All Questions
Tagged with linear-algebra ac.commutative-algebra
60 questions with no upvoted or accepted answers
14
votes
0
answers
603
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
10
votes
0
answers
312
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
8
votes
0
answers
292
views
Image of multiplication map in tensor powers of finite-dimensional ring
Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...
- 148k
7
votes
0
answers
226
views
Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,287
6
votes
0
answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
5
votes
0
answers
138
views
Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)
Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
- 32.5k
5
votes
0
answers
220
views
Rank of matrix over UFD polynomial ring
I have a matrix, $M$, size of approximately $20\times 25$, over a polynomial ring $\mathbb{Q}[x_1, \cdots, x_n]$ and a number $r$ such that I would like to test whether or not there exist values for ...
- 51
4
votes
0
answers
219
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Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
4
votes
0
answers
216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
4
votes
0
answers
211
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Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
4
votes
0
answers
312
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Dimension of a commuting nilpotent variety
Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
- 768
3
votes
0
answers
181
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Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
3
votes
0
answers
70
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Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
- 31
3
votes
0
answers
71
views
Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
- 1,766
3
votes
0
answers
249
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Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
3
votes
0
answers
120
views
Checking the generic rank of a matrix
Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values ...
- 16.9k
3
votes
0
answers
155
views
Bass theorem on non-affine scheme
A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...
- 3,472
3
votes
0
answers
56
views
Regular sequence of binary forms in a subspace
Let $A_d=\mathbb{C}[x_1,\dots,x_n]_d$ denote the vector space of homogeneous polynomials of degree d. And assume $U\subset A_d$ is a subspace of codimension $d-1$ such that there is no point in $\...
- 51
3
votes
0
answers
97
views
Minimal localization need it to "diagonalize" a matrix
Let $A$ be an $n\times n$-matrix over $\mathbb Z[t^\pm]$. In general doesn't exist $P,Q\in GL(n,\mathbb Z[t^\pm])$ such that $PAQ$ is a diagonal matrix (this happens cause $\mathbb Z[t^\pm]$ is not a ...
3
votes
0
answers
288
views
When is a polynomial ring free over a graded subalgebra?
Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...
user78172
3
votes
0
answers
211
views
the annihilator of cokernel in a particular case
Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...
- 2,232
3
votes
0
answers
112
views
Nullspace of a matrix modulo an ideal
Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.
Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$.
Thus, $M$ has a non-...
- 31
3
votes
0
answers
474
views
Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
3
votes
0
answers
473
views
Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
2
votes
0
answers
112
views
Invariant factors and commuting matrices over a discrete valuation ring
$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Ker{Ker}$Let $A$ be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T$, $Y^T$ be ...
2
votes
0
answers
164
views
Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
- 4,058
2
votes
0
answers
97
views
How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices
Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
- 131
2
votes
0
answers
115
views
Linear algebra. commutative algebra, a matrix having the maximal rank
I found a matrix that looks to have the maximal rank for my research.
I would really appreciate if some of you could give me any comments or suggestions.
Thanks in advance.
Sincerely, Yong-Su Shin
Let ...
- 21
2
votes
0
answers
238
views
Computing the codimension of the variety defined by a system of quadratic forms
Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...
- 3,625
2
votes
0
answers
60
views
Linearity of a canonical morphism related to scalar extension and coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
- 6,700
2
votes
0
answers
197
views
Full-rank factorization property of integer-valued matrices
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\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\...
- 128k
2
votes
0
answers
87
views
How to prove this matrix is idempotent and that it obeys a telescoping identity
Let $P_k$ be the size $k$ leading principal submatrix of the real-valued $N \times N$, invertible matrix $M$, and let $Q_k$ be the size $N$ matrix having $P_k^{-1}$ in the top left corner, and zeroes ...
- 51
1
vote
0
answers
37
views
Bounding the length of an R-module of matrices
Loosely related to this: Bounding the length in a module of evaluated skew polynomials
Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
- 223
1
vote
0
answers
60
views
Bounding the length in a module of evaluated skew polynomials
Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
- 223
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
1
vote
0
answers
24
views
One-step vectorial recurrence into multi-step scalar recurrence on commutative rings with boundary conditions
Introduction over unbounded domain
Consider the forward time shift $\mathsf{z}$ acting on a discrete function of time (sequence) $f=(f^n)_{n\in\mathbb{N}}$ as $(\mathsf{z} f)^{n} = f^{n+1}$. Also ...
- 11
1
vote
0
answers
45
views
Lifting module homomorphisms imposing conditions on characteristic polynomials
Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
- 573
1
vote
0
answers
186
views
Proving the non-existence of canonical isomorphisms
From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...
- 1,561
1
vote
0
answers
41
views
A bi-variate polynomial interpolation question
Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...
- 11
1
vote
0
answers
45
views
A "spectral theorem" to SVD reduction for every commutative *-ring
Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is ...
- 4,943
1
vote
0
answers
145
views
Has anyone studied this possible generalisation of the Singular Value Decomposition to all commutative $*$-rings?
I don't know any abstract algebraists personally, which is why I'm asking this question here.
Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated ...
- 4,943
1
vote
0
answers
108
views
When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?
$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
- 573
1
vote
0
answers
33
views
Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial
Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...
- 573
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
1
vote
0
answers
1k
views
Are the integers a vector space or algebra over "some" field or over "some" ring?
Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
- 180
1
vote
0
answers
95
views
Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
1
vote
0
answers
59
views
How to associate the following two kinds of real polynomials?
Suppose the following real polynomial of $n$ variables
$$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$
is easy or familiar to us, but I need to deal with ...
- 1,997
1
vote
0
answers
1k
views
Algebraic Independence of Polynomials in n Variables with Real Coefficients
I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
- 11