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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
80 views

Computing the height of an ideal...?

I hope I'm not overbearing in this site. Yes, I'm still struggling. If you can, I have a question about primary decomposition that still needs help, you can find it in my page. Now I wanted to find ...
Goffredo Valenza's user avatar
6 votes
0 answers
170 views

Real-valued dimension

Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$. Given a commutative ring with unity $R$, $R\operatorname{-Mod}$ denotes the category of $R$-modules. Given $R$-modules $A$ ...
Smiley1000's user avatar
1 vote
0 answers
41 views

misunderstanding on real algebraic varieties

Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find ...
Hair80's user avatar
  • 393
1 vote
0 answers
23 views

Proof verification: dimension of inverse limit of modules

I'm looking for clarification as to whether or not the following proof is valid. I am unsure in justifying the last paragraph. Thank in advance as always, M Claim: Let $R$ be a ring and $(M_{k} , q_{...
mathieu_matheux's user avatar
1 vote
0 answers
29 views

Rank preservation of inverse limit of modules

I am asking about the properties of rank over an inverse limit. Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be ...
mathieu_matheux's user avatar
1 vote
1 answer
101 views

To calculate the dimension of a vector space [closed]

Let $E$ and $F$ be two subspaces of $\mathbb{R}^n$, and let $$G = \{\begin{pmatrix} X \\ Y \end{pmatrix}\in \mathbb R^{2n} \mid X+Y \in E, Y \in F\}$$. I am trying to calculate the dimension of $G$, ...
Lou16's user avatar
  • 145
1 vote
1 answer
48 views

Is the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq \dim_RU_1+\dim_RU_2+\dim_RU_3$

If V is the linear space over the field R of real numbers, and $U_1, U_2, U_3$ are subspaces of this space. Is the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq \dim_RU_1+\dim_RU_2+\dim_RU_3$ ...
ugjumb's user avatar
  • 205
2 votes
2 answers
165 views

Tensor product of simple $sl_2$ modules

I am working on the following problem: Let $M(n)$ be the finite-dimensional, simple $\mathfrak{sl}_2(\mathbb{C})$-module with highest weight $n\in\mathbb{N}_0$. Show that the module $M(l)\otimes_{\...
john_psl1298's user avatar
2 votes
1 answer
97 views

Classical Krull Dimension of Commutative Rings

I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische ...
Dave's user avatar
  • 1,363
1 vote
2 answers
184 views

Dimension of $\mathcal{O}_X(X-\{P\})$.

Let $X$ be a smooth projective and irreducible curve and $P\in X$. I am asked to show that the dimension (as a $k$-vector space where $k$ is a algebraically closed field) of $\mathcal{O}_X(X-P)$ is ...
ASP's user avatar
  • 388
-1 votes
2 answers
120 views

Is the dimension of this subspace 1?

"Let $V=M_{2\times2}(\mathbb{R})$ denote the vectors space of all $2\times2$ matrices with real number entries. Determine which of the following subsets are subspaces of $V$. If it is a subspace, ...
raffaello.sanzio's user avatar
1 vote
1 answer
46 views

Dimension of a Matrix subspace

What is the dimension and the number of basis vectors for a subspace of 3×3 symmetric matrices? Earlier my professor told us that the dimension and the number of basis vectors for a subspace are the ...
Shadow Nik's user avatar
0 votes
0 answers
22 views

About the regular representation of weak hopf algebra

In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
popo's user avatar
  • 11
0 votes
1 answer
38 views

Rank and dimension of matrix in a linear map

Let's say I map a $3 \times 1$ vector $\underline v=(x, y, z)$ by multiplying it with a $3 \times 3$ matrix of rank $2$. Would I be correct in thinking that it transforms all points in 3D space into a ...
michelle tan's user avatar
-1 votes
1 answer
107 views

Suppose U is a subspace of V such that V/U is finite dimensional. Can we say that V is finite dimensional?

Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U. I think we cannot show that V is finite ...
Harry's user avatar
  • 23
1 vote
0 answers
141 views

What is the dimension and nature of this variety?

Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n \...
S Huntsman's user avatar
0 votes
0 answers
94 views

VC dimension of indicator functions is equal to pseudo dimension

I am reading the "Foundation of machine learning" by Mehryar Mohri (https://cs.nyu.edu/~mohri/mlbook/). In the proof of Theorem 11.8, it said the following statement, which I can not ...
Harry's user avatar
  • 699
-1 votes
1 answer
122 views

Find subset of vectors which form basis

Question Let W be the subspace of $R^5$ spanned by$ u_1 = (1, 2, –1, 3, 4)\\ u_2 = (2, 4, –2, 6, 8) \\ u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$ Find a subset of the ...
Sandeep's user avatar
  • 27
0 votes
1 answer
77 views

Question about dimension of a vector space

I'm reading Exercises in Classical Ring Theory (T.Y.Lam) and there is a exercise: I'm not sure how to determine the left (right) dimension of the vector spaces (red underline in the above). My though ...
Na Man's user avatar
  • 135
0 votes
1 answer
67 views

What is the $\mathbb{F}_p$ dimension of $\mathbb{F}_p[G]$?

I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is ...
stillconfused's user avatar
0 votes
1 answer
64 views

Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are spanned by given vectors

I have got the following entrance exam question. Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are respectively spanned by $$\{(1,1,1,...
Srijan's user avatar
  • 12.5k
2 votes
1 answer
67 views

Non-zero counts in increasing dimensions

I am working on a presentation that shows the exponential increase as one increases the number of dimensions, and I'm trying to figure out a way to calculate all non-zero or null counts, which I'll ...
Faeborn's user avatar
  • 21
0 votes
1 answer
342 views

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 ...
Adithya's user avatar
  • 11
1 vote
0 answers
68 views

If sum of subspaces is $V$, show $V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff \dim U_1 + \dim U_2 +\ldots + \dim U_r = n$ [duplicate]

Let $V$ be a vector space with $\dim V = n <\infty$ and $U_1, U_2,\ldots, U_r$ subspaces of $V$, whose sum space is all of $V$. Prove that $V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff \dim U_1 + \...
annnna's user avatar
  • 49
3 votes
1 answer
70 views

Compute $\dim_{\mathbb{Q}}(\mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,s^2t-st+s-1))$

In the paper the homological algebra of Artin groups by Craig C. Squire it is stated that the following ring: $$R=\mathbb{Z}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,(st+1)(s-1))$$ seen as an abelian group has ...
Marcos's user avatar
  • 1,860
1 vote
0 answers
75 views

Questions about the proof of Theorem 13.4 in Matsumura‘s Commutative Ring Theory

I'm reading the Matsumura's Commutative Ring Theory and I'm trying to understand theorem 13.4. Suppose $A$ is Noetherian semilocal ring, $M$ is a finite $A$-module, $\mathfrak{m}$ is the Jacobson ...
Kevin's user avatar
  • 395
3 votes
0 answers
49 views

Computing the global and Hochschild dimensions of a free product of direct products of fields.

Let $k$ be a field (algebraically closed of characteristic 0, but I do not expect it to make a difference). Consider the algebra $A=k^{n+1}\ast k^{m+1}$, where $\ast$ denotes the free product of ...
Sergey Guminov's user avatar
0 votes
0 answers
81 views

Eisenbud commutative algebra corollary 10.7

The $\leq$ direction is clear to me, but I do not know how the other direction works: I understand all the steps but I do not understand how this proofs anything in terms of dim R.
ashold7's user avatar
  • 167
0 votes
1 answer
53 views

Codimension of variety versus linear independence of defining homogeneous polynomials

Let $p_1, ..., p_m \colon \mathbb{R}^n \to \mathbb{R}$ be $m < n$ linearly independent homogeneous polynomials each of degree exactly $d$, and assume they have nonzero intersection. Is the ...
ccriscitiello's user avatar
0 votes
1 answer
83 views

Homological Methods in commutative Algebra Reference | lecture notes/ video lectures

I am studying Homological Methods in Commutative Algebra,TIFR Bombay pamphlet (this). Can anyone suggest any good reference/ notes/ video lectures for this? I am feeling lost. Thanks in advance.
Lemon's user avatar
  • 9
4 votes
2 answers
304 views

how to determine the dimension of a vector space given linear transformations

I was dealing with the following problem: Let $V,W,X$ be vector spaces and let $T:V\to W$ and $S:W\to X$ be linear transformations. (i) Prove Sylvester's Rank Inequality: $ rank(T) + rank(S) - dim(W) \...
pongdini's user avatar
  • 121
1 vote
1 answer
199 views

Localization of Finitely Generated $k$-algebra is Catenary (Vakil FOAG Exercise 11.2.F)

A ring $A$ is catenary if for any two prime ideals $\mathfrak{p} \subset \mathfrak{q}$ of $A$, any maximal chain of prime ideals between them has the same length. The exercise asks to show that any ...
Ray's user avatar
  • 1,280
2 votes
1 answer
214 views

Questions about Theorem 11.4.1 and Exercise 11.4.C in Vakil's FOAG

Background. I am trying to solve Exercise 11.4.C in Vakil's Foundations of Algebraic Geometry (November 18, 2017 draft) (Exercise 11.4.C) Suppose $\pi: X \to Y$ is a proper morphism to an irreducible ...
WLOG's user avatar
  • 1,296
2 votes
0 answers
52 views

Dimension property of certain rings involving localization and quotient by primes?

Following this answer, let us make the following definition: Definition: We say a comm ring $R$ has the "DIM property" iff for every prime ideal $p \subset R$, we have $$ \mathrm{dim}(R_p) +...
Indraneel Tambe 2's user avatar
4 votes
1 answer
52 views

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. ...
Indraneel Tambe 2's user avatar
2 votes
0 answers
49 views

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} } ...
Alberto Pipitone Federico's user avatar
1 vote
0 answers
187 views

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 ...
user302934's user avatar
  • 1,698
1 vote
1 answer
221 views

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 ...
user302934's user avatar
  • 1,698
3 votes
1 answer
156 views

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 ...
Henrique Augusto Souza's user avatar
0 votes
0 answers
94 views

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 ...
rileygrey65's user avatar
0 votes
1 answer
121 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 $k$...
Takuto's user avatar
  • 41
1 vote
0 answers
47 views

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 $\ ...
Murat's user avatar
  • 11
2 votes
1 answer
68 views

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 ...
Dan Sims's user avatar
  • 515
0 votes
0 answers
26 views

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 ...
M.K's user avatar
  • 541
1 vote
1 answer
256 views

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) \...
Niko Gruben's user avatar
2 votes
0 answers
59 views

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 ...
Maman's user avatar
  • 3,300
1 vote
0 answers
79 views

"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 ...
hasManyStupidQuestions's user avatar
1 vote
1 answer
87 views

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 ...
S. M.'s user avatar
  • 1
3 votes
2 answers
231 views

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 \...
Desura's user avatar
  • 2,011
0 votes
1 answer
143 views

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 $...
Pont's user avatar
  • 5,895