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  • HAILONG DAO (a1) and PHAM HUNG QUY (a2)

Let $R$ be a commutative Noetherian ring of prime characteristic $p$ . In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$ -representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

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[1] Asadollahi, J. and Schenzel, P., Some results on associated primes of local cohomology modules , Jpn. J. Math. 29 (2003), 285296.
[2] Bahmanpour, K. and Quy, P. H., Localization at countably infinitely many prime ideals and applications , J. Algebra Appl. 15 (2016), 1650045 (6pages).
[3] Bhatt, B., Blickle, M., Lyubeznik, G., Singh, A. and Zhang, W., Local cohomology modules of a smooth ℤ-algebra have finitely many associated primes , Invent. Math. 197 (2014), 509519.
[4] Brodmann, M. and Faghani, A. L., A finiteness result for associated primes of local cohomology modules , Proc. Amer. Math. Soc. 128 (2000), 28512853.
[5] Brodmann, M. and Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.
[6] Hochster, M. and Núñez-Betancourt, L., Support of local cohomology modules over hypersurfaces and rings with FFRT , Math. Res. Lett. 24 (2017), 401420.
[7] Huneke, C., “ Problems on local cohomology ”, in Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, UT, 1990), Res. Notes Math. 2 , Jones and Bartlett, Boston, MA, 1992, 93108.
[8] Huneke, C., Katz, D. and Marley, T., On the support of local cohomology , J. Algebra 322 (2009), 31943211.
[9] Huneke, C. and Sharp, R. Y., Bass numbers of local cohomology modules , Trans. Amer. Math. Soc. 339 (1993), 765779.
[10] Katzman, M., An example of an infinite set of associated primes of a local cohomology module , J. Algebra 252 (2002), 161166.
[11] Kunz, E., Characterization of regular local rings for characteristic p , Amer. J. Math. 91 (1969), 772784.
[12] Lyubeznik, G., Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra) , Invent. Math. 113 (1993), 4155.
[13] Lyubeznik, G., F-modules: applications to local cohomology and D-modules in characteristic p > 0 , J. Reine Angew. Math. 491 (1997), 65130.
[14] Marley, T., The associated primes of local cohomology modules over rings of small dimension , Manuscripta Math. 104 (2001), 519525.
[15] Nagel, U. and Schenzel, P., “ Cohomological annihilators and Castelnuovo–Mumford regularity ”, in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, Contemp. Math. 159 , Amer. Math. Soc., Providence, RI, 1994, 307328.
[16] Núñez-Betancourt, L., Local cohomology properties of direct summands , J. Pure Appl. Algebra 216 (2012), 21372140.
[17] Patakfalvi, Z. and Schwede, K., Depth of F-singularities and base change of relative canonical sheaves , J. Inst. Math. Jussieu 13(1) (2014), 4363.
[18] Quy, P. H., A remark on the finiteness dimension , Comm. Algebra 41 (2014), 20482054.
[19] Quy, P. H. and Shimomoto, K., F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0 , Adv. Math. 313 (2017), 127166.
[20] Singh, A. K., p-torsion elements in local cohomology modules , Math. Res. Lett. 7 (2000), 165176.
[21] Singh, A. K. and Swanson, I., Associated primes of local cohomology modules and of Frobenius powers , Int. Math. Res. Not. IMRN 33 (2004), 17031733.
[22] Smith, K. E. and Van den Bergh, M., Simplicity of rings of differential operators in prime characteristic , Proc. Lond. Math. Soc. (3) 75 (1997), 3262.
[23] Takagi, S. and Takahashi, R., D-modules over rings with finite F-representation type , Math. Res. Lett. 15 (2008), 563581.
[24] Yao, Y., Modules with finite F-representation type , J. Lond. Math. Soc. (2) 72 (2005), 5372.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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