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PreprintCC BY 4.0arXiv

Weak and Serre Lifting of Cyclic Modules and the Liftability of Auslander Transposes

  • Shashi Ranjan Sinha
  • Amit Tripathi
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Abstract

Let Q be a local ring, f a nonzerodivisor, and R=Q/(f). We characterize weak and Serre liftability of cyclic R-modules under suitable hypotheses, obtaining complete-intersection criteria for Serre liftability in small height. We prove that the tensor product of two weakly liftable cyclic modules remains weakly liftable when their first Tor module vanishes, and give sufficient conditions for tensor products to preserve Serre liftability. We construct an Artinian Gorenstein quotient Q/I, with Q regular, that is perfect over R and Serre liftable but not weakly liftable to Q, showing that the negative answer to a question of Jorgensen persists in this more restrictive setting. Finally, we characterize when a lift induces a lift of the Auslander transpose and obtain a freeness criterion related to the Auslander-Reiten conjecture, recovering a theorem of Ghosh and Samanta.

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Cite

BibTeXRIS
BibTeX
@misc{sinha2026weak,
  title = {{Weak and Serre Lifting of Cyclic Modules and the Liftability of Auslander Transposes}},
  author = {Shashi Ranjan Sinha and Amit Tripathi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.37967},
  eprint = {2609.37967},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37967},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Weak and Serre Lifting of Cyclic Modules and the Liftability of Auslander Transposes
AU  - Shashi Ranjan Sinha
AU  - Amit Tripathi
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.37967
UR  - https://arxiv.org/abs/2609.37967
AB  - Let Q be a local ring, f a nonzerodivisor, and R=Q/(f). We characterize weak and Serre liftability of cyclic R-modules under suitable hypotheses, obtaining complete-intersection criteria for Serre liftability in small height. We prove that the tensor product of two weakly liftable cyclic modules remains weakly liftable when their first Tor module vanishes, and give sufficient conditions for tensor products to preserve Serre liftability. We construct an Artinian Gorenstein quotient Q/I, with Q regular, that is perfect over R and Serre liftable but not weakly liftable to Q, showing that the negative answer to a question of Jorgensen persists in this more restrictive setting. Finally, we characterize when a lift induces a lift of the Auslander transpose and obtain a freeness criterion related to the Auslander-Reiten conjecture, recovering a theorem of Ghosh and Samanta.
ER  -