All Questions
6,056 questions
1
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4
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717
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Given an integer $N$, find solutions to $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$
Given an integer $N > 0$ with unknown factorization, I would like to find nontrivial solutions $(X, Y, Z)$ to the congruence $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$. Is there any algorithmic way ...
- 131
1
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0
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55
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Induced exact sequence on symmetric bilinear forms on abelian groups
Let $F=\operatorname {Hom}(S^2(\,\_\,),\mathbb{Q}/\mathbb{Z}):\mathsf{Ab} \to \mathsf{Ab}$ be the functor that sends an abelian group to the group of symmetric bilinear forms on this group. As far as ...
- 1,813
2
votes
0
answers
67
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Restriction of a dominant map to a hypersurface
Suppose there is a dominant morphism $H: \mathbb{A}^1_k \times W \to X$ such that $H(0, -) \neq H(1, -)$ as morphisms from $W$ to $X$. Here $W$ and $X$ are varieties over a field $k$. Assume that $...
- 21
7
votes
3
answers
969
views
Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
- 63.9k
2
votes
0
answers
161
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Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
CommunityBot
- 1
5
votes
2
answers
445
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For a finite-type $\mathbb{Z}$-algebra $A$, is the intersection of all ideals $I$ such that $A/I$ is finite and local necessarily zero?
Background: A referee has suggested a shorter proof of one of my results, but I'm having trouble justifying one of their assertions. The setting is that $A$ is a commutative ring, and the referee's ...
- 7,893
3
votes
4
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874
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Extension of Tate's result regarding Tor
In a 1957 paper (Link), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H_i X=0$ except $H_0 X= R/I$ (I guess R should ...
- 2,821
1
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0
answers
329
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Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
- 63.9k
7
votes
2
answers
588
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Representation theory of the full linear monoid
The full linear monoid $M_N(k)$ of a field $k$ is the set of $N \times N$ matrices with entries in $k$, made into a monoid with matrix multiplication.
A representation of $M_N(k)$ on a vector space $V$...
- 11.2k
4
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0
answers
74
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Is each TS-topologizable group TG-topologizable?
Definition 1. A topology $\tau$ on a group $X$ is called
$\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$;
$\bullet$ a ...
- 42k
4
votes
0
answers
170
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Corollaries of Kleene's Theorem (Regular Languages)
Kleene's theorem that finite automata (specifically, nondeterministic) are expressively equivalent to regular expressions seems to be a powerful and not immediately obvious tool for untangling the ...
- 429
3
votes
0
answers
263
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Higher Artin-Schreier homomorphism?
For the additive group over a characteristic $p$ field one has a short exact sequence of abelian algebraic groups
$$\{1\} \to {\mathbb Z}/p \to {\mathbb G}_a \to {\mathbb G}_a \to \{1\},$$
where the ...
- 63.9k
5
votes
0
answers
137
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A particular family of symmetric functions (sums of powers of sums of subsets)
For any $m,k$ define
$$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$
Do these symmetric polynomials have a name and any theory?
- 37.7k
4
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0
answers
157
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On skew monoid rings and skew ordered series rings
To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
- 10.5k
4
votes
3
answers
321
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Examples of integral ring extensions that $\operatorname{ht}P \lt \operatorname{ht}P\cap A$
$\DeclareMathOperator\ht{ht}$All rings are commutative Noetherian with identity.
Exercise 9.8 of Matsumura's book Commutative ring theory: Let $A$ be a ring and $A\subset B$ an integral extension. If $...
- 1,355
4
votes
1
answer
164
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Any ideal as an intersection of ideals primary to maximal ones
The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that
$\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$
Is it also true that we ...
- 7,893
48
votes
4
answers
4k
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Are there more Nullstellensätze?
Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. ...
- 30.3k
21
votes
1
answer
1k
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A Krull-like Theorem and its possible equivalence to AC
A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is ...
- 15.9k
7
votes
1
answer
498
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Basic example of derived descent
I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a ...
- 5,760
3
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0
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43
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Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
- 63.9k
6
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0
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237
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Functorial criterion for local complete intersection morphisms?
Let me state the question for rings (rather than schemes) for simplicity. Let $R$ be a commutative ring with unit and $A$ an $R$-algebra of finite presentation. Recall that $R\to A$ is called a ...
- 4,492
8
votes
3
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921
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Generic Noether normalisation
Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
- 63.9k
25
votes
2
answers
1k
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The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 42k
9
votes
2
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781
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Additivity of projective dimensions, or, help me lower my blood pressure
Sorry for the shameless title. I'm rather stuck on a lemma in commutative algebra - namely, I have both a proof and a counterexample! I have tried rather strenuously and frustratingly to find the ...
- 2,821
6
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0
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190
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The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 42k
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1
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148
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Why is the natural map $\hom(A,\mathbb{R}/\mathbb{Z})\to K/A$ an isomorphism, $K/\mathbb{Q}_p$ unramified, $A=\mathcal{O}_K$?
While looking at an analogue of Pontryagin duality for compact Discrete Valuation Rings (DVRs), I came about the observation that generally one should have an isomorphism of $A$-modules
$$\hom_{\...
- 2,902
3
votes
1
answer
470
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Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...
- 149k
0
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0
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41
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
- 42k
6
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1
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394
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On the Artin-Rees Lemma for non-commutative rings
Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
- 541
6
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0
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225
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Iterating exact triangles (particularly in Floer homology)
There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
- 101
1
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0
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53
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The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
4
votes
4
answers
3k
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Subtle examples of morphisms that are finite but not flat
Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be ...
- 12.9k
2
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0
answers
181
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So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
2
votes
1
answer
178
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Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents
Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ $X_s$ is a point with residue field $\kappa(s)$ } \}$$
Is $E$ a constructible set?
The basic ...
- 19.6k
4
votes
1
answer
345
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Relative valuative criteria of properness for flat morphisms
Let $f: X\rightarrow S$ be a flat quasi-projective morphism, where $X$ is a smooth variety, and $S$ is a discrete valuation ring. Then we know that $f$ is proper morphism if and only if it satisfies ...
- 3,290
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0
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62
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
3
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0
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67
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Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...
- 10.5k
12
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1
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576
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Factoring a polynomial into linear factors by ring extension
The following sounds so natural, I'm surprised I have never asked it before:
Question 1. Let $R$ be a commutative ring. Let $P \in R\left[X\right]$ be a polynomial. Can we find a commutative ring $S$ ...
- 33.8k
1
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0
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250
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Local cohomology: Polynomial ring vs Power series ring
I study algebraic topology and am currently examining the applications of local (co)homology in algebraic topology. We have the canonical inclusion of rings $\mathbb{Z}[x_1,\cdots,x_n]\subset \mathbb{...
CommunityBot
- 1
11
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1
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840
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Which cluster algebras are coordinate rings of double Bruhat cells?
Background
A uselessly vague paragraph follows. A cluster algebra is a commutative algebra $A$ with a distinguished set of generators called cluster variables. These cluster variables are grouped ...
- 2,821
1
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0
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37
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Valuations of coefficients of minimal polynomials for tuples
Suppose you are given two valued fields $(K,v) \subseteq (L,w)$ and a tuple $a \in L^n$. What kind of restrictions do we have on the valuation of the coefficients of polynomials $q \in K[x_1,\dots x_n]...
- 161
1
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0
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83
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Injectivity of tuple of incomplete elementary symmetric polynomials
For $1\leq k,j \leq n$ and $a=(a_1,\ldots,a_n)\in {\mathbb R}^n$, denote by $s_{k,j}(a)$ the $k$-th symmetric polynomial in the $n-1$ variables obtained when $a_j$ is removed from $a_1,\ldots,a_n$. ...
- 621
12
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1
answer
624
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Stone–Čech compactification as a semigroup
Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
- 38.6k
19
votes
2
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836
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Does the rational power series ring $\mathbb{Q}[[X]]$ embed as a ring into the field of real numbers?
The title says it all. I'm wondering if the power series ring $\mathbb{Q}[[X]]$ (with rational coefficients) embeds as a ring into the field of real numbers. There are various topologies one might ...
- 118k
8
votes
1
answer
437
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Noetherian but not strongly Noetherian
What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...
- 466
7
votes
1
answer
408
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Homological dimensions of rings of smooth functions
What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
- 63.9k
2
votes
1
answer
194
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Structure of reflexive modules over regular local rings
Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in Reflexive modules over a 2-dimensional ...
- 301
4
votes
1
answer
359
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Surjectivity of natural map of rings
$\DeclareMathOperator\Hom{Hom}$Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ then is the following natural map
$$\Hom_A(P,A)\otimes_A B\to \Hom_A(P,B)$$ surjective?
...
- 585
3
votes
0
answers
31
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Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup
A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...
- 42k
3
votes
1
answer
213
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Ideals whose quotient rings have a certain property
There are some well-known properties of ideals which are equally well-known to correspond to properties of their respective quotient rings. For example:
An ideal $p$ of a ring $R$ is prime iff $R/p$ ...
- 18.7k