On a question of Takahashi concerning direct summands of syzygies

@misc{ferraro2026question,
  title = {{On a question of Takahashi concerning direct summands of syzygies}},
  author = {Luigi Ferraro},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01630},
  eprint = {2611.01630},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01630},
}

TY  - UNPB
TI  - On a question of Takahashi concerning direct summands of syzygies
AU  - Luigi Ferraro
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01630
UR  - https://arxiv.org/abs/2611.01630
AB  - For a finitely generated module M of finite Gorenstein dimension over a commutative noetherian local ring (R,m,Bbbk), we prove that the natural map Ext_R(Bbbk,M)⟶Ext_R(Bbbk,M/m M) is zero unless R is Gorenstein. More generally, the same conclusion holds whenever Ext_R^i(M,R)=0 for i≫0. This gives a Gorenstein analogue of a regularity criterion of Avramov and Iyengar. The proof uses the right Ext_R(Bbbk,Bbbk)-module structure of bounded cohomology studied by the author in a previous work. As an application, we show that R is Gorenstein if a nonzero quotient of a finite direct sum of syzygies of Bbbk has finite Gorenstein dimension. In particular, this answers affirmatively a question of Takahashi concerning direct summands of syzygies of the residue field.
ER  -