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Seminario: Krull dimension of the power series ring over nonSFT domains

Giovedì 28 Maggio 2015, ore 16:00 - Aula 2AB40 - Byung Gyun Kang

ARGOMENTI: Seminari

Seminario

Giovedì 28 Maggio 2015 alle ore 16:00 in Aula 2AB40, Byung Gyun Kang (Pohang University of Science and Technology (POSTECH), South Korea) terrà un seminario dal titolo "Krull dimension of the power series ring over nonSFT domains".

Abstract
There is a large class of rings called SFT rings. A ring $A$ is an SFT ring if there does not exist an infinite sequence of elements $a_1,\dotsc,a_n,\dotsc$ such that $(a_{(n+1)})^{(n+1)}$ is not contained in $(a_1,\dotsc,a_n)$ for every $n$.
Examples of nonSFT rings are rings with the nonNoetherian spectrum, finite dimensional nondiscrete valuation domains, Prufer domains which are not generalized Dedekind domains, nonNoetherian almost Dedekind domains, and integral domains with nonzero idempotent prime ideal, etc.
In 1982, J.T. Arnold showed that the Krull dimension of $D[\![x]\!]$, $D$ a nonSFT ring, is infinite, in fact '$\aleph_0$'. After that several authors have improved the results.
In 2014, Kang, Park, Loper, Lucas, Toan showed that the Krull dimension of $D[\![x]\!]$, $D$ a nonSFT ring, is 'at least $\aleph_1$'.
In 2013, Loper and Lucas showed that the Krull dimension of $D[\![x]\!]$, $D$ a nonNoetherian almost Dedekind domain, is at least $2^{(\aleph_1)}$.
In this paper, we generalize all the previous results to show that the Krull dimension of $D[\![x]\!]$, where $D$ is a nonSFT domain, is at least $2^{(\aleph_1)}$.
We also present an example, which demonstrates that $2^{(\aleph_1)}$ is the greatest lower bound for the Krull dimension.

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