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Questions tagged [dimension-theory-algebra]

For questions about notions of dimension, rank, or length used in abstract algebra (e.g. Krull dimension, homological dimensions, composition length, Goldie dimension). Questions about dimension of vector spaces, and rank of linear transformations are better placed under the [linear-algebra] tag.

For questions about notions of dimension, rank, or length used in abstract algebra (e.g. Krull dimension, homological dimensions, composition length, Goldie dimension). Questions about dimension of vector spaces, and rank of linear transformations are better placed under the tag.

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147 questions
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A proof for $\dim(R[T])=\dim(R)+1$ without prime ideals?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial…
Martin Brandenburg
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Is $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$ true?

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ refers to the Krull dimension of a ring?…
Zhen Lin
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What is the "dimension" of a locally ringed space?

Let $(X,\mathscr{O}_X)$ be a locally ringed space. If it is a scheme, the natural notion of dimension is the dimension of the subjacent topological space (the size of the biggest chain of irreducible closed subsets). But if $X$ is a manifold, I…
Gabriel
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Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$, how does one…
math-visitor
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Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, or, the Krull dimension of $R$. A few minutes…
Derek Allums
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$\mathbb C[y_1,\cdots, y_\ell]/I,$ where $I$ is generated by the relation $\sum_j(-1)^je_j h_{m-j}$ of symmetric polynomials, is $\ell!$-dimensional

Let $Y_\ell = \mathbb C[y_1,\cdots, y_\ell]$ be an unital associative commutative algebra in $\ell$ variables $y_1,\cdots, y_\ell$. From the theory of symmetric polynomials, we know that the elementary symmetric polynomials $e_j(X_1,\cdots, X_n)$…
user2345678
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Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct…
user521337
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The height of a principal prime ideal

A formal consequence of Krull's principal ideal theorem is the following: If $A$ is a Noetherian ring, and $I$ is an ideal generated by $r$ elements, then any prime ideal which is minimal among those that contain $I$ has height at most $r$. This…
Rankeya
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How many times harder is to think in 3D as compared with 2D?

I'm a chemist, currently going through a course about molecular symmetry and group theory applied to Chemistry. This subject is very demanding in terms of visualization in 3D space. To really grasp the subject, one must be able to see, for example,…
ksousa
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Transcendence degree of $K[X_1,X_2,\ldots,X_n]$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically…
Mohan
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Proving that irreducible components of Zariski closed subsets are minimal prime ideals

So the question is to prove that if $X = Z(\mathfrak{U})$ is a Zariski-closed subset in $A^n$, then $Y = Z(\mathfrak{P})$ is an irreducible component of $X$ if and only if $\mathfrak{P}$ is a minimal prime ideal containing $\mathfrak{U}$ in…
Matt Bright
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6
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Final step of exercise 11.7 from Atiyah-Macdonald ($\dim A[x]=\dim A+1$)

Ex. 11.7 from Atiyah-Macdonald is basically to prove $\dim A[x]=\dim A+1$ for $A$ noetherian. From exercise 11.6, we get $\dim A[x]\geq\dim A+1$, so we are left to prove "$\leq$". I've followed the hint and proved that $\text{ht}(p[x])=\text{ht}(p)$…
rmdmc89
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Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalence does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory defines it as $\dim M=\dim…
QED
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Proof verification: determining the dimension of a polynomial ring from the going up theorem.

I decided to prove that for any field $k$, dim $k[x_1, \ldots, x_n] = n$. Every proof I've seen follows either of these two approaches: Noether normalisation (first prove that if $A$ is a finitely generated domain over $k$, then $\dim A =…
Harambe
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Lower bound on dimension of fibres of a dominant mophism of irreducible affine varieties

Whilst doing exercise $11.4.B$ of Ravi Vakil's "Foundations of Algebraic Geometry", I got stuck with the following problem (although I think that many of the hypotheses are unnecessary and a more general statement can be proved by reducing to…
Tom Oldfield
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