Catenarity and Gelfand-Kirillov dimension in Ore extensions
Catenarity and Gelfand-Kirillov dimension in Ore extensions
复制标题
矿石延伸中的悬链线和 Gelfand-Kirillov 维数
DOI：
10.1016/0021-8693(89)90261-5
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发表时间：
1989
影响因子：
0.9
通讯作者：
G. Sigurdsson
中科院分区：
文献类型：
--
作者：
Allen D. Bell;G. Sigurdsson
如果R是交换泛域（例如，一个不可约代数变量的坐标环），则R中任何素理想的高度与相应因子环的维数之和等于R的维数。这意味着，如果Q和P是具有q2 P的R的素理想，则从P到Q的任何饱和素理想链的长度为dim R/P-dim R/Q，因此R是链线。Schelter， Gabber等人已经证明了一些非交换仿射环r的类似命题。本文研究了交换环的Ore扩展是链线时的问题，给出了正负结果。我们证明了对于交换仿射环的局部有限维Ore扩展，如果我们使用Gelfand-Kirillov维数，上述维数陈述仍然成立。
If R is a commutative afline 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 2 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 finitedimensional Ore extensions of commutative affine rings, the above dimension statements are still true if we use the Gelfand-Kirillov dimension.