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<p>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.</p>

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Auslander–Reiten annihilators

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月17日

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  • Özgür Esentepe

摘要

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.