Betti numbers of modules of essentially monomial type

Let R R be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of R R -modules of essentially monomial type. These modules are defined with respect to various R R -regular sequences. For example, finite length modules of monomial type over regular local rings of dimension n n are modules of essentially monomial type with respect to R R -regular sequences of length n n . If a module is of essentially monomial type with respect to an R R -regular sequence of length n n , then the rank of its i i -th syzygy is at least ( n − 1 i − 1 ) \binom {n-1}{i-1} and its i i -th Betti number is at least ( n i ) \binom ni .