Asymptotic prime divisors over complete intersection rings
  Let $A$ be a local complete intersection ring. Let $M,N$ be two finitely generated $A$-modules and $I$ an ideal of $A$. We prove that   \[   \bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}\mathrm{Ass}_A\left(\mathrm{Ext}_A^i(M,N/I^n N)\right)   \] is a finite set. Moreover, we prove that there exist $i_0,n_0\geqslant 0$ such that for all $i\geqslant i_0$ and $n \geqslant n_0$, we have   \[   \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right),   \]   \[   \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i+1}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right).   \]   We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity $\mathrm{cx}_A(M,N/I^nN)$ is constant for all sufficiently large $n$. 
Dipankar Ghosh and Tony J. Puthenpurakal
math.AC