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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19 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
31 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
35 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
52 views

Dimension of a vector space/ subspace with a finite basis

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
16 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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2answers
29 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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14 views

direct sum of modules properties and proof

We have V vector space above field F, and $W_1, ..., W_t$ in $V$ vector sub-spaces . How do i prove that the following are equivalent: $\sum _{n=1}^t\left(W_i\right)$ is direct sum $\forall v\:in\:\...
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34 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
58 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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34 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
26 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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5 views

Question about multi dimension concepts

In a multidimension space, we have the concept of hypersurface - the "area" , and hypervolume - the "volume" of some shape in the dimension Specifically, considering a shape in n-dimension, the ...
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1answer
276 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
81 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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67 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
105 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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52 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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78 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
64 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
28 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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27 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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18 views

Common direct complement of the same-dimensional subspaces

Prove that any two subspaces of the same dimension (of a finite-dimensional vector space) have the same direct common complement. Show that the statement isn't true for the three subspaces of the ...
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1answer
57 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
61 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
42 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 ...
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1answer
46 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
156 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
103 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 ...
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85 views

projective and injective dimension of an abelian group $\mathbb Z$ [closed]

How do I compute the projective and injective dimension of an abelian group $\mathbb Z$ viewed as a $\mathbb Z-$module ?
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1answer
74 views

If $a_1,…,a_r$ is an $M$-regular sequence of maximal length, $M/(a_1,…,a_r)M$ has finite length.

Let $M$ be a finitely generated module over a Noetherian local ring $R$, and $a_1,...,a_r$ be an $M$-regular sequence of maximal length. Then, $M/(a_1,...,a_r)M$ has finite length? I guess it is not ...
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1answer
60 views

The multiplicity of the intersection of two curves is the dimension of a certain vector space.

I am reading Fulton's book of algebraic curves and on page 37 there is a theorem 3, which says: The truth is that the proof seems very long and difficult to follow (I am a beginner in algebraic ...
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1answer
109 views

Help Understanding Proof about Dimension Theory

I am reading through Atiyah and Macdonald's Dimension theory, chapter, but I can't understand a step in the proof. The relevant definitions are included below. $\lambda$ is hte length function. Why ...
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3answers
179 views

$V(I)$ consists of a finite set of points if and only $k[x_1,\ldots,x_n]/I$ has Krull dimension zero

I am trying to prove the following: Let $I\subset k[x_1, ..., x_n]$ be an ideal. Show that $V(I)$ consists of a finite set of points if and only if $k[x_1,..., x_n]/I$, seen as k-space vector, is an ...
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110 views

Typo in Introduction to Commutative Algebra by Atiyah-Macdonald?

Proposition 11.4. Let $A$ be a Noetherian local ring, $m$ its maximal ideal, $q$ an $m$-primary ideal, $M$ a finitely generated $A$-module, ($M_n$) a stable $q$-filtration of $M$. Then i) $M/M_n$ is ...
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1answer
26 views

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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235 views

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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1answer
136 views

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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112 views

Hartshorne II.3.22c following the hint

I know this exercise can be solved in several ways, such as Ravi Vakil's proof or this answer, but I'd like to try to follow the given hint if possible. The exercise says the following: Let $f:X→Y$ ...
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101 views

Dimension image of morphism of projective varieties

Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a rational map. Then there exists $U \subset \mathbb{P}^n$ open so that $f_{|U}$ is a morphism. What can we say about the dimension of $\overline{f(U)}$? We ...
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1answer
299 views

Hartshorne II-3.22(b)

Let $f:X\to Y$ be a dominant morphism of integral schemes of finite type over a field $k$. Let $e=\dim(X)-\dim(Y)$. For any point $y\in f(X)$, show that every irreducible component of the fibre $...
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1answer
38 views

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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1answer
78 views

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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122 views

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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33 views

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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48 views

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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109 views

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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1answer
156 views

Exercise 1.8 from Hartshorne

(Hartshorne 1.8) Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \...
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44 views

How to define $\operatorname{dim}(\{0\})$ and $\operatorname{ht}(A)$?

Matsumura's "Commutative Algebra", Chapter 5, Page 72. It follows from the definition that $\operatorname{ht}(\mathfrak p)=\operatorname{dim}(A_{\mathfrak p})\quad (\mathfrak p\in \operatorname{...
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1answer
355 views

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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1answer
143 views

Extending a morphism to a projective scheme

Let $C$ be a reduced affine Noetherian scheme of pure dimension 1 (all its irreducible components have dimension 1) and $p \in C$ a regular closed point. Suppose we have a morphism $C \backslash p \to ...