Quadratic algebras with few relations

Throughout V will be a finite dimensional vector space over a field k and T(V) will denote the tensor algebra over V . For simplicity the symbol ⊗ will be omitted in the writing of the elements of T(V) . Let be a basis of V ordered by X i < X i+1 for all i . Then we order the non-commutative monomials and 1 ≤ i s ≤ n for s = 1,…, l} lexicographically from the left. D. Anick 1, p. 652 defines the high term of an element b in T(V) to be the highest monomial appearing in b . As a consequence of 1,3.2, if the set of the high terms of homogeneous relations is combinatorically free in the sense of no overlap ambiguities, then the connected algebra has global dimension 2. The purpose of this note is to prove this result and more for quadratic algebras under other hypotheses on the relations.