Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-29T03:28:21.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifting Projectives

Published online by Cambridge University Press:  22 January 2016

Jan R. Strooker
Jan R. Strooker*
Affiliation:
University of Utrecht
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.

Let R be a ring with radical (all rings have a unit element, all modules are unital). Often, one wishes to lift modules modulo , that is, to a given, say, left R/-module U find a left R-module E with the property that E/E ≃ U. This is of course not always possible. Here I prove, roughly, that if a finitely generated projective U can be lifted at all, it can be lifted to a projective. Or rather, if U can be lifted to an E satisfying a certain mild condition, then E is projective (Lemma).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 2 , July 1966 , pp. 747 - 751
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Auslander, M., On the dimension of modules and algebras III, Nagoya Math. J. 9 (1955), 67–77.CrossRefGoogle Scholar
[2] Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
[3] Bourbaki, N., Eléments de mathématique, Algèbre commutative, Chap. 8, Hermann, Paris, 1958.Google Scholar
[4] Bourbaki, N., Eléments de mathématique, Algèbre commutative, Chap. 1–4, Hermann, Paris, 1961.Google Scholar
[5] Eilenberg, S., Homological dimension and syzygies, Ann. of Math. (2) 64 (1956), 328–336.CrossRefGoogle Scholar
[6] Grothendieck, A., Eléments de géométrie algébrique, IV 1, Publ. Math. I.H.E.S., no. 20, Bures-sur-Yvette, 1964.CrossRefGoogle Scholar
[7] Kaplansky, I., The theory of homological dimension, C.I.M.E. conference on “Some aspects of ring theory”, Varenna, August 1965 (to appear).Google Scholar
[8] Strooker, J. R., Faithfully projective modules and clean algebras, Doctoral thesis, University of Utrecht, April 1965.Google Scholar
[9] Grothendieck, A., Eléments de géométrie algébrique, IV 3, Publ. Math. I.H.E.S. no. 28, Bures-sur-Yvette, 1966.CrossRefGoogle Scholar