Bounds on Global and Finitistic Dimension for Finite Dimensional Algebras with Vanishing Radical Cube
✍ Scribed by B.Z. Huisgen
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 807 KB
- Volume
- 161
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
It is shown that, given any left artinian ring (A) which has vanishing radical cube and (n) isomorphism classes of simple left modules, the global dimension of (A) is either infinite or bounded above by (n^{2}-n), and the left finitistic dimension of (A) is always less than or equal to (n^{2}+1); in fact, sharper bounds are obtained, but these are not as easily described. Moreover, a tight grid for the "distribution" of the projective dimensions of the simple left (A)-modules, again in terms of (n), is set up. The key to these estimates is a sequence of matrix groups which stores homological information on (A) in an efficient form. Additional applications of this machinery include lower bounds on the projective dimensions of the simple left (\Lambda)-modules and hence on the global dimension of (A). 1993 Academic Press. Inc.