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Localization at countably infinitely many prime ideals and applications. (English) Zbl 1353.13017

In the present paper, authors gave a good flat morphism between Noetherian rings. That is, let \(R\) be a Noetherian ring and \(\mathfrak p_1\), \(\mathfrak p_2\), …are countably many prime ideals such that \(\mathfrak p_i \not\subset \mathfrak p_j\) whenever \(i \neq j\). Then there is a flat \(R\)-algebra \(S\) such that \(\mathfrak p_1 S\), \(\mathfrak p_2 S\), …are all the maximal ideals of \(S\).
Authors used it to study associated primes of local cohomology modules. Let \(R\) be a Noetherian ring, \(I\) an ideal and \(M\) a finitely generated \(R\)-module. It is not known whether \(\text{Ass} H_I^i(M)\) is a finite set. However they showed that \(\# \{\mathfrak p \in \text{Ass} H_I^i(M) \mid \text{ht} \mathfrak p/I \leq 1\} < \infty\).

MSC:

13D45 Local cohomology and commutative rings
13B30 Rings of fractions and localization for commutative rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
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References:

[1] 1. K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra321 (2009) 1997-2011. genRefLink(16, ’S0219498816500456BIB001’, ’10.1016 · Zbl 1168.13016
[2] 2. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge University Press, Cambridge, 1998). genRefLink(16, ’S0219498816500456BIB002’, ’10.1017 · Zbl 0903.13006
[3] 3. D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra121 (1997) 45-52. genRefLink(16, ’S0219498816500456BIB003’, ’10.1016
[4] 4. A. Grothendieck, Local Cohomology, Notes by R. Hartshorne, Lecture Notes in Mathematics, Vol. 862 (Springer, New York, 1966).
[5] 5. R. Hartshorne, Affine duality and cofiniteness, Invent. Math.9 (1970) 145-164. genRefLink(16, ’S0219498816500456BIB005’, ’10.1007
[6] 6. C. Huneke, D. Katz and T. Marley, On the support of local cohomology, J. Algebra322 (2014) 3194-3211. genRefLink(16, ’S0219498816500456BIB006’, ’10.1016
[7] 7. C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc.110 (1991) 421-429. genRefLink(16, ’S0219498816500456BIB007’, ’10.1017
[8] 8. K.-I. Kawasaki, On the finiteness of Bass numbers of local cohomology module, Proc. Amer. Math. Soc.124 (1996) 3275-3279. genRefLink(16, ’S0219498816500456BIB008’, ’10.1090 · Zbl 0860.13011
[9] 9. G. Leuschke and R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs, Vol. 181 (American Mathematical Society, Providence, RI, 2012).
[10] 10. H. Matsumura, Commutative Ring Theory (Cambridge University Press, Cambridge, 1986). · Zbl 0603.13001
[11] 11. L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra285 (2005) 649-668. genRefLink(16, ’S0219498816500456BIB011’, ’10.1016
[12] 12. L. Melkersson, Cofiniteness with respect to ideals of dimension one, J. Algebra372 (2012) 459-462. genRefLink(16, ’S0219498816500456BIB012’, ’10.1016
[13] 13. K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J.147 (1997) 179-191. genRefLink(128, ’S0219498816500456BIB013’, ’A1997YD35100010’); · Zbl 0899.13018
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