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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: [email protected], [email protected]
TONY J. PUTHENPURAKAL
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. e-mail: [email protected], [email protected]
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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

REFERENCES

[1]Avramov, L. L. and Buchweitz, R.-O.Support varieties and cohomology over complete intersections. Invent. Math. 142 (2000), 285–318.Google Scholar
[2]Brodmann, M.Asymptotic stability of Ass(M/InM). Proc. Amer. Math. Soc. 74 (1979), 16–18.Google Scholar
[3]Eisenbud, D.Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260 (1980), 35–64.Google Scholar
[4]Gulliksen, T. H.A change of ring theorem with applications to Poincaré series and intersection multiplicity. Math. Scand. 34 (1974), 167–183.CrossRefGoogle Scholar
[5]Katz, D. and West, E.A linear function associated to asymptotic prime divisors. Proc. Amer. Math. Soc. 132 (2004), 1589–1597.Google Scholar
[6]Melkersson, L. and Schenzel, P.Asymptotic prime ideals related to derived functors. Proc. Amer. Math. Soc. 117 (1993), 935–938.CrossRefGoogle Scholar
[7]Puthenpurakal, T. J.On the finite generation of a family of Ext modules. Pacific J. Math. 266 (2013), 367–389.Google Scholar
[8]Vasconcelos, W. V.Cohomological degrees of graded modules. pp. 345–392 in Six lectures on commutative algebra (Bellaterra, 1996), edited by Elias, J.et al.Progr. Math. 166 (Birkhäuser, Basel, 1998).Google Scholar
[9]West, E.Primes associated to multigraded modules. J. Algebra 271 (2004), 427–453.CrossRefGoogle Scholar