All Questions
6,055 questions
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144
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Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 73
0
votes
1
answer
216
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A Newton identity and the primes--the Faber partition polynomials and modular arithmetic
[Edit, July 6, 2022: Removed erroneous characterization of Faber polynomials as an Appell sequence.]
Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I ...
- 10.5k
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votes
1
answer
63
views
Changing base field for sum of polynomials
Let $L/\mathbb{Q}$ be a finite extension and $f_{1},\dotsc,f_{n}\in L[x_{1},\dotsc,x_{k}]$ be degree $d$ homogeneous polynomials. Is there a way to find homogenous degree $d’$ polynomials $g_{1},\...
- 3
0
votes
1
answer
116
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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
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votes
1
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139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 1
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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 ...
- 225
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votes
1
answer
252
views
Injectivity of Keller maps
Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$,
$(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$
satisfying $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Such a polynomial map is ...
- 2,837
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votes
1
answer
81
views
Non-fractions in faithfully flat extension were non-fractions
All rings are commutative with unity.
Let $\phi:A \to B$ be a faithfully flat ring homomorphism. Let $f \in A$, $g = \phi(f) \in B$, and $\psi:A_f \to B_g$ the induced homomorphism on the ...
- 81
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1
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94
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$k[h(x),y] \subseteq k[h(x),y] + \langle h(x),y \rangle \subseteq k[x,y]$
Let $k$ be a field of characteristic zero.
Let $h=h(T) \in k[T]$ with $\deg(h)=d \geq 2$ and $h(0)=0$ (namely, $h$ has zero constant term).
Consider the following chain of $k$-algebras:
$$k \subseteq ...
- 2,837
0
votes
1
answer
200
views
The rank of a semigroup
Let $S$ be a finite noncommutative semigroup(without identity) with a subset $M$ such that $\langle M \rangle =S$. If every element of $M$ is indecomposable in $M$, i.e. for any $a \in M$, there are ...
0
votes
1
answer
296
views
Multiplicative monoid of ring modulo units
Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\...
- 943
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1
answer
54
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Writing a set of all possible (symmetric) products condensely? [closed]
I have a set of elements $\{a_1, a_2, a_3...\}$ and $\{b_1, b_2, b_3...\}$ and I want to condensely formally write the set of all possible products of these elements, where the ordering does not ...
- 1,465
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votes
1
answer
364
views
Koszul complex of $xy$, $yz$ and $xz$
Has anyone computed the homology of the sequence $xy$, $yz$ and $xz$ in $\mathbb{C}[x,y,z]$?
- 17
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1
answer
139
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Change chain of prime ideals so that $a \in P_1$
A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof.
Lemma: Let $n\in\mathbb{N}_{>0}$, and let $P_0\subsetneq P_1\subsetneq\cdots\...
- 545
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votes
2
answers
244
views
Power series ring and monomials
Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables.
For a given positive number $\epsilon > 0$ we call a monomial $X_{...
- 563
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votes
1
answer
149
views
Power series rings and the formal generic fibre
Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
- 563
0
votes
1
answer
670
views
Symmetric polynomials in two sets of variables
Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
user133644
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1
answer
354
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Counting the number of poles for rational functions in a coordinate ring of a curve
I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
- 133
0
votes
1
answer
254
views
When does a subspace of the affine space form a regular sequence in a ring of regular functions?
Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$.
Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...
- 1,227
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votes
1
answer
92
views
Homomorphisms between structure sheaves of subvarieties
Let $X$ be a non-singular (complex) variety of dimension $n>1$. Let $D$ and $D'$ be distinct integral divisors in $X$. It seems to me that one should have $$\mathcal{Hom}_{\mathcal O_X}(\mathcal ...
- 1,463
0
votes
1
answer
187
views
How to prove this by extension group
Let R be a commutative ring.$r\in R$.Let M and N are R-modules such that $rM=0$ and there is a short exact sequence $0\rightarrow N\xrightarrow r N\rightarrow N/rN \rightarrow 0$.
if X is arbitrary ...
- 496
0
votes
1
answer
260
views
Analytic spread of localization of an ideal
Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
Q) Are there ...
- 1,713
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votes
1
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63
views
Monoid morphisms satisfying a decomposition condition
Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research:
For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1....
- 327
0
votes
1
answer
226
views
Lifting sections to completion of closed subschemes
Let $R$ be a reduced finite type $\bar{k}$-algebra, a projective morphism $\pi \colon V \rightarrow \mathrm{Spec}(R)$ and ideals $I, J \subseteq R$. Assume there is a split $s_{IJ} \colon \mathrm{Spec}...
- 189
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1
answer
209
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Reduction of ideal in noetherian local ring
Let $R$ be a noetherian local ring and $I$ an ideal with $\operatorname{ht}I=\mu(I)$. Prove that $I$ is basic. (Recall that an ideal $I$ is basic when it has no proper reduction.)
- 19
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1
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83
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Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials
Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...
- 173
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votes
1
answer
107
views
Generalization of a Result about degree bounds of invariant rings
A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or ...
- 928
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1
answer
152
views
Is it possible to generalize a result of Wang?
Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...
- 2,837
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votes
1
answer
82
views
Some references for f-ring
A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there
is a $b\in I$ such that $ab = ...
- 1
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votes
1
answer
464
views
Irreducible components of a cone
Suppose $B=A\oplus S^1\oplus S^2\dots$ is a graded ring, $B$ is generated by $S^1$, $C=\textrm{Spec}B$ is called a cone over $X=\textrm{Spec}A$. We have natural projection $\pi\colon C\to X$. Moreover,...
user39380
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1
answer
110
views
How to find ideals of finite length in a power series ring with special properties?
Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.
Assume we are given two ideals $I,J$ of finite length in $A$ such that:
$xJ\subseteq I\subseteq J$
Is it possible to find ideals of finite ...
- 1,025
0
votes
1
answer
184
views
Change of grading used in the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry
I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here.
Let $T=\...
- 1,713
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votes
1
answer
237
views
When every module is a scalar extension?
Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and $B$...
- 2,837
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votes
1
answer
598
views
Reference for a lemma on étale maps
The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ B\...
- 1,838
0
votes
1
answer
381
views
Samuel multiplicity
Let $X$ be the hyper-surface defined by
$$f:=\sum_{i=1}^k x_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x_1^{n-1},\dots , x_k^{n-1}) $$
What is ...
- 2,384
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1
answer
541
views
Properties of Integral Closure [closed]
Definition(Integral closure): Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisfies a monic equation
$x^n + i_1x^{n−1} + ··· + i_n = 0$ such that $...
- 381
0
votes
1
answer
97
views
Homologue of the Inertia group and of the Frobenius theorem for the group of values of a valuation
As I said previously, I have some problems in the theory of valuations and places.
Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L.
I denote by l and k the residue ...
- 687
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votes
1
answer
533
views
Valuations and places - decomposition and inertia group
I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is ...
- 687
0
votes
1
answer
286
views
A condition on isolated singularity
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
- 1,207
0
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1
answer
264
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Bounded Index of Nilpotency of $R[x]$
A ring $R$ is called with bounded index (of nilpotency) $n$ if $n$ is the smallest natural number such that $a^n=0$ for all nilpotent $a \in R$.
Now let $R$ be a commutatitve ring with bounded index $...
- 101
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1
answer
164
views
Extending derivations to the superposition closure
Let $X$ be a set and $\mathcal{F}\subseteq {\mathbb{R}^X}$ an arbitrary family of functions.
The superposition closure of $\mathcal{F}$ is defined as
$$
\overline{\mathcal{F}}=\{ H\circ(f_1\times\...
- 2,527
0
votes
1
answer
225
views
Depth formula in CM-ring involving canonical module
In this article by Iyama and Wemyss there is the following formula:
Let $R$ be a Cohen-Macaulay ring with canonical module $\omega$, let $X$ be a finitely generated $R$-module. Then
$$\mbox{depth}(X)=\...
- 73
0
votes
1
answer
340
views
reduction of an admissible filtration
Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration
1) if $m\geq n$ then $I_m\subset I_n.$
2) for all $m,n,$ $...
- 127
0
votes
1
answer
156
views
The Existence of Pure Resolutions, Given a Degree Sequence?
I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
user24421
0
votes
1
answer
244
views
Finite extension of a field [closed]
Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ ...
- 169
0
votes
1
answer
263
views
$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The ...
- 21
0
votes
2
answers
602
views
Embedded associated prime and non zero divisor
$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is equidimensional and $M$ does not have any embedded prime.
Given $x\in I$, where $I$ is an ideal of $A$, and $\dim G(M)/x^*G(...
- 31
0
votes
1
answer
330
views
About regular local rings and Socles
Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...
- 11
0
votes
1
answer
133
views
Ideal membership (concerning polynomial invariants of orthogonal groups)
Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials
$$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$
I'm ...
0
votes
1
answer
327
views
Gluing free modules to get a finitely generated free module
Hi, maybe this is a stupid question, however none of my mathematicians colleagues could answer it properly. It's know that a finitely generated projective $A$-module $M$ is free if $A$ is a local ring....
- 163