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THE STRUCTURE OF BALANCED BIG COHEN–MACAULAY MODULES OVER COHEN–MACAULAY RINGS

Part of: Commutative algebra: Homological methods Theory of modules and ideals

Published online by Cambridge University Press:  10 June 2016

HENRIK HOLM
HENRIK HOLM*
Affiliation:
Department of Mathematical Sciences, Universitetsparken 5, University of Copenhagen, 2100 Copenhagen Ø, Denmark e-mail: [email protected]
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Abstract

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Over a Cohen–Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a “structure theorem” for balanced big CM modules. (2) Every finitely generated module has a pre-envelope with respect to the class of finitely generated maximal CM modules. This result is, in some sense, dual to the existence of maximal CM approximations, which has been proved by Auslander and Buchweitz.

MSC classification

Primary: 13C14: Cohen-Macaulay modules
Secondary: 13D05: Homological dimension 13D07: Homological functors on modules (Tor, Ext, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 549 - 561
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Adámek, J. and Rosický, J., Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
2. Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.), vol. 38 (Gauthier-Villars, Paris, 1989), 5–37, Colloque en l'honneur de Pierre Samuel (Orsay, 1987).Google Scholar
3. Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
4. Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993).Google Scholar
5. Christensen, L. W., Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747 (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
6. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22 (5) (1994), 1641–1674.CrossRefGoogle Scholar
7. Eckmann, B. and Schopf, A. H., Über injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78.CrossRefGoogle Scholar
8. Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer-Verlag, New York, 1995).Google Scholar
9. Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, de Gruyter Expositions in Mathematics, vol. 30 (Walter de Gruyter & Co., Berlin, 2000).CrossRefGoogle Scholar
10. Enochs, E. E., Jenda, O. M. G. and Torrecillas, B., Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (3) (1993), 1–9.Google Scholar
11. Enochs, E. E., Jenda, O. M. G. and Xu, J., Foxby duality and gorenstein injective and projective modules, Trans. Amer. Math. Soc. 348 (8) (1996), 3223–3234.CrossRefGoogle Scholar
12. Fossum, R. M., Griffith, P. A. and Reiten, I., Trivial extensions of abelian categories Homological algebra of trivial extensions of abelian categories with applications to ring theory, Lecture Notes in Mathematics, vol. 456 (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar
13. Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules, de Gruyter Expositions in Mathematics, vol. 41 (Walter de Gruyter GmbH & Co. KG, Berlin, 2006).CrossRefGoogle Scholar
14. Govorov, V. E., On flat modules, Sibirsk. Mat. Ž. 6 (1965), 300–304.Google Scholar
15. Griffith, P., A representation theorem for complete local rings, J. Pure Appl. Algebra 7 (3) (1976), 303–315.CrossRefGoogle Scholar
16. Hochster, M., Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Conference on Commutative Algebra–1975, Queen's Univ., Kingston, Ont., 1975, Queen's Papers on Pure and Applied Math., No. 42 (Queen's Univ., Kingston, Ont., 1975), pp. 106–195.Google Scholar
17. Hochster, M., Topics in the homological theory of modules over commutative rings, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975, Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24–28, 1974, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24.CrossRefGoogle Scholar
18. Hochster, M., Big and small Cohen-Macaulay modules, Module theory (Proc. Special Session, Amer. Math. Soc., Univ. Washington, Seattle, Wash., 1977), Lecture Notes in Math., vol. 700 (Springer, Berlin, 1979), 119–142.Google Scholar
19. Holm, H. and Jørgensen, P., Covers, precovers, and purity, Illinois J. Math. 52 (2) (2008), 691–703.CrossRefGoogle Scholar
20. Ischebeck, F., Eine dualität zwischen den funktoren ext und Tor, J. Algebra 11 (1969), 510–531.CrossRefGoogle Scholar
21. Iyengar, S. and Sather-Wagstaff, S., G-dimension over local homomorphisms. Applications to the Frobenius endomorphism, Illinois J. Math. 48 (1) (2004), 241–272.CrossRefGoogle Scholar
22. Lazard, D., Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.CrossRefGoogle Scholar
23. Lenzing, H., Homological transfer from finitely presented to infinite modules, Abelian group theory (Honolulu, Hawaii, 1983), Lecture Notes in Math., vol. 1006 (Springer, Berlin, 1983), 734–761.Google Scholar
24. Matsumura, H., Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Translated from the Japanese by M. Reid (Cambridge University Press, Cambridge, 1989).Google Scholar
25. Rada, J. and Saorin, M., Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26 (3) (1998), 899–912.CrossRefGoogle Scholar
26. Rotman, J. J., An introduction to homological algebra, second ed., (Universitext, Springer, New York, 2009). 2455920 (2009i:18011)CrossRefGoogle Scholar
27. Sharp, R. Y., Cohen-Macaulay properties for balanced big Cohen-Macaulay modules, Math. Proc. Cambridge Philos. Soc. 90 (2) (1981), 229–238.CrossRefGoogle Scholar
28. Simon, A.-M., Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring, Algebr. Represent. Theory 12 (2–5) (2009), 385–400.CrossRefGoogle Scholar
29. Strooker, J. R., Homological questions in local algebra, London Mathematical Society Lecture Note Series, vol. 145 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
30. Xu, J., Flat covers of modules, Lecture Notes in Mathematics, vol. 1634 (Springer-Verlag, Berlin, 1996).CrossRefGoogle Scholar
31. Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar