Minimal Resolutions of k-divisible and (k−1)-dominant Ideals

@misc{welte2026minimal,
  title = {{Minimal Resolutions of $k$-divisible and $(k-1)$-dominant Ideals}},
  author = {Patrick Welte and Guillermo Alesandroni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20565},
  eprint = {2610.20565},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20565},
}

TY  - UNPB
TI  - Minimal Resolutions of k-divisible and (k−1)-dominant Ideals
AU  - Patrick Welte
AU  - Guillermo Alesandroni
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20565
UR  - https://arxiv.org/abs/2610.20565
AB  - We introduce invariants we call k-divisibility and d-dominance, which are combinatorial constraints on the minimal generators of monomial ideals. We then prove that k (respectively, d) is an upper bound (respectively, lower bound) for the projective dimension of a k-divisible (respectively, d-dominant) monomial ideal. This leads us to an explicit minimal resolution of k-divisible, d-dominant ideals when d=k−1 via a Lyubeznik resolution. This framework allows us to prove a conjecture made by Alesandroni. Using these invariants, we also provide a characterization of monomial ideals whose minimal free resolution is attainable from the Taylor resolution by performing exactly one consecutive cancellation. We also prove that for all q,k,d in the range 1≤ d<k ≤ q−1, there exist monomial ideals that are k-divisible and d-dominant with q minimal generators.
ER  -