Abstract: In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t 1 ,t 2 ,…, t n ) be a minimal system of generators for M . By a complex R<T 1 . . ., T p > we mean an R -algebra * obtained by the adjunction of the variables T 1 . . ., T p of degree 1 which kill t 1 ,…, t p . The main purpose of this paper is, among other things, to construct an R -algebra resolution of the field K , so that we can investigate the relationship between the homology algebra H (R < T 1 ,…, T n > ) and the homological invariants of R such as the algebra Tor R (K, K) and the Betti numbers B p = dim k Tor R (K, K) of the local ring R . The relationship was initially studied by Serre [5].