All Questions
5,985 questions
5
votes
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Computing the Second Exterior Power of Certain Ideals in $\mathbb{Z}[\sqrt{-5}]$ and $\mathbb{Z}[\sqrt{5}]$ as Modules
I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings $R_1 = \mathbb{Z}[\sqrt{-5}]$ and $R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as ...
- 7,893
4
votes
2
answers
284
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Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 15.8k
1
vote
0
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58
views
Universal formulas for polynomials with prescribed jets
Let $A$ be a commutative ring and $f\in A[x]$ a split monic. When $f$ is separable with roots $\mathrm Z(f)= \{ a_1,\dots ,a_k \}$, the Chinese remainder theorem (CRT) ensures that evaluation is an $A$...
- 10.5k
5
votes
1
answer
248
views
Integral closure in characteristic 0
Let A be a Noetherian domain of characteristic 0, K be its field of fractions. Is the integral closure of A in K always finitely generated as A-module?
- 12.9k
4
votes
0
answers
211
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When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
4
votes
0
answers
140
views
Can an ideal in the ring of holomorphic functions on the complex plane be non-finitely generated?
Let $( I )$ be an ideal in the ring $( R )$ of all holomorphic functions of a single complex variable on the complex plane. I am interested in understanding whether it is possible for $( I )$ to be ...
- 6,357
1
vote
1
answer
73
views
In a ring with a $p$-derivation every $p$-power-torsion element is nilpotent
Let $p$ be a prime. The definition of $p$-derivation on a ring (aka $\delta$-ring structure) can be read in [K, Definition 2.1.1]. In short, a $\delta$-ring is a commutative ring with unity $A$ plus a ...
3
votes
1
answer
165
views
Finite flat maps
Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
- 19.6k
2
votes
1
answer
191
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Does Serre's condition $S_k$ depend only on codimension $\leq k$ points?
Recall a locally Noetherian scheme $X$ has Serre's condition $S_k$ if for every $x\in X$ we have $\mathrm{depth}(\mathcal{O}_{x,X})\geq \mathrm{min}(k,\mathrm{dim}(\mathcal{O}_{x,X}))$.
Let $X$ be a ...
- 4,111
5
votes
0
answers
176
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Example of a Boolean Ring with infinite injective dimension over itself
It is known that Boolean rings can have infinite global dimension (free Boolean algebra on a large enough number of generators) [ see The Global Dimension of Boolean Rings by Pierce]. Are there any ...
- 5,407
2
votes
1
answer
505
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Family of curve singularities whose generic members are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
CommunityBot
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0
votes
0
answers
95
views
Conditions for regularity in a covering
Let $V$ be a DVR of mixed characteristic, whose residue field is a finite field of characteristic $2$. Let $R$ be a flat, finitely generated algebra over $V$, which is regular. Let $a\in R^*$ be an ...
- 4,713
2
votes
0
answers
128
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 63.9k
2
votes
1
answer
388
views
When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?
For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...
CommunityBot
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1
vote
1
answer
115
views
$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
CommunityBot
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2
votes
1
answer
198
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Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
- 148k
3
votes
0
answers
180
views
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
0
votes
1
answer
362
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Derivations and ideals
Let $R$ be a regular local ring and $I$ and ideal of $R$. If $D$ is a derivation of $R$, let
$$\lambda_D:I/I^2\to R/I$$
be the composition of the restriction of $D$ to $I$ and the quotient map $R\to R/...
CommunityBot
- 1
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 63.9k
9
votes
1
answer
2k
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Well founded induction attributed to Noether
What I know as well founded induction, namely the rule
$$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$
whose validity is the ...
- 8,481
6
votes
1
answer
2k
views
$\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$
Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\...
CommunityBot
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4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
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8
votes
2
answers
538
views
Local Profinite Ring
I haven't received any substantial responses to a similar question on math.stackexchange, so let me try here.
Let $R$ be a profinite ring (that is a projective limit of finite rings). Assume
that ...
- 5,407
30
votes
2
answers
3k
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Even XOR Odd Infinities?
Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
(http://en.wikipedia.org/wiki/Peano_axioms#First-...
- 6,357
5
votes
0
answers
216
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 128k
1
vote
0
answers
47
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Examples of graded subrings of $\mathbb Q(T)$
The following question came up in some discussion on some very unrelated matters.
A graded algebra $A$ is an algebra $A$ with a decomposition $A = \oplus_{i \in \mathbb Z} A_i$ such that $A_i A_j \...
- 2,029
-2
votes
1
answer
77
views
integral ring extension implies algebraicity of their fraction fields extension?
$\DeclareMathOperator\Fr{Fr}$There is something I don't get about the following :
Start with the well known fact that if $A\subset B\subset C$ are rings with $B$ the integral closure of $A$ in $C$, ...
- 12.9k
0
votes
1
answer
222
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 12.9k
1
vote
1
answer
108
views
Gorenstein property from initial ideal
My question is:
If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
1
vote
0
answers
110
views
How large can the Krull dimension of the Rees algebra be?
Let $d$ be a natural number. How large can the Krull dimension of the Rees algebra $A[It]$ be, where $A$ is a commutative ring of Krull dimension $d$, and $I$ is an ideal of $A$.
Currently, I know the ...
- 76
1
vote
1
answer
331
views
Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
CommunityBot
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3
votes
0
answers
91
views
Quillen-Suslin theorem for non-strict polydiscs in the sense of Berkovich
Let $K$ be a complete non-archimedean field of mixed characteristic $(0,p)$. Choose $\rho_1,\dots,\rho_n\in \mathbb{R}_{>0}$ and let $P$ be a finite projective module over $K\langle\rho_1^{-1}t_1,\...
- 63
2
votes
1
answer
327
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 148k
50
votes
0
answers
2k
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How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 3,065
0
votes
1
answer
372
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The growth of the Hilbert function of a graded ring
Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$.
In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...
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5
votes
1
answer
156
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The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
CommunityBot
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3
votes
1
answer
3k
views
Tensor product of field extensions
Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ ...
- 3,432
11
votes
1
answer
511
views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 15.8k
3
votes
2
answers
255
views
Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
If we consider any differential graded algebra $A^\bullet$, then its homology is a graded algebra, since the tensor product interacts well with homology.
A sufficient condidtion for the homology to be ...
- 4,254
7
votes
0
answers
202
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Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.1k
5
votes
0
answers
128
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What are the conditions for the dual of the exterior algebra to be isomorphic to the exterior algebra of the dual?
The exterior algebra $\Lambda^*_kM$ can be defined for a $k$-module $M$, where $k$ is a commutative ring. A number of sources mention, without condition or proof, a (canonical) isomorphism $$(\Lambda^*...
- 12.9k
4
votes
0
answers
175
views
What is the equivalent of Artin gluing for quasicoherent sheaves?
Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \...
- 15.9k
3
votes
0
answers
120
views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
3
votes
0
answers
95
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History of the notion of integral ring extension?
[I asked that question in "history of science and mathematics" but didn't get any answer so I take my chance here. I hope this is not out of context]
Can anyone give me references about the ...
- 1,031
6
votes
1
answer
306
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Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 55.9k
4
votes
1
answer
131
views
Zero dimensional complete intersection ring of length a power of $p$
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C$ be a $k$-algebra of finite dimension over $k$ such that $k[C^p]=k$. Under these hypothesis it is known by results of L. ...
- 16.2k
3
votes
2
answers
271
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Orbits under the automorphism group of projective space
Let $\mathbb{P}^d_K$ be projective space of dimension $d\geq 1$ over an infinite field $K$. Let $x\in\mathbb{P}^d_K$ with $\dim\overline{\lbrace x\rbrace}=n\leq d-1$.
My question: is the set $\lbrace ...
- 21
7
votes
0
answers
769
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Artin-Schreier Theorem for Rings
This has been in my mind for quite some time. Looking at the Artin-Schreier Theorem for fields:
If $L$ is a field and $K$ its algebraic closure and if $1< [K:L] < \infty$ then $K=L[i]$ and $L$ ...
- 3,065
0
votes
1
answer
468
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Finite extensions of residue fields of Henselian DVRs
Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
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