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Minimal injective resolutions with applications to dualizing modules and gorenstein modules

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References

  1. M. Auslander andM. Bridger, Stable module theory,Memoirs Amer. Math. Soc., no. 94 (1969).

  2. M. Auslander andD. Buchsbaum, Homological dimension in local rings, II,Trans. Amer. Math. Soc.,85 (1957), 390–405.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Auslander andO. Goldman, The Brauer group of a commutative ring,Trans. Amer. Math. Soc.,97 (1960), 367–409.

    Article  MathSciNet  Google Scholar 

  4. G. Azumaya, On maximally central algebras,Nagoya Math. J.,2 (1951), 119–150.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. M.-J. Bertin, Anneaux d’invariants d’anneaux de polynômes, en caractéristiquep, C. R. Acad. Sci. Paris,264 (1967), 653–656.

    MATH  MathSciNet  Google Scholar 

  7. H. Cartan andS. Eilenberg,Homological Algebra, Princeton University Press, 1956.

  8. I. S. Cohen, On the structure and ideal theory of complete local rings,Trans. Amer. Math. Soc.,59 (1946), 54–106.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. A. Eagon andD. G. Northcott, Ideals defined by matrices and a certain complex associated with them,Proc. Royal Soc. A,269 (1962), 188–204.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. A. Eagon, Examples of Cohen-Macaulay rings which are not Gorenstein,Math. Z.,109 (1969), 109–111.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Ferrand andM. Raynaud, Fibres formelles d’un anneau local noethérien,Ann. Sci. Éc. Norm. Sup. (4),3 (1970), 295–311.

    MATH  MathSciNet  Google Scholar 

  12. R. Fossum andP. Griffith, A complete local factorial ring of dimension 4 which is not Cohen-Macaulay,Bull. Amer. Math. Soc.,81 (1975), 111–113.

    MATH  MathSciNet  Google Scholar 

  13. R. Fossum, P. Griffith andI. Reiten, Trivial Extensions of Abelian Categories,Lecture Notes in Mathematics, No. 456, Berlin-Heidelberg-New York, Springer-Verlag, 1975.

    MATH  Google Scholar 

  14. H.-B. Foxby, On theμ i in a minimal injective resolution,Math. Scand.,29 (1971), 175–186.

    MathSciNet  Google Scholar 

  15. ——, Gorenstein modules and related modules,Math. Scand.,31 (1972), 367–384.

    MathSciNet  Google Scholar 

  16. ——,n-Gorenstein rings,Proc. Amer. Math. Soc.,42 (1974), 67–72.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Gabriel, Des catégories abéliennes,Bull. Soc. Math. France,90 (1963), 323–448.

    MathSciNet  Google Scholar 

  18. A. Grothendieck, Théorèmes de dualité pour les faisceaux algébriques cohérents,Séminaire Bourbaki, Exp. 149, Mai 1957.

  19. ——, Local Cohomology,Lecture Notes in Mathematics, No. 41, Berlin-Heidelberg-New York, Springer, 1967.

    MATH  Google Scholar 

  20. ——, Le groupe de Brauer, I,Dix Exposés sur la cohomologie des schémas, Amsterdam, North-Holland Pub. Co. (1969), 46–65.

    Google Scholar 

  21. T. Gulliksen, Massey operations and the Poincaré series of certain local rings,J. Algebra,22 (1972), 223–232.

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Hartshorne, Residues and duality,Lecture Notes in Mathematics, No. 20, Berlin-Heidelberg-New York, Springer, 1966.

    MATH  Google Scholar 

  23. R. Hartshorne andA. Ogus, On the factoriality of local rings of small embedding codimension,Communications in Algebra,1 (1974), 415–437.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Herzog andE. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings,Lecture Notes in Mathematics, No. 238, Berlin-Heidelberg-New York, Springer, 1971.

    MATH  Google Scholar 

  25. M. Hochster,Deep local rings, Preprint Series No. 8 (1973/74), Aarhus, Denmark, Aarhus Universitets Mathematiske Institut.

  26. Birger Iversen, Generic Local Structure in Commutative Algebra,Lecture Notes in Mathematics, No. 310, Berlin-Heidelberg-New York, Springer, 1973.

    Google Scholar 

  27. I. Kaplansky,Commutative Rings, Boston, Allyn and Bacon, Inc., 1970.

    MATH  Google Scholar 

  28. M.-A. Knus andM. Ojanguren, Sur quelques applications de la théorie de la descente à l’étude du groupe de Brauer,Comm. Math. Helv.,47 (1972), 532–542.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Matlis, Injective modules over noetherian rings,Pacific J. Math.,8 (1958), 511–528.

    MATH  MathSciNet  Google Scholar 

  30. H. Matsumura,Commutative Algebra, New York, W. A. Benjamin, Inc., 1970.

    MATH  Google Scholar 

  31. M. P. Murthy, A note on factorial rings,Arch. Math.,15 (1964), 418–420.

    Article  MATH  MathSciNet  Google Scholar 

  32. D. G. Northcott, Some remarks on the theory of ideals defined by matrices,Quart. J. Math. Oxford (2),14 (1963), 193–204.

    Article  MATH  MathSciNet  Google Scholar 

  33. C. Peskine andL. Szpiro, Dimension projective finie et cohomologie locale,Publ. Math. I.H.E.S.,42 (1973), 49–119.

    Google Scholar 

  34. I. Reiten, The converse to a theorem of Sharp on Gorenstein modules,Proc. Amer. Math. Soc.,32 (1972), 417–420.

    Article  MATH  MathSciNet  Google Scholar 

  35. I. Reiten andR. Fossum, Commutativen-Gorenstein rings,Math. Scand.,31 (1972), 33–48.

    MATH  MathSciNet  Google Scholar 

  36. J.-E. Roos, Sur les dérivés de lim. Applications,C. R. Acad. Sci. Paris,252 (1961), 3702–3704.

    MATH  MathSciNet  Google Scholar 

  37. R. Y. Sharp, Gorenstein modules,Math. Z.,115 (1970), 117–139.

    Article  MATH  MathSciNet  Google Scholar 

  38. ——, Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings,Proc. London Math. Soc.,25 (1972), 303–328.

    Article  MATH  MathSciNet  Google Scholar 

  39. ——, On Gorenstein modules over a complete Cohen-Macaulay local ring,Quart. J. Math. (Oxford) (2),22 (1971), 425–434.

    Article  MATH  MathSciNet  Google Scholar 

  40. ——, The Cousin complex for a module over a commutative Noetherian ring,Math. Z.,112 (1969), 340–356.

    Article  MATH  MathSciNet  Google Scholar 

  41. ——, The Euler characteristic of a finitely generated module of finite injective dimension,Math. Z.,130 (1973), 79–93.

    Article  MATH  MathSciNet  Google Scholar 

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Additional information

During the preparation of this paper the first, third and fourth authors were partially supported by the United States National Science Foundation.

The third author was supported by the Alfred P. Sloan Foundation and the fourth author by Norges Almenvitenskapelige Forskningsraad. We take this opportunity to thank the various universities which have provided hospitality, support and stimulating research atmosphere, namely Brandeis University, University of Illinois, Aarhus Universitet and Københavns Universitet.

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Fossum, R., Foxby, HB., Griffith, P. et al. Minimal injective resolutions with applications to dualizing modules and gorenstein modules. Publications Mathématiques de L’Institut des Hautes Scientifiques 45, 193–215 (1975). https://doi.org/10.1007/BF02684302

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  • DOI: https://doi.org/10.1007/BF02684302

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