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<p>Let <i>R</i> be an associative ring with identity. We establish that the generalized Auslander–Reiten conjecture implies the Wakamatsu tilting conjecture. Furthermore, we prove that any Wakamatsu tilting <i>R</i>-module of finite projective dimension that is tensorly faithful is projective. By utilizing this result, we show the validity of the Wakamatsu tilting conjecture for <i>R</i> in two cases: when <i>R</i> is a left Artinian local ring or when it is the group ring of a finite group <i>G</i> over a commutative Artinian ring.</p>

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On the Wakamatsu tilting conjecture

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Kamran Divaani-Aazar,
  • Ali Mahin Fallah,
  • Massoud Tousi

摘要

Let R be an associative ring with identity. We establish that the generalized Auslander–Reiten conjecture implies the Wakamatsu tilting conjecture. Furthermore, we prove that any Wakamatsu tilting R-module of finite projective dimension that is tensorly faithful is projective. By utilizing this result, we show the validity of the Wakamatsu tilting conjecture for R in two cases: when R is a left Artinian local ring or when it is the group ring of a finite group G over a commutative Artinian ring.