2 00 4 Hochschild ( co ) homology dimension ∗

In 1989 Happel conjectured that for a finite-dimensional algebra A over an algebraically closed field k, gl.dim.A < ∞ if and only if hch.dim.A < ∞. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel’s conjecture. They found a family of pathological algebra Aq for which gl.dim.Aq = ∞ but hch.dim.Aq = 2. These algebras are pathological in many aspects, however their Hochschild homology behaviors are not pathological any more, indeed one has hh.dim.Aq = ∞ = gl.dim.Aq. This suggests to pose a seemingly more reasonable conjecture by replacing Hochschild cohomology dimension in Happel’s conjecture with Hochschild homology dimension: gl.dim.A < ∞ if and only if hh.dim.A < ∞ if and only if hh.dim.A = 0. The conjecture holds for commutative algebras and monomial algebras. In case A is a truncated quiver algebras these conditions are equivalent to the quiver of A has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off in case the underlying field is characteristic 0. Mathematics Subject Classification (2000): 16E40, 16E10, 16G10 Let k be a fixed field. All algebras considered here are of the form A = kQ/I where Q is a finite quiver and I is an admissible ideal of the path algebra kQ. We refer to [ARS] for the theory of quivers and their representations. It is well-known that in case k is an algebraically closed field, up to Morita Project 10201004 supported by NSFC.