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On a category of cofinite modules which is Abelian

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Abstract

Let A be a noetherian local ring, and I an ideal of A of dimension one. Our purpose of this paper is to prove that the category \({\mathcal{M} (A, I)_{cof}}\) of cofinite modules is Abelian. Consequently, this assertion answers affirmatively the questions given by Hartshorne (Invent Math 9:147, 1970; cf. Sect. 2. Four Questions and One theorem) for an ideal of dimension one over a local ring.

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References

  1. Atiyah M.F., Macdonald I.G.: Introduction to Commutative Algebra, Addison-Wesley Series in Mathematics. Addison-Wesley publishing company, Reading (1969)

    Google Scholar 

  2. Divaani-Aazar K., Sazeedeh R.: Cofiniteness of generalized local cohomology modules. Colloq. Math. 99((2), 283–290 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bass H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourbaki N.: Algèbra Commutative, Chapters 1 and 2. Hermann, Paris (1961)

    Google Scholar 

  5. Delfino D.: On cofiniteness of local cohomology modules. Math. Proc. Camb. Philos. Soc. 115(1), 79–84 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delfino D., Marley T.: Cofinite modules and local cohomology. J. Pure Appl. Algebra 121(1), 45–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gelfant S.I., Manin Yu.I.: Methods of Homological Algebra. Springer, Berlin (1996)

    Google Scholar 

  8. Grothendieck A.: Cohomologie locale des faisceaux cohérants et théorèmes de Lefschetz locaux et globaux (SGA 2). Amsterdam, North-Holland (1968)

    Google Scholar 

  9. Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9, 145–164 (1969/1970)

    Google Scholar 

  10. Hartshorne R.: Residue and Duality. Springer Lecture note in Mathematics, No. 20. Springer, New York (1966)

    Google Scholar 

  11. Hartshorne R.: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer, New York (1977)

    Google Scholar 

  12. Huneke C., Koh J.: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110(3), 421–429 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kawakami S., Kawasaki K.-i.: On the finiteness of Bass numbers of generalized local cohomology modules. Toyama Math. J. 29, 59–64 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Kawasaki K.-i.: On the finiteness of Bass numbers of local cohomology modules. Proc. Am. Math. Soc. 124, 3275–3279 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kawasaki K.-i.: Cofiniteness of local cohomology modules for principal ideals. Bull. Lond. Math. Soc. 30, 241–246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kawasaki K.-i.: On finiteness properties of local cohomology modules over Cohen-Macaulay local rings. Ill. J. Math. 52(3), 727–744 (2008)

    MathSciNet  Google Scholar 

  17. Kawasaki, K.-i.: On a characterization of confinite complexes, preprint

  18. Lipman, J.: Lectures on Local Cohomology and Duality, Local Cohomology and its Applications. Lecture notes in Pure and Applied Mathematics, vol. 226, pp. 39–89. Marcel Dekker, Inc., New York (2002)

  19. Matsumura H.: Commutative Algebra, 2nd edn. Benjamin/Cummings, Reading (1980)

    MATH  Google Scholar 

  20. Matsumura H.: Commutative Ring Theory, Cambridge Studies in Advance Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  21. Melkersson L.: Properties of cofinite modules and applications to local cohomology. Math. Proc. Camb. Philos. Soc. 125(3), 417–423 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Melkersson L.: Modules cofinite with respect to an ideal. J. Algebra 285(2), 649–668 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roberts P.: Homological Invariants of Modules over Commutative Rings. Séminaire de Mathématiques Supérieures, vol. 72 , Les Presses de l’Université de Montréal, Montréal (1980)

  24. Rotman J.: An introduction to homological algebra. Pure and applied mathematics, vol. 226. Academic press, Inc., Harcourt Brace Company, Boston (1979)

    Google Scholar 

  25. Yassemi S.: Cofinite modules. Commun. Algebra 29(6), 2333–2340 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yoshida K.-i.: Cofiniteness of local cohomology modules for dimension one ideals. Nagoya J. Math. 147, 179–191 (1995)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nara University of Education, Takabatake-cho, Nara, 630-8528, Japan

    Ken-ichiroh Kawasaki

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  1. Ken-ichiroh Kawasaki
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Correspondence to Ken-ichiroh Kawasaki.

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To the memory of Professor Yokitoshi Honohara.

K.-i. Kawasaki was supported in part by grants from the Grant-in-Aid for Scientific Research (C) \({\sharp}\) 20540043.

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Kawasaki, Ki. On a category of cofinite modules which is Abelian. Math. Z. 269, 587–608 (2011). https://doi.org/10.1007/s00209-010-0751-0

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