Notes on Matlis reflexive modules
Abstract
Let M denote a module over a local Noetherian ring (R,m) with residue field Bbbk and its injective hull ER. Let DR(·) = HomR(·,ER) denote the Matlis duality functor. Then M is called Matlis reflexive whenever the injection M → DR2(M) is an isomorphism. The following results are proved: (1) For an ideal of R a reflexive module is quasi-complete, generalizing part of Belshoff's results (see \cite{Br}). (2) We improve Zöschinger's numerical criterion for Matlis reflexivity (see \cite{Zh2}). (3) As shown by Enochs (see \cite{Ee}) for a prime ideal and the injective hull ER(R/p) it follows that DR(ER(R/p)) is the completion of a free Rp-module of rank τp, the Enochs number of p. We characterize when τp = 1, extending author's result (see \cite{Sp16}).
Related
BibTeX
@misc{schenzel2026notes,
title = {{Notes on Matlis reflexive modules}},
author = {Peter Schenzel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25167},
eprint = {2609.25167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25167},
note = {CC0},
}
RIS
TY - UNPB
TI - Notes on Matlis reflexive modules
AU - Peter Schenzel
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25167
UR - https://arxiv.org/abs/2609.25167
AB - Let M denote a module over a local Noetherian ring (R,m) with residue field Bbbk and its injective hull E_R. Let D_R(·) = Hom_R(·,E_R) denote the Matlis duality functor. Then M is called Matlis reflexive whenever the injection M → D_R^2(M) is an isomorphism. The following results are proved: (1) For an ideal of R a reflexive module is quasi-complete, generalizing part of Belshoff's results (see \cite{Br}). (2) We improve Zöschinger's numerical criterion for Matlis reflexivity (see \cite{Zh2}). (3) As shown by Enochs (see \cite{Ee}) for a prime ideal and the injective hull E_R(R/p) it follows that D_R(E_R(R/p)) is the completion of a free R_p-module of rank τ_p, the Enochs number of p. We characterize when τ_p = 1, extending author's result (see \cite{Sp16}).
ER -