On the module of effective relations of a standard algebra

Abstract

Let A be a commutative ring. We denote by a standard A -algebra a commutative graded A -algebra U =[oplus ] n [ges ]0 U n with U 0 = A and such that U is generated as an A -algebra by the elements of U 1 . Take x a (possibly infinite) set of generators of the A -module U 1 . 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 → U be the graded free presentation of U induced by the x . For n [ges ]2, we will call the module of effective n-relations the A -module E ( U ) n = ker f n / V 1 · ker f n . The minimum positive integer r [ges ]1 such that the effective n -relations are zero for all n [ges ] 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 ([Rscr ]( I )) n and rt( I )=rt( [Rscr ] ( I )), where [Rscr ]( I )= [oplus ] n [ges ]0 I n t n ⊂ 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 Andre–Quillen homology we have that E ( U ) n = H 1 ( 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 = H 1 ([ 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 ]).