Hostname: page-component-784d4fb959-mv4gh Total loading time: 0 Render date: 2025-07-15T23:12:05.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Toward the construction of big Cohen-Macaulay modules

Published online by Cambridge University Press:  22 January 2016

Yuji Yoshino
Yuji Yoshino*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 103 , October 1986 , pp. 95 - 125
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[B] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8–28.Google Scholar
[BDLD] Becker, J., Denef, J., Lipshits, L. and Dries, van den, Ultraproducts and approximation in local rings I, Invent. Math., 51 (1979), 189–203.Google Scholar
[CE] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton, New Jersey, Princeton University Press, 1956.Google Scholar
[EGA] Grothendieck, A. and Dieudonne, J., Element de Geometrie Algebrique, Chapter Oiv.i, Inst. Hautes Etudes Sci. Publ. Math, no. 20 (1964).Google Scholar
[F] Foxby, H.-B., A homological theory of complexes of modules, Kobenhavens Univ. Math. Inst., Preprint Series no. 196, September, 1981.Google Scholar
[H1] Hochster, M., Topics in the homological theory of modules over commutative rings, C. B. M. S. Regional Conference Series in Math. no. 24, Amer. Math. Soc, Providence, R. I., 1975.,Google Scholar
[H2] Hochster, M., Contracted ideals from integral extension of regular local rings, Nagoya Math. J., 51 (1975), 25–43.Google Scholar
[H3] Hochster, M., Cohen-Macaulay modules, Proc. Kansas Commutative Algebra Conference, Lecture Note in Math., vol. 311, Springer Verlag, Berlin-Heiderberg-New York, 1973, 120–152.CrossRefGoogle Scholar
[H4] Hochster, M., Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors, Queen’s papers in Pure Appl. Math., 42 (1975), 106–195.Google Scholar
[H5] Hochster, M., Canonical elements in local cohomology modules and the direct summand conjecture, preprint.Google Scholar
[I] Iversen, B., Amplitude inequalities for complexes, Ann. Scient. Ec. Norm. Sup., (4) 10 (1977), 547–558.CrossRefGoogle Scholar
[M] Matlis, E., Infective modules over Noetherian rings, Pacific J. Math., 8 (1958), 511–528.Google Scholar
[Ma] Matsumura, H., Commutative Algebra, Benjamin, New York, 1970.Google Scholar
[MP] Mateescu, C. and Popescu, D., Ultraproducts and big Cohen-Macaulay modules, preprint.Google Scholar
[N] Nagata, M., Local Rings, Interscience Tracts in Pure nad Appl. Math., no. 13, Interscience, New York, 1962.Google Scholar
[NR] Northcott, D. G. and Rees, D., Principal systems, Qurt. J. Math. Oxford, (2)8 (1957),119–127.CrossRefGoogle Scholar
[PS1] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, I.H.E.S. Publ. Math., no. 42, Paris, 1973, 323–395.Google Scholar
[PS2] Peskine, C. and Szpiro, L., Syzygy et multiplicités, C. R. Acad. Sci. Paris, Serie A-B 278 (1974), 1421–1424.Google Scholar
[R1] Roberts, P., Two applications of dualizing complexes over local rings, Ann. Scient. Ec. Norm. Sup., (4) 9 (1976), 103–106.Google Scholar
[R2] Roberts, P., Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture, J. Algebra, 66 (1980), 202–225.Google Scholar
[S] Strooker, J. R., The monomial and direct summand conjectures, a draft for Chapter 13 of a book on the homological conjectures, Univ. of Utrecht, June, 1981.Google Scholar
[Y1] Yoshino, Y., On the homological conjectures of local rings, Proc. of the 9th International Symposium, Div. of Math, of the Taniguchi Foundation, Conference on Commutative Algebra, Katata, September, 1981, 51–56.Google Scholar
[Y2] Yoshino, Y., On Northcott-Rees theorem on principal systems, Nagoya Math. J., 95 (1984), 41–50.Google Scholar