On the generalized Nakayama conjecture and the Cartan determinant problem

Fuller, K. R.; Zimmermann-Huisgen, B.

doi:10.1090/S0002-9947-1986-0825739-2

On the generalized Nakayama conjecture and the Cartan determinant problem
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by K. R. Fuller and B. Zimmermann-Huisgen PDF

Trans. Amer. Math. Soc. 294 (1986), 679-691 Request permission

Abstract:

For Artin algebras allowing certain filtered module categories, the Generalized Nakayama Conjecture is shown to be true; our result covers all positively graded Artin algebras and those whose radical cube is zero. For the corresponding class of left artinian rings we prove that finite global dimension forces the determinant of the Cartan matrix to be 1.

References

Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975), 69–74. MR 389977, DOI 10.1090/S0002-9939-1975-0389977-6
W. D. Burgess, K. R. Fuller, E. R. Voss, and B. Zimmermann-Huisgen, The Cartan matrix as an indicator of finite global dimension for Artinian rings, Proc. Amer. Math. Soc. 95 (1985), no. 2, 157–165. MR 801315, DOI 10.1090/S0002-9939-1985-0801315-7
V. P. Camillo and K. R. Fuller, On graded rings with finiteness conditions, Proc. Amer. Math. Soc. 86 (1982), no. 1, 1–5. MR 663852, DOI 10.1090/S0002-9939-1982-0663852-5
Samuel Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310–319. MR 65544, DOI 10.1007/BF02566937
Robert Gordon and Edward L. Green, Graded Artin algebras, J. Algebra 76 (1982), no. 1, 111–137. MR 659212, DOI 10.1016/0021-8693(82)90240-X
Edward L. Green, William H. Gustafson, and Dan Zacharia, Artin rings of global dimension two, J. Algebra 92 (1985), no. 2, 375–379. MR 778455, DOI 10.1016/0021-8693(85)90127-9

On

rings

Bruno J. Müller, The classification of algebras by dominant dimension, Canadian J. Math. 20 (1968), 398–409. MR 224656, DOI 10.4153/CJM-1968-037-9
Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740
George V. Wilson, The Cartan map on categories of graded modules, J. Algebra 85 (1983), no. 2, 390–398. MR 725091, DOI 10.1016/0021-8693(83)90103-5
Dan Zacharia, On the Cartan matrix of an Artin algebra of global dimension two, J. Algebra 82 (1983), no. 2, 353–357. MR 704760, DOI 10.1016/0021-8693(83)90156-4