Asymptotic prime divisors over complete intersection rings

2014/03/31 by Dipankar Ghosh, Tony J. Puthenpurakal · 1 citation
Mathematics · #math.AC #msc:13A02 #msc:13A15 #msc:13D07 #msc:13H10

paper · pdf · doi:10.1017/s0305004115000778

published as Math. Proc. Camb. Phil. Soc. 160 (2016) 423-436 · 17 pages, final version

arxiv created 2016/11/11 · arxiv updated 2016/11/14

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}Ass_A(Ext_Aⁱ(M,N/Iⁿ N)) is a finite set. Moreover, we prove that there exist i₀,n₀\geqslant 0 such that for all i\geqslant i₀ and n \geqslant n₀, we have Ass_A(Ext_A²ⁱ(M,N/IⁿN)) = Ass_A(Ext_A^{2 i₀}(M,N/I^n₀N)), Ass_A(Ext_A²ⁱ⁺¹(M,N/IⁿN)) = Ass_A(Ext_A^{2 i₀ + 1}(M,N/I^n₀N)). We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity cx_A(M,N/IⁿN) is constant for all sufficiently large n.

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• Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules

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