complete intersection ring \(A\) and \(M, N\) are finitely generated \(A\)-modules, then for \(t=0,1\), the values \(\mathrm{depth\, Ext}^{2i+t}(M,N/I^nN)\) become independent of \(i, n\) for all sufficiently large \(n,i\). Finally, they show that if \(\mathfrak{p}\) is a prime ideal in \(A\), then the \(j\)-th Bass numbers \(\mu_j(\mathfrak{p}, \mathrm{Ext}^{2i+t}(M,N/I^nN)\) have polynomial growth in \((n,i)\) with rational coefficients for all sufficiently large \((n,i)\).