Bounds on Annihilator Lengths in Families of Quotients of Noetherian Rings

In this paper we study lengths of annihilators of m-primary ideals, J, in quotients of finitely generated modules, M, over local rings, (R, m), modulo m-primary ideals generated by a sequence of ring elements each raised to a power, IN = (fN1,…,fNs), as a function of this power. The motivation for studying these lengths arose initially from tight closure theory. However, the function we define to organize this study is closely related to the Hilbert–Kunz and Hilbert–Samuel functions. Using new observations on the implications of the structure of complete local rings and subsequent homological algebra results, we give an upper bound, a constant times Nd − 2, for the length of AnnM/INMJ when the dimension of M is d.