Auslander-Reiten annihilators

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.

arXiv commentSeveral typos fixed and more references added

References in corpus (10)

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• The radius of a subcategory of modules
• Auslander-Reiten conjecture and Auslander-Reiten duality
• Local rings with quasi-decomposable maximal ideal
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• Auslander-Reiten conjecture and finite injective dimension of Hom
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