Asymptotic prime divisors over complete intersection rings

Abstract

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 $i_0,n_0⩾0$ such that for all $i⩾i_0$ and $n⩾n_0$, 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 ${\mathrm{cx}}_A(M,N/I^{n}N)$ is constant for all sufficiently large $n$.

17 pages, final version

Cited by in corpus (2)

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