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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

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