Finitistic dimensions of formal power series rings

@misc{zhang2026finitistic,
  title = {{Finitistic dimensions of formal power series rings}},
  author = {Xiaolei Zhang and Weiheng Xia},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20060},
  eprint = {2610.20060},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20060},
}

TY  - UNPB
TI  - Finitistic dimensions of formal power series rings
AU  - Xiaolei Zhang
AU  - Weiheng Xia
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20060
UR  - https://arxiv.org/abs/2610.20060
AB  - We consider the big finitistic projective and flat dimensions of a commutative ring and of its formal power series ring. When the ring is Noetherian, both invariants increase by one. Coherence of the power series ring does not force either equality to hold. For every n≥0 and every d with n+1≤ d≤∞, the flat pair (n,d) can be realized. The finite projective pairs are realized when n≥1, and also for (0,1). For every n≥0, one family realizes (n,∞) for both invariants at the same time. The seed is a zero-dimensional perfect local ring. Its power series ring has countably presented modules whose flat dimension is r and whose projective dimension lies between r and r+1, for every r≥1. The remaining finite projective case is controlled by a dichotomy for perfect coefficient rings. If the Jacobson radical is nilpotent, the power series value is one; if it is not nilpotent, there is a module of projective dimension exactly three. In particular, the pair (0,2) is impossible. For one fixed countable perfect ring, the power series ring has finitistic flat dimension one. Its finitistic projective dimension is m+2 when 2^ℵ_0=ℵ_m with 1≤ m<∞, and is infinite when 2^ℵ_0>ℵ_ω.
ER  -