All Questions
6,056 questions
2
votes
0
answers
312
views
Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
- 923
0
votes
1
answer
130
views
Levi-Civita field in unusual basis
Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}...
- 28.2k
8
votes
3
answers
1k
views
How to prove that two univariate polynomials are always algebraically dependent?
How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...
- 251
3
votes
0
answers
150
views
Finite commutative group schemes whose exponent coincides with its rank
In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
- 329
1
vote
1
answer
128
views
Koszul complex of equations defining a stabilizer
Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.
Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
- 39.3k
4
votes
0
answers
258
views
Cotangent complex of a formal thickening
Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...
- 858
1
vote
1
answer
206
views
Question about linear functionals over C-infinity modules
A remark in Nelson's "Tensor Analysis" implies there are no non-trivial linear functionals on the module of continuous vector fields on a manifold, when considered as a module over the ring ...
- 5,407
1
vote
2
answers
158
views
An example of a commutative ring with a non-zero nil ideal that is idempotent
Recall that an ideal of a commutative ring is said to be a nil ideal if each of its elements is nilpotent. I am looking for a non-zero nil ideal of a commutative ring that is idempotent.
- 9,650
1
vote
0
answers
39
views
When nilradical belongs to a Gabriel filter
Recall that a Gabriel filter of ideals $\mathscr{I}_\sigma$ of a commutative ring $R$ is a non–empty filter of ideals satisfying that every ideal
$I$ of $R$, for which there exists an ideal $J\in\...
- 147
0
votes
1
answer
154
views
Nullstellensatz and nilpotence of a module
Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated.
The rational abelianization of $N$, $H_1(N)$ is a $\mathbb{C}[H]$-module, ...
- 149k
3
votes
1
answer
452
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 6,453
0
votes
0
answers
218
views
Cohen-Macaulay modules and connections to Mirror Symmetry
Let $ R $ be a local Noetherian Gorenstein domain. Suppose a module $ M $ fits into an exact sequence $$ 0 \rightarrow K \rightarrow R^n \rightarrow M \rightarrow 0 $$ Then we write $ K = tM $. A ...
- 936
1
vote
0
answers
61
views
When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory
Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under ...
- 147
2
votes
1
answer
253
views
Strict henselianization and branches of explicit curve at singularity
Let $A$ be a local ring, which we can assume is reduced. Let $k$ be the residue field of $A$.
In the Stacks project (https://stacks.math.columbia.edu/tag/06DT), I have learned some notion of the ...
- 921
6
votes
1
answer
245
views
Reference request for results that involve the transcendence degree
Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
- 123
0
votes
2
answers
299
views
A question about localization of commutative rings
Given a commutative ring $ R $ and a multiplicatively closed subset $ S $ of $ R $, there are two ways to consturct $ S^{-1}R $:
define an equivalence relation $ \sim $ on $ R\times S $ and then take ...
- 1,424
2
votes
0
answers
235
views
Formally étale maps of animated $k$-algebras
In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
- 301
3
votes
0
answers
292
views
modules over principal ideal rings
Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory ...
- 429
10
votes
3
answers
3k
views
Sum of radical ideals
Let $A$ be a commutative ring and endow the closed subsets of $\operatorname{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This ...
- 12.9k
2
votes
0
answers
115
views
When $k(f(x), g(x)) = k(x)$? In other words, when is a given polynomial parametrization of an affine planar (rational) curve proper?
If $f, g \in k[x]$, where $k$ is a field, then $k(f, g) = k(h)$ for some rational function $h \in k(x)$ (this is a special case of Lüroth's theorem).
Question 1: Under what conditions does the above ...
- 5,349
0
votes
1
answer
128
views
About cluster variables obtained by (sequentially) mutating at exchangeable variables from an initial cluster
Let $\Sigma=(X,ex,B)$ be a seed, $\mathcal{A}(\Sigma)$ a corresponding geometric cluster algebra and $\mathcal{X}_{\Sigma}$ the set of all cluster variables of $\mathcal{A}(\Sigma)$. We call a ...
- 3,587
4
votes
0
answers
356
views
Jet Nestruev's proof that the exterior derivative $d$ on a real line is not a Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}$
The relation between the exterior derivative $d:C^\infty(\mathbb{R})\to\Omega^1\mathbb(\mathbb{R})$ and the Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}:C^\infty(\mathbb{R})\to\Omega_{C^\...
- 2,385
11
votes
0
answers
214
views
Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
- 4,713
1
vote
2
answers
311
views
A variation on Abhyankar–Moh–Suzuki theorem
The well-known theorem of Abhyankar–Moh–Suzuki says the following:
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \...
- 5,349
5
votes
0
answers
276
views
Analysis proof of dual number spectral theorem
Let the dual numbers be $\mathbb R[\varepsilon]/(\varepsilon^2)$. Write a general dual number as $a + b \varepsilon$ (where $\varepsilon^2 = 0$). Given a symmetric matrix $M$ over the dual numbers (i....
- 4,943
1
vote
1
answer
203
views
Special cases of the embedding problem
Embedding problem. Let $I$ be the ideal of polynomial algebra $A=K^{[n]}$, such that $A/I$ is also a polynomial algebra with smaller number $k$ of variables. Is it true that $I$ is generated by $n-k$ ...
CommunityBot
- 1
10
votes
1
answer
327
views
Proving that polynomials belonging to a certain family are reducible
In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.
Assume that $\mathbb F_p$ is ...
- 22.5k
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
6
votes
1
answer
442
views
If power of an ideal is locally free then it is locally free
Let $X$ be a noetherian scheme and $\mathcal{I} \subset \mathcal{O}_X$ a coherent sheaf of ideals. Suppose that $\mathcal{I}^d$ is locally-free for some power $d$. Then the blowing up $\mathrm{Bl}_{\...
- 149k
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 12.9k
3
votes
0
answers
234
views
Non-flat base change
Let $(R,\mathfrak{m},k,E)$ be a Noetherian local ring with $E$ an injective hull of $k$, let $M$ be a finitely generated $R$-module and $M^{\vee}=\mathrm{Hom}_{R}(M,E)$. Is there a non-flat $R$-...
- 63.9k
2
votes
0
answers
132
views
Pushforward of a Cohen-Macaulay module
Let $f \colon R \to S$ be a homomorphism between (local Noetherian) rings which turns $S$ into a finitely generated $R-$module and let $M$ be a finitely generated over $S$.
Is $M$ is Cohen-Macaulay ...
9
votes
2
answers
417
views
Over which (graded) rings are all modules decomposable into indecomposables?
A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
- 35.2k
2
votes
0
answers
112
views
Abelian approximation of fields
Given a field extension $K/k$ of finite degree and a norm $d$ on $\overline{k}$, what is the smallest real number $\alpha_{d}^K$ such that for every element $z$ of $K$ there is an element $z^a$ of $k^...
- 21
2
votes
0
answers
161
views
Descent of codimension of a closed subscheme
Let $X$ be an affine integral normal scheme, $Z\subset X$ a constructible closed subscheme of codimension $\geq 2$. We can write $X$ as a limit of schemes $(X_{\lambda})_{\lambda\in \Lambda}$ of ...
- 119
4
votes
0
answers
85
views
Do generators of the radical of an ideal generated by polynomials with rational coefficients have rational coefficients?
Let $I\in\mathbb{C}[x_1,\dots,x_n]$ be a an ideal generated by polynomials with rational coefficients. Is $\sqrt{I}$ also generated by polynomials with rational coefficients?
- 41
4
votes
2
answers
493
views
Explicit Bézout cofactors
$\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of a question I asked recently. We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization....
- 12.9k
16
votes
2
answers
740
views
Do power sums determine the variables?
In my analysis research, I came across the following problem. Given $n$ positive real numbers $x_1,\dots,x_n$, consider the $n$-many power sums
$$ p_3 = x_1^3 + x_2^3 + \dots + x_n^3 , $$
$$ p_5 = ...
- 12.9k
3
votes
0
answers
68
views
Finding generators and relations for special commutative algebras with a computer
Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$.
Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
- 26.5k
3
votes
1
answer
396
views
A criterion for whether an ideal is contained in a principal ideal
Let $R$ be a ring and $I$ an ideal. I am interested under which conditions the following holds:
Claim. Suppose that any two elements in $I$ have a non-trivial $\operatorname{gcd}$. Then $I$ is ...
- 149k
1
vote
0
answers
82
views
Ring structure of coinvariant of $W(U(4))$
I want to know the ring structure of the coinvariant of $W(U(4))$, where $W(G)$ is the Weyl group of G.
I know that the ring structure of the coinvariant of $W(U(3))$ is $\mathbb{Z}[x_1,x_2,x_3]$ with ...
- 63.9k
4
votes
1
answer
428
views
Idempotents in group rings of finite cyclic groups
For which fields $K$ and integers $n>1$ does the group ring $K(\mathbb{Z}/n\mathbb{Z})$ have idempotents distinct from $0$ and $1$?
- 44.4k
2
votes
0
answers
184
views
Finding étale slices
I'm trying to understand Luna's étale slice theorem by computing some examples. The theorem is usually phrased as an existence result. I wondered if there was a natural way to figure out the slice at ...
- 12.9k
2
votes
1
answer
276
views
Structure theorem for finitely generated $\Lambda$-modules - uniqueness part
In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}_p[[T]]$.
If $M$ is a finitely generated torsion ...
- 149k
4
votes
1
answer
434
views
Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
- 32.5k
2
votes
0
answers
730
views
What algebraic condition corresponds to injectivity of a morphism of varieties?
$\DeclareMathOperator{\Spec}{Spec}$
Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are ...
- 1,141
0
votes
1
answer
116
views
Order 2 matrices with entries in the polynomial ring over a field are diagonalisable
This is a variant on the question posed here, in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}_n(A)$, where $A$ is a $k$-algebra for some field $k$ ...
- 721
1
vote
0
answers
43
views
Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
2
votes
0
answers
287
views
Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 63.9k
5
votes
1
answer
265
views
Is every matrix involution over a UFD diagonalisable?
Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...
- 2,928