Predicting syzygies over monomial relations algebras

We show that all projective resolutions over a monomial relations algebra Λ simplify drastically at the stage of the second syzygy; more precisely, we show that the kernel of any homomorphism between two projective left Λ-modules is isomorphic to a direct sum of principal left ideals generated by paths. As consequences, we obtain: (a) a tight approximation of the finitistic dimensions of Λ in terms of the (very accessible) projective dimensions of the principal left ideals generated by paths; (b) a basis for comparison of the ‘big’ and ‘little’ finitistic dimensions of Λ, yielding in particular that these two invariants cannot differ by more than 1 and that they are equal in ‘most’ cases; (c) manageable algorithms for computation of finitistic dimensions.