A study on v-numbers of Ext^i_R(L,M/I^nN) and Tor_i^R(L,M/I^nN)

Abstract

Let R be a Noetherian ℕ-graded ring. Assume that L and M are finitely generated ℤ-graded R-module, and I a homogeneous ideal of R. Let N be a graded submodule of M. Fix k∈ ℕ. Denote H:=Ext^k_R(L,M) (or Tor_k^R(L,M)) and H_n:=Ext^k_R(L, M/IⁿN) (or Tor_k^R(L,M/IⁿN)). When I is generated by elements of positive degree, we show that the local v-number of H_n either coincides with that of colon submodule (0:_H I) or is a polynomial in n of degree one whose leading coefficient is one of the degrees of the generators of I. This extends the main result of Ghosh-Pramanik [J. Algebra 671 (2025), 61-74], where asymptotic linearity was shown under the assumption that (0:_H I)=0.

Cite

BibTeX RIS

BibTeX

@misc{islam2026study,
  title = {{A study on v-numbers of $\operatornameExt^i_R(L,M/I^nN)$ and $\operatornameTor_i^R(L,M/I^nN)$}},
  author = {Sayed Sadiqul Islam and Lisa Mandal and Samarendra Sahoo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01156},
  eprint = {2611.01156},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01156},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A study on v-numbers of Ext^i_R(L,M/I^nN) and Tor_i^R(L,M/I^nN)
AU  - Sayed Sadiqul Islam
AU  - Lisa Mandal
AU  - Samarendra Sahoo
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01156
UR  - https://arxiv.org/abs/2611.01156
AB  - Let R be a Noetherian ℕ-graded ring. Assume that L and M are finitely generated ℤ-graded R-module, and I a homogeneous ideal of R. Let N be a graded submodule of M. Fix k∈ ℕ. Denote H:=Ext^k_R(L,M) (or Tor_k^R(L,M)) and H_n:=Ext^k_R(L, M/I^nN) (or Tor_k^R(L,M/I^nN)). When I is generated by elements of positive degree, we show that the local v-number of H_n either coincides with that of colon submodule (0:_H I) or is a polynomial in n of degree one whose leading coefficient is one of the degrees of the generators of I. This extends the main result of Ghosh-Pramanik [J. Algebra 671 (2025), 61-74], where asymptotic linearity was shown under the assumption that (0:_H I)=0.
ER  -