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Monoid gradings on algebras and the cartan determinant conjecture*

Published online by Cambridge University Press:  20 January 2009

Manuel Saorín
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Abstract

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In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 3 , October 1998 , pp. 539 - 551
Copyright
Copyright © Edinburgh Mathematical Society 1998

Footnotes

*

Work partially supported by the D.G.E.S. of Spain (PB96-0961-C02-02)

References

REFERENCES

1. Anick, D. J. and Green, E. L., On the homology of quotients of path algebras, Comm. Algebra 15(1) (1987), 309–341.Google Scholar
2. Auslander, M., Representation theory of Artin algebras I, Comm. Algebra 1 (1974), 177–268.Google Scholar
3. Bautista, R., Gabriel, P., Rojter, A. V. and Salmeron, L., Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), 217–285.Google Scholar
4. Belzner, T., Burgess, W. D., Fuller, K. R. and Schulz, R., Examples of ungradable algebras, Proc. Amer. Math. Soc. 114 (1992), 1–9.Google Scholar
5. Burgess, W. D., The graded Cartan matrix and global dimension of O-relation algebras, Proc. Edinburgh Math. Soc. 30 (1987), 351–362.CrossRefGoogle Scholar
6. Eilenberg, S., Algebras of cohomologically finite dimension, Comment. Math. Helv. 28 (1954), pp. 310–319.CrossRefGoogle Scholar
7. Fuller, K. R., Algebras from diagrams, J. Pure Appl. Algebra 48 (1987), 23–37.Google Scholar
8. Fuller, K. R., The Cartan determinant conjecture and global dimension of Artinian rings, Contemp. Math. 124 (1992), 51–72.CrossRefGoogle Scholar
9. Fuller, K. R. and Zimmermann-Huisgen, B., On the generalized Nakayama conjecture and the Cartan determinant problem, Trans. Amer. Math. Soc. 294(2) (1986), 679–691.Google Scholar
10. Gabriel, P., Auslander-Reiten sequences and representation-finite algebras, in I.C.R.A. I Proceedings (Carleton Univ. 1979, Springer LNM 831, 1980).Google Scholar
11. Gilmer, R., Commutative semigroup rings (University of Chicago Press, 1984).Google Scholar
12. Saorin, M., Isomorphisms between representations of algebras, Publ. Mat. 36 (1992), 955–964.Google Scholar
13. Saorin, M., Gradability of algebras (Marcel Dekker L.N. 140, 1993), 285–301.Google Scholar
14. Wilson, G. V., The Cartan map on categories of graded modules. J Algebra 85 (1983), 390–398.Google Scholar