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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31 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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34 views

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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2answers
42 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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25 views

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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1answer
31 views

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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1answer
29 views

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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54 views

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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57 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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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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60 views

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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62 views

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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103 views

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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271 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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1answer
405 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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39 views

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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1answer
22 views

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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1answer
47 views

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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78 views

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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1answer
174 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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1answer
190 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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1answer
60 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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56 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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171 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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35 views

Gabber's analogue of Bernstein inequality for category $\mathcal{O}$

Let $\mathsf{k}$ be an algebraically closed field of zero characteristic. It is a well known result due to Gabber that for any finite dimensional algebraic Lie algebra $\mathfrak{g}$ and every ...
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1answer
49 views

Does locally DVR implies Dedekind Domain when it is 1-dimensional, semi-local domain but Noetherian not given

Let R be a semi-local integral domain of dimension 1 such that $\forall P \in Spec{R} $ such that $P \ne 0$ we have, $R_P$ to be a Discrete Valuation Ring. Then prove that $R$ is a Dedekind Domain? ...
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1answer
44 views

Is it true that $\dim_k R/(IJ) \leq \dim_k R/I + \dim_k R/J$ for ideals $I,J$ of the $k$-algebra $R$ of Krull dimension one?

Let $R$ be a $k$-algebra of Krull dimension one where $k$ denotes a field. Let $I,J \subseteq R$ be two ideals of $R$ of dimension zero (that is $R/I$ has Krull dimension zero). Is it true that $$\...
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2answers
127 views

Dimension of a vector space/ subspace with a finite basis [closed]

Is the dimension of a vector space/subspace with a finite basis always the same as the number of elements in each vector and if so how can I derive that from the definition of a dimension?
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1answer
24 views

norm and projections on inner product space

How do I show that if $\Vert Px-Qx \Vert <\Vert x \Vert$ for any $x\in V$ not $0$, then $\dim\left(M\right)=\dim\left(N\right)$. $V$ is an inner product space and $M, N$ are sub-spaces of $V$.$P$ ...
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1answer
31 views

We have to prove that $H=${$(x_1,…,x_n)\in{\mathbb{R^n}}|a_1x_1+…+a_nx_n=0$} is an hyperplane of $\mathbb{R^n}$.

I've got $\space$ $V$ $K$ - vectorial space, and $H$ which is a subspace of $V$. We say that $H$ is a hyperplane when $dimH=n-1$. If we've got $\space a_1,a_2,...,a_n\in{\mathbb{R}}$ which are not ...
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63 views

Krull dimension of affine algebra is equal to maximum of transcendence degrees

I'm going through some introductory books on commutative algebra and I'm struggling with the following problem: Let $A$ be a non-trivial affine algebra over the field $K$. Since $A$ is noetherian, we ...
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1answer
68 views

Finite dimension quotient ring

Let $R=C[x_1,...,x_n]$ and $M$ be a maximal ideal of $R$ such that $R/M$ is a finite dimensional $C-$algebra. Can we deduce that $R/M^n$ for n>1 is also finite dimensional $C$-algebra? We know that $...
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1answer
38 views

If each number line also has the Imaginary number line in it, does that mean x, y, z is six dimensions?

Are we to assume that x is two dimensional? I can somewhat picture this, but I'm having trouble with a number line for i (square root of negative one) with more than one dimension. i is not a ...
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1answer
55 views

Dimension, graph of functions of several variable and it's visualization.

To visualise a scalar function of $n$ variables we consider its graph in $(n + 1)$ dimensional space. If $\mathit{f} :U \subset \mathbb{R}^n \to \mathbb{R}$ is a function of n variable its graph ...
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1answer
317 views

Using rank-nullity for rankA + rank(adj(A)) = n iff col(adj(A)) = nullA

Let A be an $n \times n$ matrix. Show that $A$ is not invertible and $\text{rank } A + \text{rank}(\text{adj}(A)) = n$ if and only if $\text{col}(\text{adj}(A)) = \text{null } A$. The rank-nullity ...
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1answer
1k views

Find the dimension of the subspace of R^4 spanned by the set {(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)}. Hence find a basis for the subspace

GIven set is not Linearly dependent hence not a basis. So should we take basis as {(1.0.0.0),(0,1,0,0),(0,0,1,0),(0,0,0,1)} and give as dim(R^4) = 4 or any other solution is expected?
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1answer
194 views

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 ...
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1answer
173 views

Hilbert-Samuel multiplicity of a local ring of positive dimension is positive?

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Then $e(R)=(d-1)!\lim_{n\to \infty} \mu (\mathfrak m^n)/n^{d-1}$. (Here $e(R)$ denotes Hilbert-Samuel multiplicity of $R$) . ...
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65 views

Explicit computation of projective dimension

I want to study the projective dimension $h_p(M)$ of an $A$-module $M,$ which it was defined as the least integer $n\in\mathbb{N}$ ($0\in\mathbb{N}$) such that exists a projective resolution of length ...
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80 views

On reduction of ideals

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Let $I$ be an $\mathfrak m$-primary ideal of $R$ i.e. $\sqrt I =\mathfrak m$ . How to show that there exists $x_1,...,x_d \in ...
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1answer
71 views

Do linear independent sets of a central (Lie-)algebra remain linear independent in scalar extensions?

Let K'/K be a field extension, L' a K'-(Lie-)algebra and L a K-(Lie-)algebra, such that $L'\subseteq \K'\otimes_K L$ (by injective embedding). Than we can review $L'=\K'\cdot L$. Consider there is a K-...
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1answer
35 views

If $R = k[T_0,\cdots,T_n]/I$ is integral, then $\text{dim Proj R}=\text{dim D}_+(T_i)$ for some $i$

Why is it that if $R = k[T_0,\cdots,T_n]/I$ is integral, $k$ a field, then $\text{dim Proj R}=\text{dim D}_+(T_i)$ for some $i$? $D_+(T_i)$ is just the set of primes of $\text{Proj R}$ which don't ...
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1answer
51 views

How can the height of a non zero prime ideal be $0$?

In my exercise sheet I am supposed to prove that the only prime ideal of height $0$ in an integral domain domain is $(0)$, and to compute the prime ideals of height $0$ in $\mathbb R[x,y]/(xy)$. I ...
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1answer
74 views

On the dimension of a linear transformation

On the general scope, I was wondering how to define the dimension of a linear transformation, if it even has sense. The only ressource I've found addressing the question is a short youtube video (c.f. ...
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1answer
116 views

Dimension of vector space of all matrices satisfying AB=BA

Let $A$ be a $55\times 55$ diagonal matrix with characteristic polynomial $(x-c_1)(x-c_2)^2(x-c_3)^3,\ldots ,(x-c_{10})^{10}$, where $c_1,c_2,\ldots ,c_{10}$ are all distinct. Let $V$ be the vector ...
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1answer
47 views

If $X$ infinite dimensional and $F\colon X\to\mathbb{F}^{n}$ linear, then $\ker(F)$ infinite dimensional subspace of $X$.

Suppose that $X$ is an infinite dimensional vectorspace over the field $\mathbb{F}$ (real or complex numbers). Let $F\colon X\to\mathbb{F}^{n}$ be a linear map. I want to prove that $\ker(F)$ is an ...
3
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1answer
97 views

Complements of a subspace

Show that a non-trivial subspace $U$ of $V$ has two virtually disjoint complements iff $dim(U)\geq \frac{dim(V)}{2}$. Definition 1:$S$ and $T$ are said to be virtually disjoint if $S\cap T=\{0\}$. ...
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1answer
302 views

Show that if the height of a prime ideal is zero, then it is a prime ideal belonging to 0

I was reading Atiyah-Macdonald p. 122, the proof of the Krull's principal ideal theorem: Let $A$ be a Noetherian ring and let $x$ be an element of $A$ which is neither a zero-divisor nor a unit. Then ...
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1answer
232 views

The associated graded ring of the localization $k[x_1,\dots,x_n]_{(x_1,\dots,x_n)}$

I was reading the Atiyah-Macdonald p. 121: Example. Let $A$ be polynomial ring $k[x_1,\dots,x_n]$ localized at the maximal ideal $\mathfrak{m}=(x_1,\dots,x_n)$. Then $G_{\mathfrak{m}}(A)$ is a ...