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.
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