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The finitistic weak dimensions of Gaussian rings
Abstract
We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring A, with nilradical N, the kernel I of A→ AN satisfies NI=I2=0. A hypothetical module M of flat dimension three gives a nonzero torsion module H=Tor2A(A/N,M) over the valuation domain A/N, with H≅Tor1A(I,M) and I⊗A/NH=0. Two carefully controlled localizations, a propagation argument in the chain ring A/D, where D={x:x2=0}, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.
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BibTeX
@misc{zhang2026finitistic,
title = {{The finitistic weak dimensions of Gaussian rings}},
author = {Xiaolei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38241},
eprint = {2609.38241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38241},
}
RIS
TY - UNPB
TI - The finitistic weak dimensions of Gaussian rings
AU - Xiaolei Zhang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38241
UR - https://arxiv.org/abs/2609.38241
AB - We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring A, with nilradical N, the kernel I of A→ A_N satisfies NI=I^2=0. A hypothetical module M of flat dimension three gives a nonzero torsion module H=Tor_2^A(A/N,M) over the valuation domain A/N, with H≅Tor_1^A(I,M) and I⊗_A/NH=0. Two carefully controlled localizations, a propagation argument in the chain ring A/D, where D={x:x^2=0}, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.
ER -