Mathematical Proceedings of the Cambridge Philosophical Society

On the module of effective relations of a standard algebra

a1 Departament Matemática Aplicada I. ETSEIB-UPC. Diagonal 647, E-08028 Barcelona; e-mail:


Let A be a commutative ring. We denote by a standard A-algebra a commutative graded A-algebra U=[oplus B: plus sign in circle]n[gt-or-equal, slanted]0Un with U0=A and such that U is generated as an A-algebra by the elements of U1. Take x a (possibly infinite) set of generators of the A-module U1. Let V=A[t] be the polynomial ring with as many variables t (of degree one) as x has elements and let f[ratio]V[rightward arrow]U be the graded free presentation of U induced by the x. For n[gt-or-equal, slanted]2, we will call the module of effective n-relations the A-module E(U)n= ker fn/V1· ker fn. The minimum positive integer r[gt-or-equal, slanted]1 such that the effective n-relations are zero for all n[gt-or-equal, slanted]r+1 is known to be an invariant of U. It is called the relation type of U and is denoted by rt(U). For an ideal I of A, we define E(I)n= E([script R](I))n and rt(I)=rt([script R](I)), where [script R](I)= [oplus B: plus sign in circle]n[gt-or-equal, slanted]0 Intn[subset or is implied by]A [t] is the Rees algebra of I.

In this paper we give two descriptions of the A-module of effective n-relations. In terms of André–Quillen homology we have that E(U)n= H1(A, U, A)n (see 2·3). It turns out that this module does not depend on the chosen [x]. In terms of Koszul homology we prove that E(U)n= H1([x], U)n (see 2·4). Using these characterizations, we show later some properties on the module of effective n-relations and the relation type of a graded algebra. Our approach has connections with several earlier works on the subject (see [2, 5–7, 9, 10, 13, 14]).

(Received May 14 1996)
(Revised December 17 1996)