All Questions
6,055 questions
1
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Exceptional Lenz-Barlotti classes IVa.3 and IVb.3
On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...
- 1,300
4
votes
1
answer
185
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Order of pole of Poincaré series
Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
- 12.9k
3
votes
0
answers
189
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How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
1
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92
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On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 6,262
2
votes
0
answers
213
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Open problems in differential algebra and affine algebraic geometry
I am currently doing a PhD in differential algebra and affine algebraic geometry at the University of Buenos Aires. I've been struggling to find a list of interesting and big open problems in these ...
- 869
3
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0
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111
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When is a ring complete with respect to its nilradical?
Let $R$ be a commutative ring and let $I$ be its nilradical. When is $R$ complete with respect to $I$?
For example, if $I$ is finitely generated, there exists $N$ such that $I^N = 0$ and thus $R$ is ...
- 2,368
1
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1
answer
149
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$F=\mathbb{C}(u,v)$ satisfying: For every $a,b \in \mathbb{C}[y],c,d \in \mathbb{C}[x]$: $\mathbb{C}(x,y)=F(ax+b)=F(cy+d)$
Let $u,v \in \mathbb{C}[x,y]$, where $u$ and $v$ are algebraically independent over $\mathbb{C}$ and $F=\mathbb{C}(u,v)$. Of course, $d:=[\mathbb{C}(x,y):F] < \infty$.
Denote the following ...
- 2,837
1
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1
answer
88
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Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 16.2k
2
votes
1
answer
112
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Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 16.2k
0
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1
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473
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A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
- 2,837
1
vote
0
answers
59
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If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$,
$F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
- 2,837
9
votes
3
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2k
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Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 11
1
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1
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215
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Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 468
0
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1
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219
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 148k
6
votes
1
answer
225
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Integral preimages of topologically noetherian, affine schemes
Let $A\to B$ be a ring homomorphism, $d\in \mathbb{Z}_{>0}$ and let $C=\bigotimes^d_A B$ the $d$-fold tensor product of $B$ over $A$. Then $\mathfrak{S}_d$, the symmetric group of $d$ elements, ...
- 1,144
2
votes
0
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93
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Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
- 480
26
votes
3
answers
2k
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Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
- 1,726
0
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0
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42
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When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)
Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$,
$1 \leq i \leq n$, $1 \leq j \leq m$, be such that
$f_i(x,0) \neq 0$ and $g_j(x,0)=0$.
Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
- 2,837
2
votes
1
answer
206
views
Localization of quasi-excellent rings are quasi-excellent?
If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ?
I think Matsumura's commutative ring theory book says that localization of ...
- 1,813
2
votes
1
answer
245
views
Characterization of a certain class of modules-broader than Noetherian
Let $R$ be a commutative ring with $1$.
An $R$-module $K$ has the 'S' property if $K/T \simeq K$ (i.e. isomorphic) implies that the submodule $T$ is trivial.
By Fitting's lemma any Noetherian module ...
- 63.9k
0
votes
0
answers
31
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Formalization of the independance of products in a (commutative) semigroup
1/ It is well known that associativity implies that the result of the product of an ordered finite set of elements in a semigroup does not depend of the order of composition of the partial products.
...
- 2,655
1
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1
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135
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Artinian Gorenstein subrings with same socle degree
I am looking for examples of Artinian Gorenstein subalgebras with the same socle degrees.
More precisely, let $A$ be an Artinian Gorenstein $k$-algebra (with $k$ algebraically closed of characteristic ...
- 50.8k
3
votes
1
answer
502
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Description of prime ideals in $\operatorname{Spec}(\mathbb{Z}[x_1, \dots, x_n])$
Edit: This seems to be wrong, as pointed out by Will Sawin in the comments.
The prime ideals of $\mathbb{Z}$ and $\mathbb{Z}[x]$ are well-known. It is also not too hard to compute the underlying set ...
- 35.5k
1
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0
answers
67
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Conormal module of a commutative Koszul algebra
Does the conormal module $I/I^2$ of a commutative Koszul algebra $A=k[x_1,\dots,x_n]/I$ have a linear minimal free resolution? Is there a formula for the Betti numbers of $I/I^2$, or, even better, a ...
- 359
2
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1
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407
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Finite generation of certain graded sequences of ideals
Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
CommunityBot
- 1
2
votes
1
answer
57
views
Are simplicial commutative inverse semigroups fibrant?
Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...
- 37.8k
1
vote
0
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80
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Localization of totally acyclic complex or projective modules remain totally acyclic?
Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic.
...
- 480
1
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0
answers
54
views
Algorithm for finding generating sets of projective modules
Suppose $R$ is a (Dedekind) domain and $M$ is a projective module of constant rank over $R$. We know that $M$ is finitely generated over $R$. I'm wondering is there any algorithms to produce a (...
2
votes
0
answers
182
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Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29
6
votes
2
answers
219
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Steinitz isomorphism theorem for non-Dedekind domains
(Cross-posted from https://math.stackexchange.com/questions/4931582/steinitz-isomorphism-theorem-for-non-dedekind-domains)
Fix a Dedekind domain $R$ and fractional ideals $I, J$. It's a classical ...
- 7,893
1
vote
0
answers
181
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Examples of semirings where the additive neutral element is not absorbing for multiplication
In the case of a non unital ring, the additive 0 must be absorbing for the multiplication because we have a⋅0 = a⋅(a − a) = a⋅a − a⋅a = 0 and similarly on the other side.
In the case of a unital ...
- 22.3k
6
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2
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1k
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Question about the sum of odd powers equation
Quite surprisingly the following question appears while studying the billiard dynamics.
Assume we have $2n$ real numbers: $ x_1, x_2,..., x_{2n}$.
Assume also that $S_k=0$ for any odd positive integer ...
- 105k
4
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1
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686
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Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 188k
1
vote
2
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471
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Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
- 63.9k
-3
votes
2
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818
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Is there a "weak" fundamental theorem of algebra for matrices?
Let $R$ be the ring of complex $n\times n$ matrices, where $n>1$.
Does every nonconstant polynomial in $R[X]$ have a root in $R$?
Note: The "strong" fundamental theorem of algebra for ...
- 16.2k
1
vote
0
answers
108
views
Do étale coordinates give rise to a regular sequence of diagonal elements?
Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
- 507
3
votes
2
answers
246
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Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
- 10.8k
10
votes
1
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561
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Can you constructively prove a univariate polynomial algebra over a Jacobson ring is itself Jacobson?
Recall the Jacobson radical of a commutative ring $\mathrm J(A)=\lbrace a\in A\mid \forall b\in A:1-ba\in A^\times\rbrace$. The Jacobson radical of a quotient by an ideal $I\vartriangleleft A$ is ...
- 35.5k
6
votes
1
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414
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Constructive treatment of Jacobson rings
Which result is closest to the classical
General Hilbert's Nullstellensatz: Finite type algebras over Jacobson rings are Jacobson.
and constructively true at the same time? And where can I find a ...
- 76
0
votes
0
answers
47
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Relationship between equation of integral dependence of an element and its inverse
Let $A$ be a reduced, Noetherian ring. Let $B$ be its integral closure. Let $b\in B$ and let $v\in B$ be its inverse. Let $b^n+\ldots a_0=0$ be an equation of integral dependence for $b$. Is there any ...
- 71
2
votes
0
answers
274
views
Can the completion of a local domain which is not a field be a field?
I would like to prove/disprove the following claim:
Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...
- 293
3
votes
1
answer
413
views
Do there exist these real polynomials?
Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients, and $n>20$ a natural number such that
$$\left(\sum\limits_{k=0}^{n} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$
I have ...
- 22.5k
84
votes
31
answers
70k
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Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
- 60.5k
1
vote
0
answers
107
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A certain subfield of $\mathbb{C}(x,y)$
Let $A=\mathbb{C}(x+y,xy)$, the subfield of symmetric polynomials with respect to the involution $\alpha: (x,y) \mapsto (y,x)$.
Denote $G_A=\{w \in \mathbb{C}(x,y) \ | \ \mathbb{C}(x+y,xy,w)=A(w)=\...
- 2,837
9
votes
1
answer
889
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Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 12.9k
11
votes
0
answers
616
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Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
4
votes
0
answers
112
views
The $K_1$-group of integer valued polynomials
Let $R=$ Int$(\mathbb{Z}) = \{f \in \mathbb{Q}[x]| f(\mathbb{Z}) \subset \mathbb{Z}\}$. I am interested to find $K_1(R)$. I list my trials below:
Let us construct a Milnor square $$\matrix{R&\...
- 141
2
votes
1
answer
152
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A commutative ring with unity which does not have relatively pseudo-injective ideals with zero intersection
Let $R$ be a ring with $1$ and $M$ and $N$ be any right $R$-modules. We say that $M$ is pseudo-$N$-injective if every $R$-monomorphism $f:X \to M$ from a submodule $X_R$ of $N_R$ can be extended to $N$...
- 14.6k
20
votes
4
answers
4k
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What is interesting/useful about Castelnuovo-Mumford regularity?
What is interesting/useful about Castelnuovo-Mumford regularity?
- 708
3
votes
0
answers
115
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English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
- 31