Let Δ be a finite quiver, that is, a finite directed graph,
k be a commutative ring, and kΔ be the semigroup ring of paths
in Δ. In this paper we compute the Hochschild homology of the following
classes of algebras:
(1) kΔ/[mfr ]n, where [mfr ] is the arrow ideal;
(2) kΔ/I where I is an ideal generated
by quadratic monomials.
In Section 2 we establish the notation, and recall a projective resolution of
kΔ0,
the degree 0 part in kΔ, over a monomial algebra which is due to Anick and Green.
This resolution is then used in Section 3 as the building block in the construction of
a resolution of a truncated or quadratic monomial algebra A, over its enveloping
algebra (in the Hochschild sense), Ae. The fact that a monomial algebra
possesses a fine grading then enables us to compute the homology.
The results obtained extend results by Liu and Zhang [3] and Geller,
Reid and Weibel [2].