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Finitistic dimensions of finite dimensional monomial algebras. (English) Zbl 0727.16003

The left (right) finitistic global dimension lFPD(R) (rFPD(R)) of a ring R is the supremum of the projective dimensions of left (right) R-modules of finite projective dimension. The finitistic dimension conjecture then says that for a finite dimensional algebra A the finitistic dimension always is finite.
The main result of the paper is the proof of the finitistic dimension conjecture for monomial algebras. A finite dimensional k-algebra A, k some field, is called monomial algebra or zero-relation algebra, if A is the path-algebra \(k\Gamma\) of some finite quiver \(\Gamma\) modulo an ideal generated by paths (of length at least two).
The proof uses the result of the first author, D. Happel and D. Zacharia [Ill. J. Math. 29, 180-190 (1985; Zbl 0551.16008)] that the minimal projective resolution of a simple (right) module S can be constructed explicitly for a monomial algebra A. This construction can be used to find a natural number n such that \(Tor^ A_ n(S,M)=0\) for all M with finite projective dimension and all simple modules S, which implies lFPD(A)\(\leq n-1\). The authors use their methods to show additionally that for each pair (m,n) of natural numbers, there exists a monomial algebra A with \(rFPD(A)=m\) and \(lFPD(A)=n\). Moreover they give lower and upper bounds for the finitistic global dimension of monomial algebras in terms of combinatorial invariants of the quiver and relations.
It should be mentioned finally that it was shown independently by K. Igusa and D. Zacharia [Proc. Am. Math. Soc. 108, 601-604 (1990; Zbl 0688.16032)] with different methods, that the finitistic dimension of a monomial algebra has \(\dim_ k(rad A)\) as an upper bound.

MSC:

16E10 Homological dimension in associative algebras
16G20 Representations of quivers and partially ordered sets
16P10 Finite rings and finite-dimensional associative algebras
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[1] Anick, D. J., On monomial algebras of finite global dimension, Trans. Amer. Math. Soc., 291, 291-310 (1985) · Zbl 0575.16002
[2] Anick, D. J., On the homology of associative algebras, Trans. Amer. Math. Soc., 296, 641-659 (1986) · Zbl 0598.16028
[3] Anick, D. J.; Green, E. L., On the homology of path algebras, Comm. Algebra, 15, 309-341 (1987) · Zbl 0608.16029
[4] Auslander, M.; Buchsbaum, D., Homological dimension in local rings, Trans. Amer. Math. Soc., 85, 390-405 (1957) · Zbl 0078.02802
[5] Auslander, M.; Reiten, I., On a generalized version of the Nakayama conjecture, (Proc. Amer. Math. Soc., 52 (1975)), 69-74 · Zbl 0337.16004
[6] Backelin, J., La serie de Poincare-Betti d’une algebre graduée de type fini a une relation est rationnelle, C. R. Acad. Sci. Paris Ser. A, 287, 843-846 (1978) · Zbl 0395.16025
[7] Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc., 95, 466-488 (1960) · Zbl 0094.02201
[8] Bass, H., Injective dimension in Noetherian rings, Trans. Amer. Math. Soc., 102, 18-29 (1962) · Zbl 0126.06503
[9] Cortzen, B., Finitistic dimension of ring extensions, Comm. Algebra, 10, 993-1001 (1982) · Zbl 0484.16014
[10] Eilenberg, S.; Nagao, H.; Nakayama, T., On the dimension of modules and algebras. IV. Dimension of residue rings of hereditary rings, Nagoya Math. J., 10, 87-95 (1956) · Zbl 0074.26003
[11] Fossum, R.; Griffith, P.; Reiten, I., Trivial Extensions of Abelian Categories (1975), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0255.16014
[12] Fuller, K. R.; Zimmermann-Huisgen, B., On the generalized Nakayama conjecture and the Cartan determinant problem, Trans. Amer. Math. Soc., 204, 679-691 (1986) · Zbl 0593.16010
[13] Green, E. L.; Happel, D.; Zacharia, D., Projective resolutions over Artin algebras with zero relations, Illinois J. Math., 29, 180-190 (1985) · Zbl 0551.16008
[14] Jensen, C.; Lenzing, H., Homological dimension and representation type of algebras under base field extension, Manuscripta Math., 39, 1-13 (1982) · Zbl 0498.16023
[15] Nagata, M., Local Rings (1962), Interscience: Interscience New York · Zbl 0123.03402
[16] Nastasescu, C.; Van Oystaeyen, F., Graded Ring Theory (1982), North-Holland: North-Holland Amsterdam · Zbl 0494.16001
[17] Mochizuki, H., Finitistic global dimension for rings, Pacific J. Math., 15, 249-258 (1965) · Zbl 0154.28603
[18] Raynaud, M.; Gruson, L., Criteres de platitude et de projectivite, Invent. Math., 13, 1-89 (1971) · Zbl 0227.14010
[19] Rotman, J., An Introduction to Homological Algebra (1979), Academic Press: Academic Press New York · Zbl 0441.18018
[20] Small, L. W., A change of rings theorem, (Proc. Amer. Math. Soc., 19 (1968)), 662-666 · Zbl 0162.33902
[21] Wiedemann, A.; Roggenkamp, K. W., Path orders of global dimension two, J. Algebra, 80, 113-133 (1983) · Zbl 0518.16001
[22] Wilson, G. V., The Cartan map on categories of graded modules, J. Algebra, 85, 390-398 (1983) · Zbl 0519.18012
[23] Zacharia, D., Graded Artin algebras, rational series, and bounds for homological dimensions, J. Algebra, 106, 476-483 (1987) · Zbl 0609.16001
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