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2018年1月24日 · Krull dimension as defined for rings involves a strictly increasing chain of prime ideals. In algebraic geometry, (for decent settings, i.e. integral finite type schemes over a field), an increasing chain of prime ideals corresponds to a strictly increasing chain of …

2020年7月9日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2020年3月22日 · What is the Krull dimension of $\mathbb{C}[x,y] / (xy)$? I believe the only prime ideals of this ring are $(x,y),(x), (y)$. The supremum of all the heights is $1$ by the sequence $(x) \subset (x,y)$, so I would say that the Krull dimension is $1$, but I am not sure. Can someone tell me if this reasoning is correct?

2022年12月6日 · As I said, whether or not Krull dimension and Zariski dimension agree for affine varieties more generally depends on your definition of "variety", and indeed, the failure of these two notions to agree for non-algebraically closed fields …

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

For graded algebras over a field, does finite Krull dimension imply finite generation? 4 Transcendence degree and Krull dimension of finitely generated algebras

2014年9月22日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2016年12月22日 · In his 1973 paper Krull Dimension in Power Series Rings, J.T.Arnold gives examples of such rings where $\dim R$ is finite, but $\dim R[[X]]$ is infinite. Share Cite

2024年12月20日 · As learning Krull dimension in commutative algebra recently, I cannot successfully construct a non noetherian ring of finite krull dimension. I think the ring must have some special conditions like ...