Coefficient modules and Ratliff-Rush closures^*

Abstract

Let $(R,\mathfrak{m})$ be a d-dimensional Noetherian local ring, M be an R-submodule of the free module $F=R^p$. In this work, in analogy to the papers of Liu in [16] and of Ratliff and Rush in [20], if we consider R a formally equidimensional ring and the R-module F/M having finite length, we prove the existence of a unique chain of modules, $M\subseteq M_{\{d+p-1\}}^F\subseteq \cdots \subseteq M_{\left\{1\right\}}^F\subseteq M_{\left\{0\right\}}^F\subseteq \overline{M}$ such that i-the Buchsbaum-Rim coefficients of M and $M_{\left\{k\right\}}^F$ are equal for $i=0,\dots ,k$, between M and its integral closure $\overline{M}$. This modules will be called Coefficient Modules of M. We also give a colon structure description of these coefficient modules, and, in addition, as consequence of this results, we obtain certain properties of the Ratliff-Rush module of M.

Keywords:

• Coefficient module
• Hilbert polynomial
• multiplicity
• Ratliff-Rush modules

2020 Mathematics Subject Classification:

• Primary: 13H15

Acknowledgments

This paper was initiated as part of process to obtain the Professor M. D. Ferrari’s Ph.D. It has been kept saved for all these years, and it is now being published as a recognition and honor of this great mathematician and advisor that we were proud to work with him. Thank you Professor Roberto Callejas Bedregal for all that we learn in your presence. Unfortunately, Professor Roberto died in 2021 during the Covid-19-Pandemic as a result the aggravation of Corona Virus disease. Your teachings will be kept saved. The authors would like to thank to the anonymous referee for her/his careful reading of this manuscript and the many deep suggestions and corrections.

Additional information

Funding

Victor H. Jorge Pérez is partially supported by CNPq-Brazil, grant 421440/2016-3. Marcela D. Ferrari is partially supported by CNPq grant 140764/2006-2 and by FAPESP grant 2010/11668-4, Brazil.