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.
164
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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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1
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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 ...
- 1
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Why does different nondimensionalizations give different results? Although the results should be the same.
I have some problems with the non-dimensionalization of the Hamiltonian of motion in a Coulomb field.
The Hamiltonian has a following form:
$$H=-\frac{\hbar^2}{2\mu^*} \Delta_r-\frac{e^2}{\epsilon_0 r}...
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If sum of subspaces is $V$, show $V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff \dim U_1 + \dim U_2 +\ldots + \dim U_r = n$ [duplicate]
Let $V$ be a vector space with $\dim V = n <\infty$ and $U_1, U_2,\ldots, U_r$ subspaces of $V$, whose
sum space is all of $V$.
Prove that
$V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff \dim U_1 + \...
- 39
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1
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Compute $\dim_{\mathbb{Q}}(\mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,s^2t-st+s-1))$
In the paper the homological algebra of Artin groups by Craig C. Squire it is stated that the following ring:
$$R=\mathbb{Z}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,(st+1)(s-1))$$
seen as an abelian group has ...
- 1,604
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0
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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 ...
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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 ...
- 3,298
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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.
- 167
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1
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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 ...
- 507
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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.
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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) \...
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1
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1
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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 ...
- 1,138
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Tools for finding real dimension of an algebraic variety
I have a system of polynomial equations with the unknowns being real numbers.
The set of solutions is infinite.
What software can I use to compute the real dimension of the solution set?
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1
answer
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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 ...
- 1,164
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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) +...
4
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1
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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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0
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63
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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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1
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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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votes
1
answer
106
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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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0
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33
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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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67
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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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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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1
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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
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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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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
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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2
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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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370
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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 ...
- 1,604
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1
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384
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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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1
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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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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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2
answers
215
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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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1
answer
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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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votes
1
answer
92
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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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1
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148
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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 $Y$, ...
user805324
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votes
2
answers
52
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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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votes
2
answers
733
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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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1
answer
37
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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
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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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1
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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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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 ...
