When are syzygies of the residue field self-dual?
Souvik Dey
TL;DR
This work addresses when reflexive syzygies of the residue field in a local ring are self-dual, building on Dao’s characterization of Arf rings. Using reflexivity theory, Auslander transposes, and depth-based homological arguments, it isolates when self-duality forces regularity or restricts the ring to be a hypersurface. The main results show that for $\operatorname{depth} R = t\ge 3$, self-duality of $\Omega_R^i k$ (with $2\le i\le t-1$) implies $R$ is regular with $t$ odd and $t=2i-1$; for non-regular rings with $t\ge 2$, $\Omega_R^i k$ is reflexive for all $i\ge t$, and a full self-duality across $i\ge t$ characterizes hypersurfaces (with equivalences when $t$ is even). A strong partial converse shows even-dimensional non-regular hypersurfaces exhibit this self-duality, and, overall, if every reflexive syzygy of $k$ is self-dual, then $R$ must be either regular of dimension $3$ or a hypersurface of dimension $2$. These results advance the understanding of duality phenomena in Cohen–Macaulay representations and stable module theory.
Abstract
Finitely generated reflexive modules over commutative Noetherian rings form a key component of Auslander and Bridger's stable module theory and are likewise essential in the study of Cohen--Macaulay representations. Recently, H. Dao characterized Arf local rings as exactly those one-dimensional Cohen--Macaulay local rings over which every finitely generated reflexive module is self-dual, and raised the general question of characterizing rings over which every finitely generated reflexive module is self-dual. Motivated by this, in this article, we study the question of self-duality of syzygies of the residue field of a local ring when they are known to be reflexive. We show that for local rings of depth at least 2, the answer is given by hypersurface or regular local rings in most cases.