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.


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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