Talk:Krull dimension

Some edits to improve the readability of the article

 * No symbol has been specified for the Krull dimension of a ring, the definition should be changed to something like:
 * We define the Krull dimension of R to be the supremum of the lengths of all chains of prime ideals in R and we denote it by $$\operatorname{dim}_{\rm K}(R)$$ (or simply $$\operatorname{dim}(R)$$ when there is no risk of confusion ).


 * Incorrect use of punctuation: (geometers call it the ring of the normal cone of I.) should be changed to (geometers call it the ring of the normal cone of I).
 * Add some links:
 * the space of prime ideals of R equipped with the Zariski topology --> the space of prime ideals of R equipped with the Zariski topology
 * The equality holds if R is finitely generated as algebra (for instance by the noether normalization lemma). ---> The equality holds if R is finitely generated as an algebra (for instance by the Noether normalization lemma).
 * In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles. ---> In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.

This follows from the following observation: for any prime ideal $$\mathfrak{p}\varsubsetneq R$$ consider the localization of $$R$$ to the multiplicative system $$S=R\setminus\mathfrak{p}$$ which we denote by $$S^{-1}R=:R_{\mathfrak{p}}$$; the natural map $$j\colon R\to R_{\mathfrak{p}}$$ induces a bijection "$\lbrace\text{prime ideals }I\subseteq\mathfrak{p}\varsubsetneq R\rbrace\rightleftharpoons\lbrace\text{prime ideals }I\varsubsetneq R_{\mathfrak{p}}\rbrace,$|undefined"
 * Rename the Notes section as References;
 * Introduce a Notes section for remarks and clarifications on the many facts listed in the article.
 * Add the following note concerning the fact the height of $$\mathfrak{p}$$ is the Krull dimension of the localization of $$R$$ at $$\mathfrak{p}$$ to the Note section

defined by $$I\mapsto I_{\mathfrak{p}}=\lbrace x/t\mid x\in I,\,t\in S\rbrace$$, with inverse $$J\mapsto j^{-1}(J)=\lbrace x\in R\mid x/1\in J\rbrace$$.
 * I am concerned with the use of both I and I to denote an ideal: not only is this confusing for the reader, but the symbol I (or $$\mathbb{I}$$) is also commonly used to denote either the set of imaginary numbers or the compact $$[0,1]\subset\mathbb{R}$$ (especially in algebraic topology).

Please, let me know what you think.--Ale.rossi91 (talk) 23:08, 1 February 2020 (UTC)

Example
It seems to me that there is an error in computation of the Krull dimension of (Z/8Z)[x,y,z] : we get a chain of prime ideals of length four by adding the (0) ideal to the chain that is given : $$(0) \subsetneq (2) \subsetneq (2,x) \subsetneq (2,x,y) \subsetneq (2,x,y,z) $$. Thus I think that the dimension is 4.

129.199.2.17 (talk) 11:38, 13 February 2009 (UTC)


 * It's correct, because we don't count (0)? (Otherwise, the field would have the dimension 1.) -- Taku (talk) 21:25, 13 February 2009 (UTC)


 * In fact, we are both wrong, and the article was correct. The ideal (0) is prime if and only if the ring is a domain. The example is not a domain, so (0) is not prime. In the case of a field, the only prime ideal is (0), because the whole ring (field) is never a prime ideal. Thus the dimension of a field is still 0.82.67.178.125 (talk) 22:33, 14 February 2009 (UTC)

Eh?
Eh?

Mr. Billion 08:47, 12 Jan 2005 (UTC)