Arf rings, simple singularities and reflexive modules
Özgür Esentepe
TL;DR
The paper investigates when self-duality phenomena for reflexive modules extend to higher-dimensional commutative Noetherian local rings, motivated by Arf rings. It develops the $n$-torsionfree/totally reflexive framework, along with Auslander transpose and related homological tools, to characterize when every reflexive module is self-dual and when a ring has finite reflexive type. The main contributions show that, under (sd), depth $t\ge 2$ forces $R$ to be a hypersurface of dimension at most $3$, and that with (frt) and (sd) plus a non-free $3$-torsionfree module, $R$ must be a simple singularity of dimension at most $3$; there is also a remark that non-Gorenstein Arf local rings cannot admit non-free $3$-torsionfree reflexive modules. Together, these results illuminate a homological parallel between Arf rings and simple singularities in low dimensions and extend the self-duality perspective beyond dimension one.
Abstract
In a pandemic era preprint, Dao showed showed two remarkable properties of Arf rings: under some mild conditions, they admit finitely many indecomposable reflexive modules up to isomorphism and every reflexive module is actually isomorphic to its own dual. In fact, the latter property characterises Arf rings. Arf rings are one dimensional rings and it is natural to wonder what happens in higher Krull dimension. In this paper, we investigate the self-dual property for commutative Noetherian local rings.