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A note on block triangular presentations of rings and finitistic dimension. (English) Zbl 1147.16301
From the introduction: In [W. D. Burgess, Proc. Edinb. Math. Soc., II. Ser. 30, 351-362 (1987; Zbl 0608.16028)] and [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 M. I. Platzeck and 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 [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 S. O. Smalø [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 𝒫 < (R) of modules of finite projective dimension, and the analogous categories over the irreducible components of R.
MSC:
16E10Homological dimensions (associative rings and algebras)
16G10Representations of Artinian rings
16G20Representations of quivers and partially ordered sets
16S50Endomorphism rings: matrix rings
16D90Module categories (associative rings and algebras); Morita equivalence and duality