Syzygy pairs in a monomial algebra

In this paper we construct the set of "syzygy pairs" of a finitedimensional monomial algebra and use it to prove the following theorem. Theorem A. (Finitistic global dimension theorem) Let A be a monomial (zero relations) algebra over a field k. Let M be a A-module with finite injective dimension. Then inj dimA M 2 is zero. Then gldim A = n = dimk rad A. We briefly recall the definition of a finite-dimensional monomial algebra. Let Q be a finite directed graph. Then kQ, the path algebra of Q, is the algebra spanned as a vector space by all the directed paths in Q. The multiplication of two paths is their composition, or zero if they are not composable. A monomial Received by the editors December 5, 1988 and, in revised form, January 18, 1989. 19'80 Mathematics Subject Classification (1985 Revision). Primary 1 6A46, 1 6A64. The first author was partially supported by the National Science Foundation. (D 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page