All Questions
6,055 questions
0
votes
1
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170
views
Tensoring with descending chain of modules
Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
- 11
0
votes
1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
0
votes
1
answer
94
views
What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1
0
votes
1
answer
202
views
lattice of subalgebras of a finite commutative algebra
(I) Suppose A is a finite commutative local algebra. Must every lattice of local subalgebras of A be a distributive lattice ?
By a subalgebra of A we mean an algebra contained in A that shares the ...
- 103
0
votes
1
answer
508
views
"thematic" algebras
I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a local ideal ...
- 103
0
votes
1
answer
149
views
Partial dehomogenization and smoothness
Let $P_1l_1+P_2l_2$ be a homogeneous degree $d$ polynomial in $\mathbb{C}[X_0,X_1,X_2,X_3]$ which defines a smooth surface in $\mathbb{P}^3$. Here $l_i$ are linear polynomials and $l_1 \not=\lambda ...
- 1,040
0
votes
1
answer
172
views
minimal spans of polynomial companions of co-prime polynomials.
Is there an algorithm to determine for given $P,Q$ in $\mathbb Z[x,x^{-1}]$ with $gcd(P,Q)=1$, the value of $min\lbrace Span(A)+Span(B): A,B\in \mathbb Z[x,x^{-1}],\ A\cdot P+B\cdot Q=1\rbrace$, where ...
- 2,390
0
votes
1
answer
223
views
Representation dimension of a special algebra
Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
- 1,755
0
votes
1
answer
303
views
Completion of a completion
Let $A$ be a commutative ring (not necessarily noetherian).
Let $I\subseteq J\subseteq A\,$ be two finitely generated ideals.
Let us denote the completion functor by $\Lambda_K (M) = \varprojlim_n M/...
- 1
0
votes
2
answers
205
views
Structure of Homomorphisms of commutative C^* algebra
Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the $...
- 45
0
votes
2
answers
207
views
Projective dimension of cohomology over regular rings
Suppose $R$ is a regular ring and $F^{\bullet}: 0\to F^0 \to F^1 \to \dots \to F^d \to 0$ is a complex of finite rank free $R$-modules. Is is true that $\mathrm{projdim}H^i(F^{\bullet}) \leq d$ for ...
- 13.1k
0
votes
1
answer
396
views
Dynamics of polynomial roots
Are there any good tools to understand the movement of roots of polynomials in single variable with real or rational coefficients? That is say the coefficients are of the form $a_{i} + M b_{i}$ where $...
- 13.9k
0
votes
1
answer
135
views
Uniqueness of Hensel factors of a polynomial (invariant to change of "basepoint")?
An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is Hensel lifting. Here's a brief, concrete example to set the stage for my question:
Let $f \...
- 2,019
0
votes
1
answer
154
views
Example of a non-liftable morphism from a smooth algebra
This precise question grew out from the question whether a smooth commutative $k$-algebra (char($k$)=$0$) is always cofibrant as a non-positively graded commutative differential graded co-chain $k$-...
- 257
0
votes
1
answer
544
views
associated prime ideal [duplicate]
Possible Duplicate:
minimal prime devisor(MinAss R)
Hello All,is This conclusion true?
$(R,m)$ be a local ring.if every associated prime ideal of $R$ be minimal then every associated prime ideal ...
- 418
0
votes
1
answer
494
views
Example: Nil radical of noetherian Rings with a map to simple noetherian rings
A basic example in commutative algebra: Let $A$ $B$ be noetherian rings, with $B$ simple noetherian. Suppose that for every element $b$ in $B$, there exists a power $b^{n}$ ...
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
0
votes
1
answer
208
views
How to consider a module over the ring Q[t,t^(-1)] to be a module over the polynomial ring Q[t]? [closed]
Can we view a module over the ring $\mathbb{Q}[t,t^{-1}]$ to be a module over the polynomial ring $\mathbb{Q}[t]$?
where $\mathbb{Q}$ denote any rational number coefficients.
- 161
0
votes
1
answer
223
views
Equivalent functors
Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong Hom(...
- 259
0
votes
1
answer
147
views
Small set of acts over a countable monoid?
Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
- 11
0
votes
1
answer
262
views
Subtleties in the construction of base change morphisms
Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of ...
- 1,391
0
votes
1
answer
465
views
What is lim⟶ I^n M?
Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits_{\begin{subarray}{c}
\longrightarrow \\
\...
- 259
0
votes
2
answers
356
views
Can all induced maps be described categorically.?. (or at least as generally as possible)
Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol.
I am pretty confused about induced maps in different areas of algebraic
topology; I do know how these induced maps are ...
- 1
0
votes
2
answers
172
views
small extensions of the free semigroup of rank 1
Let N denote the free semigroup of rank 1. Say that a semigroup T is a small extension
of N if N embeds in T and |T - N| is finite. Is there some kind of classification
of small extensions of N? ...
- 91
0
votes
0
answers
94
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
0
votes
0
answers
48
views
Integral graded algebra of finite type is approximable
The following is the definition of approximable algebra.
An integral graded $K$-algebra $\oplus_{n\geqslant 0}B_n$ is said to be approximable if
1.$$rk_K(B_n)<+\infty,\forall n\in \mathbb{N}, $$and ...
- 1
0
votes
0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
0
votes
0
answers
168
views
Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
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 ...
- 1
0
votes
0
answers
114
views
Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
0
votes
0
answers
176
views
$\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, with each pair of $\{f,g,x\}$ not generating $\mathbb{C}(x,y)$
Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$.
Write,
$f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$
and
$g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$,
for some $n,m ...
- 2,837
0
votes
0
answers
53
views
A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 341
0
votes
0
answers
42
views
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
0
votes
0
answers
31
views
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
0
votes
0
answers
47
views
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
0
votes
0
answers
97
views
Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
- 341
0
votes
0
answers
76
views
Largest set of monomials whose span is "co-prime" to a given polynomial
Let $K$ be a number field, and let $F \in K[x_1, \cdots, x_n]$ be a polynomial. For a positive integer $d \geq 3$, define $M(F;d)$ to be the largest positive integer such that there exists a set $S$ ...
- 26.9k
0
votes
0
answers
113
views
Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 341
0
votes
0
answers
82
views
Integer valued polynomials and divided power algebra
Let $T\subset \mathbb Q[x]$ be the ring of integer valued polynomials, i.e. the polynomials $f$ with $f(\mathbb Z)\subset \mathbb Z$. In his wonderful book ”Commutative algebra with a view toward ...
- 1,907
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votes
0
answers
95
views
Which algebraic structure characterizes the set of non-trivial qudratic residues in a finite field?
I understand this question may be too naive to ask, but I am unable to figure it out.
Suppose, $\mathbb{QR^*}$ denotes the set of all quadratic residues in a finite field except the identity element $...
0
votes
0
answers
57
views
Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
- 21
0
votes
0
answers
100
views
Shedding faces and decomposability in simplicial complexes
Definition:
A pure d-dimensional complex
$\Delta$ is $k$-decomposable if either $\Delta$ is a $d-$simplex or $\Delta$ contains a face $F$ such that
$\dim(F) \leq k$
both $\Delta \setminus F$ and $\...
- 381
0
votes
0
answers
87
views
Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
- 219
0
votes
0
answers
184
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
0
votes
0
answers
70
views
Hensel lifting of roots of a biquadratic polynomial
Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
- 381
0
votes
0
answers
60
views
Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
- 21
0
votes
0
answers
111
views
Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
- 223
0
votes
0
answers
109
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
0
votes
0
answers
329
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149
0
votes
0
answers
126
views
Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
- 25.4k