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Asymptotic prime divisors over complete intersection rings

Published online by Cambridge University Press:  02 February 2016

DIPANKAR GHOSH  and
TONY J. PUTHENPURAKAL
DIPANKAR GHOSH
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. e-mail: dipankar@math.iitb.ac.in, tputhen@math.iitb.ac.in
TONY J. PUTHENPURAKAL
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. e-mail: dipankar@math.iitb.ac.in, tputhen@math.iitb.ac.in
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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

$$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$
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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 3 , May 2016 , pp. 423 - 436
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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