The Cartan matrix as an indicator of finite global dimension for Artinian rings

The Cartan matrix as an indicator of finite global dimension for Artinian rings
HTML articles powered by AMS MathViewer

by W. D. Burgess, K. R. Fuller, E. R. Voss and B. Zimmermann-Huisgen PDF

Proc. Amer. Math. Soc. 95 (1985), 157-165 Request permission

A left serial ring has finite global dimension if and only if its Cartan matrix has determinant equal to 1.

References

Philip J. Davis, Circulant matrices, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 543191
Samuel Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), 310–319. MR 65544, DOI 10.1007/BF02566937
Samuel Eilenberg, Masatoshi Ikeda, and Tadasi Nakayama, On the dimension of modules and algebras. I, Nagoya Math. J. 8 (1955), 49–57. MR 69157
Kent R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248–260. MR 232795, DOI 10.1007/BF01110273
K. R. Fuller and Joel Haack, Rings with quivers that are trees, Pacific J. Math. 76 (1978), no. 2, 371–379. MR 498683
Robert Gordon and Edward L. Green, Modules with cores and amalgamations of indecomposable modules, Mem. Amer. Math. Soc. 10 (1977), no. 187, viii+145. MR 453815, DOI 10.1090/memo/0187

Global dimension in serial rings

J. P. Jans and Tadasi Nakayama, On the dimension of modules and algebras. VII. Algebras with finite-dimensional residue-algebras, Nagoya Math. J. 11 (1957), 67–76. MR 86824
G. J. Janusz, Some left serial algebras of finite type, J. Algebra 23 (1972), 404–411. MR 302703, DOI 10.1016/0021-8693(72)90139-1
Herbert Kupisch, Beiträge zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung, J. Reine Angew. Math. 201 (1959), 100–112 (German). MR 104707, DOI 10.1515/crll.1959.201.100
Ichiro Murase, On the structure of generalized uniserial rings. I, Sci. Papers College Gen. Ed. Univ. Tokyo 13 (1963), 1–22. MR 156875
Tadasi Nakayama, Some studies on regular representations, induced representations and modular representations, Ann. of Math. (2) 39 (1938), no. 2, 361–369. MR 1503413, DOI 10.2307/1968792
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