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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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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0 votes
1 answer
29 views

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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Is every parameter outside of the union of minimal primes?

Proposition 1: Let $R$ be a commutative, Noetherian ring and $\ \mathfrak{p} \in \operatorname{Spec}(R)$. If $\operatorname{ht}(\mathfrak{p})=h$, then there exist $y_1, \ldots, y_h \in R$ such that $\ ...
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2 votes
1 answer
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What is the dimension of $T$?

Let $U$, $V$ and $W$ be finite dimensional linear spaces (with dimensions $l$, $m$ and $n>0$, respectively) over field $\mathbb{F}$, and let $f\in{\rm Hom}$ $(U,V)$, $g\in{\rm Hom}$ $(U,W)$ such ...
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The Dimension of the Solution Set of a Set of Non-affine commutative Polynomial Equations

I am reading the webpage to learn how to determine the dimension of the solution set of a set of polynomial equations. For the first case (the commutative polynomials), it is mentioned that if it is ...
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1 vote
1 answer
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Direct sum of images of endomorphisms

Let $f, g: V\to V$ be $K$-linear functions, $V$ a $K$-vector space for some field $K$. Show, that if $V=\text{im}(f)+\text{im}(g)=\ker(f)+\ker(g)$ and $V$ has finite dimension, then $V=\text{im}(f) \...
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Visualizing a 4D object in 3D space

out of curiosity i am trying to visualize a 4D object in 3D space in Blender (3D modeling software). Using Python i have access to all the generated 4D vertices, 4D Edges and 4D polygons of the 4D ...
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1 vote
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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 ...
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1 vote
1 answer
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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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How to compute the dimension of $V(f,g)$ in $\mathbb Z[x_1,...,x_n]$

Consider the ring $R = \mathbb Z[x_1,...,x_n]$ and let $\text{Spec}(R) = \{q \subset R : q \text{ a prime ideal}\}$ be the set of prime ideals of $R$. For $I \subset R$ an ideal, define $V(I) = \{q \...
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Dimension of ring by $\dim_k (m^i/m^{i +1})$ for all $i(2≦i<∞)$ is the same as the embedding dimension?

Let $A$ be a Noetherian local ring. Define $m$ be it's maximal ideal and $A/m$ be residue field. Then we can define embedding dimension of $A$ by $\dim_k(m/m^2)$, here $\dim_k m/m^2$ is dimension of $...
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1 answer
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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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Dimension of $R\left/aR\right.$.

In my algebra course I was asked to solve the following problem: Let $R$ a finite type $K$-algebra and suppose $R$ is an integral domain. If $0\neq a\in R$ is not invertible show that $\dim (R\left/aR\...
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Dimension of Quot scheme

In the paper of Mukai & Sakai https://link.springer.com/article/10.1007/BF01171494, give a smooth projective curve $C$ and a vector bundle $E$ of rank $r$, they use the Grassmannian bundle $\...
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1 vote
1 answer
118 views

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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8 votes
1 answer
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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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1 answer
37 views

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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2 votes
2 answers
187 views

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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2 votes
1 answer
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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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0 votes
1 answer
90 views

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 $Y$, ...
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0 votes
2 answers
38 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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Derivative of Multidimensional Rosenbrock Function

im having trouble understanding the derivative of the multidimensional Rosenbrock Function as shown here: Derivative I understand how to derive multidimensional functions in generel like ...
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2 answers
92 views

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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  • 505
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0 answers
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Volume of a Hyper-sphere in n dimension

Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space. [Hint: The formulas are different for n even and n odd.]
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1 answer
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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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0 answers
26 views

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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1 vote
1 answer
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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 ...
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  • 1,951
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A question on a result.

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 ...
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3 votes
0 answers
64 views

$X, Y$ subspaces of $V$, $\dim(X) \geq \dim(Y) \implies \exists f$ linear s.t. $f(X)=Y$

Suppose that $V$ is a $K$-vector space and $X, Y$ vector subspaces of $V$ with $\dim(X) \geq \dim(Y) \implies \exists$ a linear map $f : V \to V$ such that $f(X)=Y$. I do not know what strategy to ...
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4 votes
0 answers
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Dimension counting in exact sequences of Hopf algebras

It is known that the category of commutative and cocommutative Hopf algebras over a field $k$ is an abelian category. So we can talk about exact sequences. Reading a paper I found the following ...
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Fibered product of varieties and dimension

I am trying to do exercise 3.1.5 in Liu's book which asks the following. Let $X,Y$ be algebraic varieties over a field $k$. Show that $\dim(X\times_k Y)=\dim X\, + \dim Y$. The hint is to use ...
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1 answer
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Does the number of elements in Vector space and the dimension of Vector space same? [closed]

The number of elements in any basis of a vector space is called the dimension of Vector space. In the Vector space F^n(F) and Null Space; the dimension of Vector Space and the number of Vectors in the ...
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2 votes
1 answer
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For $0 \xrightarrow{f_0} \mathbb{C}^n \xrightarrow{f_1} \mathbb{C}^\ell \xrightarrow{f_2} \mathbb{C}^r \xrightarrow{f_3} 0$, what is $\ell$?

So the question is as follows: Let $0, \mathbb{{C}}^{n}, \mathbb{{C}}^{\ell}$ and $\mathbb{{C}}^r$ be $\mathbb{{C}}$-vector spaces (where $0$ is the trivial vector space), and let $f_i$, $(i=0,…,3)$, ...
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1 vote
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Prove that the sequence $(\dim{\ker(f^{k+1})}-\dim\ker(f^k))_k$ is decreasing?

Let $E$ a $\mathbb{K}$-space vector with $\dim(E)=n$. Let $f\in \mathcal{L}(E)$. I have to prove that the sequence $(\dim{\ker(f^{k+1})}-\dim\ker(f^k))_k$ is descreasing. I do not want to use quotient ...
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6 votes
2 answers
516 views

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 ...
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16 votes
1 answer
480 views

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 ...
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Does the notion of projective/injective dimension exist for chain complexes?

Let $\mathcal{A}$ be an abelian category, and let $Ch_*(\mathcal{A})$ denote the associated category of chain complexes. We define projective resolutions in this general case: For an object $X \in Ob(...
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What is the dimension of $\mathbb{C}[X,Y,Z] / \langle X^{c-2} , Y^c , Z^c \rangle$ as a vector space over $\mathbb{C}$?

What is the dimension of $\mathbb{C}[X,Y,Z] / \langle X^{c-2} , Y^c , Z^c \rangle$ as a vector space over $\mathbb{C}$? I suspect this is really just a question of combinatorics as it seems my problem ...
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1 vote
1 answer
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A set with co-dimension less continuum

Define the dimension of a vector space to be the cardinality of any basis for the vector space. Also every subspace $S$ of $\mathbb R$ also has a codimension, which is the dimension of $\mathbb{R}/S$, ...
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If $I\subset S$ is the homogeneous ideal of a finite set of points in $\mathbb P^n$, then $\dim S/I=1$.

Let $k$ be an algebraically closed field and $X=\{p_1,\ldots,p_r\}$ be a set of $r$ distinct points in $\mathbb P^n(k)$. Write $I$ for the homogeneous ideal of $X$ in $S=k[x_0,\ldots,x_n]$. I read ...
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1 vote
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Proof of Hilbert-Serre [Atiyah-Macdonald thm 11.1]

I was very confused, for proving the Hilbert-Serre thm. The following is our assumptions. $A=\bigoplus_{n=0}^{\infty}A_n$ is graded Noetherian ring, so $A$ is finitely generated $A_0$-algebra, ...
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4 votes
1 answer
228 views

Proof of Proposition 11.20 of Atiyah-Macdonald

I struggle with verifying the pole order inequality asserted in the proof of proposition 11.20. (Full statement and proof of the proposition can be found here: Atiyah-Macdonald 11.20 and 11.21) My ...
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3 votes
1 answer
252 views

Exercise 11.1.G Vakil FOAG

I am trying to solve Ex 11.1.G from Ravi Vakil's FOAG. It says if $X$ is an affine scheme over $k$, a field and $K|_k$ is an algebraic field extension, then $X$ is of pure dimension $n$ iff $X_K:=X\...
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  • 3,902
3 votes
1 answer
110 views

Is the dimension of a Noetherian local ring equal to its associated graded ring?

For a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$, let $I$ be a primary ideal in $A$, the associated graded ring is $$ \bigoplus_{n=0}^{\infty} I^n/I^{n+1}$$
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3 votes
1 answer
140 views

Proving second injective change of rings theorem

I am trying to prove the second injective change of rings theorem:$\DeclareMathOperator{\id}{id}$ Let $R$ be a ring, $A$ be an $R$-module, $x\in R$ a central non-unit non-zerodivisor such that $xa\...
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4 votes
0 answers
287 views

Is there a thing as a "Negative Dimensional Space"?

I am wondering if there is a pure mathematical, abstract, extension of the Euclidean space for negative dimensions. Do you know any studies in topology, regarding this matter? I did a little research ...
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