Year: 2026
Type: preprint
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 $A\to A_N$ 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 $H\cong\Tor_1^A(I,M)$ and $I\otimes_{A/N}H=0$. 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.
Source: arXiv (Cornell University)
Author: Zhang, Xiaolei
Institution: Tianshui Normal University
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215049165