# 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.

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### Linear independence of characters implies statement about algebraic independence of regular functions?

Part of proposition 3.1.9 of Geck's Algebraic Geometry and Algebraic Groups has the following setup. Let $G$ be a connected affine algebraic group over $k$ an algebraically closed field. If there ...
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### Dimension of the orthogonal algebra?

The following is on page 3 of Introduction to Lie Algebras and Representation Theory by Humphreys: Here the author claims that the dimension of the orthogonal algebra is $2l^2+l$; but I think the ...
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### Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi-projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
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### Definition of codimension of variety

Let $X$ be a variety over field $k$. A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$. So I want to know the definition of codimension of a ...
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### Dimension of affine affine algebras as a module

Suppose that $A\cong \mathbb{R}[f_1,\dots,f_d]$ is a (commutative) affine $\mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $\mathbb{R}[x_1,\dots,x_N]$. When is $A$ a finite-...
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### a.c.c. and d.c.c. on radical ideals in commutative ring of dimension zero

Let $R$ be a commutative ring with unity of dimension zero (i.e. every prime ideal is maximal). Does any of the following two conditions imply the other : 1) $R$ satisfies a.c.c. on radical ideals ...
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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 ...
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### Does $\operatorname{length}(M/xM) \leq \operatorname{rank}(M) \cdot \operatorname{length}(R/(x))$ hold over non-integral rings $R$?

$\DeclareMathOperator{\len}{length} \DeclareMathOperator{\rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says: The basic result of this section expresses ...
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### Axioms that characterize the notion of dimension [duplicate]

Let $\mathcal{C}$ be a class of sets/spaces/structures among which we have a dimension. Namely a map $d:\mathcal{C}\rightarrow \mathbb{N}$ defined in a certain manner that motivated the appellative ...
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### Dimension of a hypersurface of $\mathbb C^n$ / of a cut by a hypersurface

Defining the domain $$\Gamma[V]=\mathbb C [\bar x ]/I(V)$$ for any irreducible variety $V\subset \mathbb C^n$ (by variety, I mean only zero set of a family of polynomials), $\Gamma(V)$ for the field ...
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### Proof of Theorem of Dimension of Fibres

I am following a lecture series on YouTube, and in the series the lecturer skipped the proof of the Theorem on the Dimension of Fibres. I tried to follow the style of the professor's proof but I ...
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### Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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### Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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### Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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### Hochschild dimension

I'm curious; if $A$ is a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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### prove $\dim\mathbb{Z}[X_1,X_2]=3$ from first principles

Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems ...
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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 ...
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### A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...