Asymptotic prime divisors over complete intersection rings

Abstract

AbstractLet 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 $$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$ is a finite set. Moreover, we prove that there exist i0, n0 ⩾ 0 such that for all i ⩾ i0 and n ⩾ n0, we have $$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$ We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity cxA(M, N/InN) is constant for all sufficiently large n.