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Zbl 1147.16301
Fuller, K.R.; Burgess, W.D.; Tonolo, A.
A note on block triangular presentations of rings and finitistic dimension. (English)
[J] Rend. Semin. Mat. Univ. Padova 105, 207-214 (2001). ISSN 0041-8994

From the introduction: In [{\it W. D. Burgess}, Proc. Edinb. Math. Soc., II. Ser. 30, 351-362 (1987; Zbl 0608.16028)] and [{\it K. R. Fuller}, Contemp. Math. 124, 51-72 (1992; Zbl 0746.16013)] we employed an equivalence relation on the indecomposable projective modules over a left Artinian ring $R$ that induces the ``finest'' presentation of $R$ as a ring of lower block triangular matrices. The equivalence classes correspond to the simply connected components of the quiver of $R$ and they contain information on such things as the Cartan determinant and the finitistic dimensions of $R$. Our main purpose here is to show that such presentations provide bounds on the finitistic dimensions of certain rings and Artin algebras that yield generalizations of some results of {\it M. I. Platzeck} and {\it F. U. Coelho} [in Bol. Soc. Mat. Mex., III. Ser. 7, No. 1, 49-57 (2001; Zbl 1011.16011) and Commun. Algebra 24, No. 8, 2515-2533 (1996; Zbl 0857.16012)]. In the process, we provide a simple proof of an old result from [{\it R. M. Fossum, P. A. Griffith, I. Reiten}, Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory. Lect. Notes Math. 456 (1975; Zbl 0303.18006)] and point out that a theorem of {\it S. O. Smal\o} [Proc. Am. Math. Soc. 111, No. 3, 651-656 (1991; Zbl 0724.16003)] provides a connection between the functorial finiteness in $R$-mod of the category $\cal P^{<\infty}(R)$ of modules of finite projective dimension, and the analogous categories over the irreducible components of $R$.

MSC 2000:: *16E10 Homological dimensions (assoc. rings and algebras)
16G10 Representations of Artinian rings
16G20 Representations of quivers and partially ordered sets
16S50 Endomorphism rings: matrix rings
16D90 Module categories (assoc. rings and algebras)

Keywords: block triangular presentations; finitistic dimensions; indecomposable projective modules; left Artinian rings; rings of block triangular matrices

Citations: Zbl 0608.16028; Zbl 0746.16013; Zbl 1011.16011; Zbl 0857.16012; Zbl 0303.18006; Zbl 0724.16003

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