All Questions
6,056 questions
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Lifting module homomorphisms imposing conditions on characteristic polynomials
Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
- 573
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60
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Size of minimal generating set of a module generated by columns of a diagonal matrix with extra structure
Let $R$ be a commutative ring with unit. Let $A \in R^{k \times k}$ be a diagonal matrix such that $A_{11} | A_{22} | \dots | A_{rr}$ for some $r \leq k$ and are non-zero, while $A_{ii}=0$ for all $i &...
- 431
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274
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Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
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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
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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
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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{...
user340793
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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
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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
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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 ...
- 10.5k
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77
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Subrings of rings of integers of quartic fields having prime index and a specific property
Let $L$ be a $S_4$ or $A_4$ quartic field, and $\mathcal{O}_L$ its ring of integers. Let $K$ be its cubic resolvent field, which necessarily has the same discriminant as $L$. If $\mathcal{O}_L$ has a ...
- 26.9k
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186
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How to calculate the periodic cyclic homology group of $\overline{\mathbb{Z}}/\mathbb{Z}$
$\newcommand{\ur}{\mathrm{ur}}$Fix a prime number $p$. We let $\overline{\mathbb{Z}}$ denote the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Z}}_p$ denote the ...
- 519
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173
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The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
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Combinatorial models of the refined inverse Eulerian numbers
If I evaluate substitution of an infinite set of indeterminates $(c_1,c_2,c_3,\cdots)$ into the infinite set of refined Eulerian polynomials $[E]$ of OEIS A145271, I obtain the Taylor series ...
- 10.5k
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54
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Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
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- 2,605
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66
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Counting commutative rngs (rings without identity)
A037289 counts the number of commutative rngs (rings without identity). It is complete up to 31, that is, the number of commutative rngs with 32 elements is not known. Is this in the literature?
...
- 9,114
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104
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Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?
Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$).
I will denote the ...
- 10.1k
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87
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When is the product of two elements in algebraic closures of rational functions a constant function?
I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields.
My question is as follows:
Let E and F be ...
- 11
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86
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Buchsbaum-Eisenbud-Horrocks conjecture for finitely generated modules
Someone know a version for conjecture of Buchsbaum-Eisenbud-Horrocks whithout the assumptions that
the $R$-module $M$ has length finite?
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134
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A question about minimal system of generators and regular sequences
Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $I$ an ideal. Suppose that:
$\mu(I)=\operatorname{grade}(I,R)+1$ and $\operatorname{pd}_R(R/I)=\operatorname{grade}(I,R)$. (Some people says $I$ is ...
- 33
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124
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Some properties for height 1 prime ideals in the local ring
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $R=\mathbb{K}[x_0,x_1,\dotsc,x_n]/I$ be the coordinate ring of an affine variety/projective variety. Also, assume that $I$ ...
- 839
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74
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Characterise set of polynomials which are zero over an ideal
This is not a specific question, but rather a question about possible techniques approaching a problem. Although this question came from research, it might not fit this forum; in which case I will ...
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113
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Cohomological dimension and height of ideals
Let $I$ be an ideal in a Noetherian ring $R$. We define the cohomological dimension of $I$ to be $\operatorname{cd}(I)=\operatorname{sup}\{i\in \mathbb N:\operatorname{H}_I^i(R)\neq0\}$ and it is ...
- 131
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332
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Meaning of "cut out (scheme-theoretically)"
Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
- 839
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470
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Source for conjectures in commutative algebra
Do you know some books/survey papers/ websites on conjectures or open problems in commutative ring theory? I want to see if
there are very famous open problems or conjectures in commutative ring ...
- 1,355
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126
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Strict henselianization of complete intersections
As far as I understand (and tbh for my purposes), one of the main points of strict henselisation of a local ring is that it computes the stalk at a point of a scheme in the étale topology. In the ...
- 4,418
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259
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Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA
I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently.
So far, I only found MAGMA with its ...
- 21
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65
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Bertini type result for torsion-freeness
Let $R$ be a local, regular $\mathbb{C}$-algebra and $\mathfrak{m}$ be the maximal ideal. Let $M$ be a finitely generated torsion-free $R$-module. Suppose there exists $f \in \mathfrak{m}$ such that $...
- 1,593
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Cohen-Macaulyness of Milnor algebra
Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...
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123
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Clarification about theorem on vanishing polynomials
The theorem below is from page 3 in the this paper on polynomials in $\mathbb{Z}_m[x]$.
Let $F$ be a polynomial in $\mathbb{Z}_m[X]$. Then $f \equiv 0$ iff $$F \equiv F_nS_n + \sum_{k=0}^{n-1}a_k(m/(...
- 11
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155
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Homogeneous deformation of isolated singularities
Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...
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186
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Proving the non-existence of canonical isomorphisms
From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...
- 1,561
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72
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Factorizable partition polynomials
Let $p(n)$ denote the number of (unrestricted) integer partition of $n$ which has the product generating function
$$\sum_{n\geq0}p(n)\,x^n=\prod_{j\geq1}\frac1{1-x^j}.$$
On the other hand, for the ...
- 43.2k
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165
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spectral sequence Ext(R/I,H^g(M)) => Ext^{p+q}(R/I,M)
I am reading papers of Local cohomology and came across some spectral sequences. I then started reading about spectral sequences from Rotman's book. I havent finished reading the chapters on spectrals ...
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76
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Determine whether a set generates a residue field of an invariant ring
Fix two positive integers $m>n$.
Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$.
Denote $R=\...
- 331
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A bi-variate polynomial interpolation question
Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...
- 11
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140
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The first syzygy module of a binomial ideal
It is known how you compute the first syzygy module of a monomial ideal but it seems an hard work to do the same for binomial ones. I don't know any procedure to aim that, so I would like kindly if ...
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Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring
Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
- 111
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What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?
Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.
Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...
- 10.1k
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130
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Stein Manifold and sheaf cohomology with support in a point
Let $X$ be a Stein manifold with analytic structure sheaf $\mathscr{O}_{X}$. Let $M$ be a coherent $\mathscr{O}_{X}$-module, $x \in X$, and $U$ a Stein open containing $x$. Write $\mathfrak{m}_{x}$ ...
- 11
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173
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Calculating multiplication in a finite dimensional algebra over $\mathbb{Q}$
Suppose $ L $ be an extension over $ \mathbb{Q} $ of degree $ n $. Let $\{e_{1},e_{2},\dots,e_{n}\} $ be a basis of this extension. Now I know the product $ e_{i}^{2} $ and $ e_{i}e_{j} $ . So we can ...
- 923
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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
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61
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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
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82
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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 ...
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43
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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
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51
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For any initial ideal $I$ of the ideal of maximal minors, is it true that $I^n = I^{(n)}$?
Let $X$ denote a generic $n \times m$ (with $n \leq m$) matrix and $R = k[X]$, where $k$ is any field. Let $J := I_n (X)$. It is well-known that $J^t = J^{(t)}$ for all $t$ (where $-^{(t)}$ denotes ...
- 553
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163
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Existence of a finite resolution
I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance.
Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
- 167
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45
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A "spectral theorem" to SVD reduction for every commutative *-ring
Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is ...
- 4,943
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72
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Equivalence between smoothly regular and analytically regular
I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...
- 803
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47
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Can the embedding dimension of a finite local algebra change after restricting to a finite subfield?
The embedding dimension of a commutative $k$-algebra is the minimum $n$ such that it is a quotient of the polynomial algebra in $n$-variables.
The embedding dimension of $\mathbb{F}_2\times \mathbb{F}...
- 325
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99
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Finding an injective envelope containing another injective envelope
Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
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