Auslander-Reiten annihilators
Özgür Esentepe
TL;DR
The paper studies a generalization of the Auslander-Reiten Conjecture by analyzing annihilators of Ext-modules attached to finitely generated modules over commutative Noetherian local rings, introducing the stable annihilator $\underline{\mathrm{ann}}_R(M)$ and the Auslander-Reiten annihilator $\mathrm{ARann}_R(M)$ and the category $\mathcal{E}(R)=\{M:\underline{\mathrm{ann}}_R(M)=\mathrm{ARann}_R(M)\}$. It develops the stability framework, proves closure properties under direct sums and syzygies, and establishes fundamental relations with Ext and stable Hom, setting the stage for high-syzygy analyses. The main results show that for several classes of rings—analytically unramified Arf rings (dimension 1), 2-dimensional local normal domains with rational singularities, and Gorenstein isolated singularities with dimension $\ge 2$—high syzygies of modules lie in $\mathcal{E}$, yielding equality $\underline{\mathrm{ann}}_R(M)=\mathrm{ARAnn}_R(M)$. The canonical module $\omega$ is treated under various hypotheses (generically Gorenstein, radical trace, and Gorenstein-order endomorphism conditions), providing Tachikawa-type equalities and extending Dao-Kobayashi-Takahashi-type results, thereby linking annihilator theory to non-free loci and conjectures in the area. These results offer new tools for understanding extensions, singularities, and conjectural frameworks in commutative algebra.
Abstract
The Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these Ext-modules and formulate a generalisation of the Auslander-Reiten Conjecture. We prove this general version for high syzygies of modules over several classes of rings including analytically unramified Arf rings, 2-dimensional local normal domains with rational singularities, Gorenstein isolated singularities of Krull dimension at least 2 and more. We also prove results for the special case of the canonical module of a Cohen-Macaulay local ring. These results both generalise and also provide evidence for a version of the Tachikawa Conjecture that was considered by Dao-Kobayashi-Takahashi.