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Virtually Gorenstein Artin algebras are weakly Gorenstein

  • Weiqing Li
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Abstract

We prove that all (possibly infinitely generated) semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This enables us to show the validities of the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture for virtually Gorenstein Artin algebras. Finally, we prove that the Auslander-Reiten Conjecture holds for virtually Gorenstein Artin algebras of finite CM-type.

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Cite

BibTeXRIS
BibTeX
@misc{li2026virtually,
  title = {{Virtually Gorenstein Artin algebras are weakly Gorenstein}},
  author = {Weiqing Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.10049},
  eprint = {2608.10049},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.10049},
}
RIS
TY  - UNPB
TI  - Virtually Gorenstein Artin algebras are weakly Gorenstein
AU  - Weiqing Li
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.10049
UR  - https://arxiv.org/abs/2608.10049
AB  - We prove that all (possibly infinitely generated) semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This enables us to show the validities of the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture for virtually Gorenstein Artin algebras. Finally, we prove that the Auslander-Reiten Conjecture holds for virtually Gorenstein Artin algebras of finite CM-type.
ER  -