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Goldman primes and higher cyclic presentations for the small finitistic dimension

  • Xiaolei Zhang
  • Hwankoo Kim
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Abstract

Let R be a commutative ring and let fPD(R) be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if R is Noetherian, the maximal ideals of R[X] detect exactly the Goldman primes of R, and this gives the formula fPD(R[X]) =1+sup{depth Rp| p∈ GSpec(R)}. We construct Noetherian local rings Rm with fPD(Rm)=0 and fPD(Rm[X])=m+1, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak (n,d)-conditions. For every n≥2 and h≥1, we construct a local idealization T such that every n-presented cyclic T-module is projective, while fPD(T)=h. The ring T has no nonzero proper finitely presented ideals, but it has a finitely generated ideal I with ExtTi(T/I,T)=0 for i<h and ExtTh(T/I,T)≠0. Thus the cyclic condition controls fPD at presentation level one, but at no higher level.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026goldman,
  title = {{Goldman primes and higher cyclic presentations for the small finitistic dimension}},
  author = {Xiaolei Zhang and Hwankoo Kim},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14326},
  eprint = {2609.14326},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14326},
}
RIS
TY  - UNPB
TI  - Goldman primes and higher cyclic presentations for the small finitistic dimension
AU  - Xiaolei Zhang
AU  - Hwankoo Kim
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14326
UR  - https://arxiv.org/abs/2609.14326
AB  - Let R be a commutative ring and let fPD(R) be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if R is Noetherian, the maximal ideals of R[X] detect exactly the Goldman primes of R, and this gives the formula fPD(R[X]) =1+sup{depth R_p| p∈ GSpec(R)}. We construct Noetherian local rings R_m with fPD(R_m)=0 and fPD(R_m[X])=m+1, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak (n,d)-conditions. For every n≥2 and h≥1, we construct a local idealization T such that every n-presented cyclic T-module is projective, while fPD(T)=h. The ring T has no nonzero proper finitely presented ideals, but it has a finitely generated ideal I with Ext_T^i(T/I,T)=0 for i<h and Ext_T^h(T/I,T)≠0. Thus the cyclic condition controls fPD at presentation level one, but at no higher level.
ER  -