A.D. Bell: Abstract

## Catenarity and Gelfand-Kirillov dimension in Ore extensions

### Author(s)

Allen D. Bell and Gunnar Sigurdsson

### Publication Information

Appeared in Journal of Algebra 127#2 (1989), pp. 409-425
Math Reviews: 91b:16027

### Abstract (in LaTex)

If \$R\$ is a commutative affine domain (for example, the coordinate
ring of an irreducible algebraic variety), the sum of the height
of any prime ideal in \$R\$ and the dimension of the corresponding
factor ring is the dimension of \$R\$.  This implies that if \$Q\$ and
\$P\$ are prime ideals of \$R\$ with \$Q\supseteq P\$, any saturated
chain of prime ideals from \$P\$ to \$Q\$ has length
\$\dim R/P - \dim R/Q\$, and so \$R\$ is catenary.  Schelter, Gabber,
and others have proven analogous statements for some noncommutative
affine rings \$R\$.  In this paper we study the question of when an
Ore extension of a commutative ring is catenary, giving both
positive and negative results.  We show that for locally finite
dimensional Ore extensions of commutative affine rings, the above
dimension statements are still true if we use the Gelfand-Kirillov
dimension.