On the ϕ-Weak Global Dimensions and ϕ-Global Dimensions
Abstract
Let R be a ϕ-ring and put N=Nil(R). We determine the ϕ-weak global dimension of every ϕ-Prüfer ring. Its only possible values are $0$, $1$, and ∞: the value is $0$ when R/N is a field, it is $1$ when R is strong and R/N is a nonfield Prüfer domain, and it is infinite when R is not strong. No nilpotence assumption on N is needed. The proof is based on the canonical kernel K=ker(R→ϕ(R)). In the local non-strong case, K is a nonzero torsion divisible module over the valuation domain R/N, and its self-Tor groups satisfy TorR2n(K,K)=0 (n≥ 1), TorR2n+1(K,K)≠ 0 (n≥ 0). It follows that fdRK=∞, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the ϕ-global dimension of ϕ-Dedekind rings. For strong ϕ-rings we obtain a change-of-rings theorem without assuming that the quotient map R→ R/N splits. If D=R/N, Q=Frac(D) and T=T(R) is the total quotient ring, then R≅ T×Q D. Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion D-module M, regarded as an R-module through R→ D, satisfies fdR(M)=fdD(M), pdR(M)=pdD(M). Consequently, for every strong ϕ-ring, ϕ−wgd(R)=wgld(R/N), ϕ−gld(R)=gld(R/N).
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BibTeX
@misc{zhang2026ϕweak,
title = {{On the $ϕ$-Weak Global Dimensions and $ϕ$-Global Dimensions}},
author = {Xiaolei Zhang and Wei Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05567},
eprint = {2610.05567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05567},
}
RIS
TY - UNPB
TI - On the ϕ-Weak Global Dimensions and ϕ-Global Dimensions
AU - Xiaolei Zhang
AU - Wei Qi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05567
UR - https://arxiv.org/abs/2610.05567
AB - Let R be a ϕ-ring and put N=Nil(R). We determine the ϕ-weak global dimension of every ϕ-Prüfer ring. Its only possible values are $0$, $1$, and ∞: the value is $0$ when R/N is a field, it is $1$ when R is strong and R/N is a nonfield Prüfer domain, and it is infinite when R is not strong. No nilpotence assumption on N is needed. The proof is based on the canonical kernel K=ker(R→ϕ(R)). In the local non-strong case, K is a nonzero torsion divisible module over the valuation domain R/N, and its self-Tor groups satisfy Tor^R_2n(K,K)=0 (n≥ 1), Tor^R_2n+1(K,K)≠ 0 (n≥ 0). It follows that fd_RK=∞, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the ϕ-global dimension of ϕ-Dedekind rings. For strong ϕ-rings we obtain a change-of-rings theorem without assuming that the quotient map R→ R/N splits. If D=R/N, Q=Frac(D) and T=T(R) is the total quotient ring, then R≅ T×_Q D. Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion D-module M, regarded as an R-module through R→ D, satisfies fd_R(M)=fd_D(M), pd_R(M)=pd_D(M). Consequently, for every strong ϕ-ring, ϕ−wgd(R)=wgld(R/N), ϕ−gld(R)=gld(R/N).
ER -