# 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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### Some exercises about vector spaces

Hey I want ot check if my solutions for this exercise are right. Can someone help me? Let $V$ be a finite dimensional $K$-vector space and $U_1, . . . , U_n$ a family of $K$-subspaces in $V$ . Show ...
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### How many 2D "slices" fit into a 3D object?

I assume that 3D objects are comprised of 2D cross-sections, but how many of these cross-sections fit into a finite 3D object? I know that infinite 2D cross sections would not be enough to have depth ...
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### How many 2D objects fit into a 3D object?

Hoe many times can you stack 2D objects before it becomes 3D? I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional ...
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### Do irreducible finite-type dimension-n schemes admit a "dimension-n atlas"?

We know (eg, see Vakil's Foundations of Algebraic Geometry, 11.1.B) that a scheme of finite dimension $n$ admits an open covering by affines of dimension $\leq n$ with equality holding at least once. ...
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### Can the tangent space dimension of an associated reduced scheme $X_{red}$ be bounded by the tangent space dimension of $X$?

Every scheme $X$ has an reduced scheme $X_{red}$ associated to it. In my understanding one constructs $X_{red}$ by getting rid of all purely infinitesimal informations or formally by dividing the out ...
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### Dimension of quotients of powers of maximal ideal

Let $R$ be a commutative ring with unit, local and Noetherain, with $\mathcal{M}$ maximal ideal. Let $K$ be the residue field. Define  \phi(n) := dim_{K} \frac{\mathcal{M}^n}{ \mathcal{M}^{n+1} } ...
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### Dimension of the quotient ring of a Noetherian local ring by a principal ideal

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $x\in \mathfrak{m}$. Then it is known that $\dim R/xR \geq \dim R-1$: The dimension modulo a principal ideal in a Noetherian local ring. If we add ...
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### Proof explanation) Finite injective dimension of the residue field of a Noetherian local ring implies regularity

I am trying to prove: Let $(R,\mathfrak m,k)$ be a Noetherian local ring. If $\operatorname{inj dim}_R k$ is finite, then $R$ is regular. In this link: Finite injective dimension of the residue ...
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### Projective dimension of $K[x,y]$ over $K[xy]$

Let $K$ be a field and $K[x,y]$ the polynomial ring in two variables $x$ and $y$ over $K$. Let $R = K[xy]$ be the subring generated as a $K$-algebra by the monomial $xy$. My question is: What is the ...
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### Subspace and Dimension of a Homomorphism/Linear Mapping

I am very confused about the following exercise: Let $V, W$ be vector spaces over a field $F$. Show that $Hom_F(V, W)$ is a vector subspace of the set of all mappings $Maps(V, W)$ from $V$ to $W$. It ...
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### About the proof of dimension formula for flat morphisms

I'm reading Algebraic Geometry written by Hartshorne, and my question is about proposition 9.5 in chapter III: Proposition. Let $f:X\to Y$ be a flat morphism of schemes of finite type over a field $k$...
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### References for bounds on dimension of particular matrix spaces?

I'm looking for references and well-known results about bounds on dimension of particular matrix spaces. For instance, the first result that came up to my mind was a Flander's theorem which explains ...
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### "Algebraic dimension" for finite-dimensional (non-associative) algebras?

Let $V$ be some finite-dimensional vector space (over some field $\mathbb{K}$), then a (possibly non-associative) algebra $A$ on $V$ corresponds to a bilinear map $V \times V \to V$. I prefer answers ...
1 vote
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### Plane versus 3D coordinates

We know that line $ax+by+c=0$ is one dimensional and the plane $ax+by+cz+d=0$ is two dimensional. My question is if line is one dimensional so why 2D points $(x, y)$ are used for line? And if plane ...
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### Set of singular points of a normal variety.

In my algebraic geometry course these result appeared: If X is a normal variety, then the set of singular points $S$ have codimension $\geq 2$. (Here normal means that $\mathcal{O}_x$ is integrally ...
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### Codimension of the singular locus.

I was studying Algebraic Geometry and I found the following result: If $X$ is a normal variety, the set of singular points $Sing(X)$ has codimension $\geq 2$. I understand this result and its proof, ...
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### Gelfand-Kirillov dimension of a finite extension.

In my commutative algebra course we learned about the Gelfand-Krillov dimension. This is the definition: $\newcommand{\GK}{\mathsf{GKdim}}$ Let $A$ be a $K$-algebra of finite type and $V$ a $K$-vector ...
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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 ...
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### Subspaces in linear algebra - Matrix

Let $S=$ {$a_{ij} \in M_{3}(\mathbb{R}):a_{11}+a_{12}+a_{13}=a_{21}+a_{22}+a_{23}=a_{31}+a_{32}+a_{33}$} $S$ is a subspace of $M_{3}(\mathbb{R})$ and dim $S = 7$ I tought I could arrive somewhere ...
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### Understanding Matousek's proof of Equiangular lines

In Miniature 9 in 33 Miniatures by Matousek, he proofs that: The largest number of equiangular lines in $\mathbb R^3$ is 6, and in general, there cannot be more than $\binom{d+1}{2}$ equiangular ...
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### What does it mean A contains a field k mapping isomorphically onto the residue field A/m in Atiyah Macdonald 11.21?

I'm reading Atiyah Macdonald 11.21. But I'm not sure what "A contains a field k mapping isomorphically onto the residue field A/m" means and have no idea to apply proposition 11.20 Can ...
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### Generalizing a previous result on height of irreducible polynomials (special case of Krull's Principal Ideal Theorem)

In my previous question I showed that the height of an irreducible polynomial $f \in \mathbb{C}[x_1, \cdots x_n]$ is $1$. This I was able to generalize to any Noetherian domain. Basically, the idea of ...
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### Dimension of quasi-affine variety

Proposition 1.10. If $Y$ is a quasi-affine variety, then $\dim Y=\dim \overline{Y}$. If $Z_{0}\subset Z_{1}\subset \cdots \subset Z_{n}$ is a sequence of distinct closed irreducible subsets of $Y$, ... 52 views

### Proving equality of column space to field

Question: Let $A$ be a matrix of $m\times n$ and $B$ matrix of $n\times m$ over field $F$. Given that $AB = I_m$ ($m\times m$ unit matrix), prove that the column space of $A$ is equal to $F^m$. My ...
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### Let $T : U → V$ and $S : V → W$ be linear transformations. How is $\operatorname{rank}(ST)$ related to $\operatorname{rank}(T)$?

Let $T : U → V$ and $S : V → W$ be linear transformations. How is $\operatorname{rank}(ST)$ related to $\operatorname{rank}(T)$? I know that $\operatorname{img}(S \circ T) = \operatorname{img}(S)$. I ...
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### Feedback to Basis and Dimension of $A:= \{x\in\mathbb{R}^4 | x_2 + 3x_3= 0, x_1=x_2\}$

My question: Because of $\mathbb{R}^4$ I assume the vector has to have $x_1,x_2,x_3,x_4$ To calculate the basis I assume that $x_1= 1$ because $x_1=x_2 as$ the problem says and $x_4= 0$ because this ...
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### Finding the dimension of the Kernel $\mathbb{R}^{3*2} \rightarrow \mathbb{R}_4[x]$

Approximate Question asked in Exam You have an $\mathbb{R}^{3*2}$ matrix with linear transformation to $\mathbb{R}_4[x]$ (polynomial of degree $5$). Now you had to find the dimension of the Kernel. I ...
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### If $l$ is an integer such that $k \le l \le m$, show that there exists subspace $X$ of $V$ such that $U \subseteq X \subseteq W$ and $\dim(X)=l$.
Let $U$ and $W$ be subspaces of a vector space $V$ such that $U \subseteq W, \dim(U)=k,$ dan $\dim(W)=m$ with $k \lt m$. If $l$ is an integer such that $k \le l \le m$, show that there exists subspace ...
I was messing around a little bit and I got his claim: Proposition: Let $X \subseteq A_n$, be an affine variety. Then $\dim X=\text{height }I(X)$, where $I(X)$ is the ideal generated by $X$. I know ...