Number of generators of a Cohen–Macaulay ideal

For a Noetherian local ring R, if is Cohen–Macaulay, then the ideal can be generated by at most (e−2)(ν−d−1)+2 elements, where ν is the embedding dimension of R and where d and e 3 are the dimension and the multiplicity of , respectively. This bound is in general much sharper than the bounds given by Sally or Boraty ski–Eisenbud–Rees in case has height bigger than 2. Moreover, no Cohen–Macaulay assumption on R is required.