Abstract
For a domain $A$ containing a field $k$ with ${\rm tr.deg}_{k} A < \infty$, we define a new transcendence degree of $A$ with respect to $k$, which is denoted by ${\rm td}_{k} A$. By using this, we generalize the theorem that for every affine domain $A$ over a field $k$ it holds that $\dim A = {\rm tr.deg}_{k} A$. For example, we show that if $A$ is a quasi-local domain containing a field $k$ with $\dim A = {\rm td}_{k} A < \infty$, then for every Noetherian local $k$-subalgebra $R$ of $A$ it holds that $\dim R = {\rm td}_{k} R$. Moreover we also generalize the theorem due to Gilmer, Nashier and Nichols.
Citation
Hiroshi Tanimoto. "The transcendence degree of an integral domain over a subfield and the dimension of the domain." Nagoya Math. J. 170 145 - 162, 2003.
Information