All Questions
6,055 questions
1
vote
0
answers
389
views
A criterion for purity
I have started reading the book "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.
"$E$
is pure if and only if all ...
- 159
1
vote
0
answers
81
views
Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
- 10.5k
1
vote
0
answers
86
views
Characterization of a finitely graded (almost) domain
Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then
$$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$
Hence ...
- 3,019
1
vote
0
answers
134
views
Linear combinations of reducible polynomials
My question concerns the sums of two reducible polynomials with bounded coefficients. In particular, for every $d \geq 2$ does there exist a number $C(d) > 0$ such that for any two co-prime ...
- 26.9k
1
vote
0
answers
148
views
Has there been proposed an extension of real numbers that connects logarithms and exponents in closed form? [closed]
Complex numbers do connect trigonometric functions with hyperbolic functions and exponents in closed form. Has anybody ever proposed an algebraic system that would connect in a similar way ...
- 10.1k
1
vote
0
answers
141
views
What are the associated prime ideals of rees ring?
Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $R[It]$ be a rees ring of $R$ with respect to $I$. Do we have $Ass R[It]=\{pR[It] : p\in Ass(R)\}$? if not what can we say?
($Ass(R)$ is the ...
- 113
1
vote
0
answers
77
views
Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
- 129
1
vote
0
answers
639
views
What is the real name of this relation and operation on a particular set of maps between cancellative monoids?
Let $A,B$ be cancellative monoids and define a transducer as a map $f\colon A \rightarrow B$ such that $f(1)=1$ and for all $a_1 ,a_2 \in A$, there exists a $b \in B$ such that $f(a_1 a_2)=f(a_1) b$. ...
1
vote
0
answers
75
views
Closure of the set of principal ideals under a certain operation
Suppose $K$ is a field, and $R$ is the polynomial ring $K[x_1, \ldots, x_n]$.
Suppose $S$ is a set of ideals of $R$ satisfying these properties:
$S$ contains all principal ideals.
If $I$, $J$, and $...
- 804
1
vote
0
answers
216
views
Destabilizing subsheaf: length of $0$ dimensional subscheme
I have a very specific question (quite elementary, sorry!)
Let $G$ be a rank $2$ torsion free sheaf on an algebraic surface $X$ (normal maybe?)
Let $L\otimes \mathcal{I}_Z$ be a Gieseker ...
- 297
1
vote
0
answers
133
views
Embedding of Gorenstein orbifold as a hypersurface
I am trying to understand if three complex dimensional orbifold singularity $\mathbb{C}^3 / \Gamma$ can be embedded as hypersurface in $\mathbb{C}^4$. The condition of being Gorenstein and having ...
- 367
1
vote
0
answers
107
views
Question on Hochschild cohomology
Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$.
Is it true that if for $\...
- 11
1
vote
0
answers
135
views
Monomial algebras and depth
Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence.
Assume $...
- 11
1
vote
0
answers
91
views
Cut ideal of two graphs?
Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
- 143
1
vote
0
answers
310
views
is the "reduction" of an effective cartier divisor on a relative curve still a cartier divisor?
Let $C\rightarrow S$ be a smooth proper morphism of relative dimension 1, where $S$ is a Noetherian normal scheme. Let $D\hookrightarrow C$ be a relative effective Cartier divisor finite over $S$. Let ...
- 7,527
1
vote
0
answers
195
views
Image of a Weil divisor (height 1 prime) under normalization map
Let $R$ be a noetherian integral domain and let $R'$ be the normalization of $R$ in a finite field extension of the fraction field of $R$. Let $\varphi:Spec(R') \rightarrow Spec(R)$ be the ...
- 11
1
vote
0
answers
179
views
Spivakovski-Popescu-Neron desingularisation
For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...
- 781
1
vote
0
answers
144
views
Explicit construction of a bielliptic curve
Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...
- 11
1
vote
0
answers
164
views
Combinatorial splitting in number rings
The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty ...
user94040
1
vote
0
answers
74
views
Macaulayfications and standard blow-ups
Let $X$ be an affine variety such that $X-p$ is Cohen-Macaulay,
and let $\pi \colon \widetilde{X} \rightarrow X$ be a standard blow-up of $X$
with respect to $\mathcal{O}_X$, centered at $p$.
It is ...
- 1,595
1
vote
0
answers
166
views
Popescu-Neron Desingularization for K[[T_1,...,T_∞]]
Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$.
Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
- 781
1
vote
0
answers
76
views
Coherence of subrings of K[[X,Y]]
Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$.
Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
- 781
1
vote
0
answers
189
views
What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]
Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...
- 255
1
vote
0
answers
357
views
Base change and geometrically generic reduced fiber
Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...
- 2,126
1
vote
0
answers
485
views
On the coherence of formal power series ring
Let $A = {\Bbb F}_p[[X_1,X_2,...]]$
be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$
$A$ consists of such formal sum elements as $\sum c_{e_1,.....
- 781
1
vote
0
answers
218
views
How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?
Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much.
For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...
- 6,201
1
vote
0
answers
683
views
Structure theorem for infinitely generated modules over a PID
This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain.
The question is all in the title: is there ...
- 1,959
1
vote
0
answers
139
views
A strong form of Bezout theorem
Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
- 2,126
1
vote
0
answers
67
views
Modification of nonfree locus
Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...
- 51
1
vote
0
answers
269
views
Transitivity for algebraic extensions of integral domains?
I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...
- 3,761
1
vote
0
answers
229
views
Cyclic decomposition of an infinitely generated module
My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance.
Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a ...
- 1,959
1
vote
0
answers
112
views
Dimension of a module (which is not necessarily finite)
Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
- 1,713
1
vote
0
answers
85
views
if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$
Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual.
Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$?
Is there a ...
- 1,355
1
vote
1
answer
110
views
A linear subspace of $\mathbb{R}[X_1,\cdots,X_n]$ and its generated set
Let $n$ be a positive integer and $W_n$ be the linear subspace of the real vector space $\mathbb{R}[X_1,\cdots,X_n]$ generated by the following set
$$S_n=\{X_1^{i_1}\cdots X_n^{i_n}:i_1+\cdots+i_n=n\ \...
- 1,997
1
vote
0
answers
201
views
Number of minimal primes for UFD
Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...
- 781
1
vote
0
answers
884
views
Invertible elements in a group algebra
Let $H$ be a torsion-free abelian group and let $\mathbb{K}$ be a field with two elements.
I would like to ask the following question:
Is the group of units of the group algebra $\mathbb{K}[H]$ ...
- 1,613
1
vote
0
answers
97
views
A property of minimal prime ideals in rings with finite chromatic number
Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
- 271
1
vote
0
answers
59
views
How to associate the following two kinds of real polynomials?
Suppose the following real polynomial of $n$ variables
$$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$
is easy or familiar to us, but I need to deal with ...
- 1,997
1
vote
0
answers
449
views
Deeply ramified implies non discrete valuation - Almost ring theory
In their book "Almost Ring Theory" (http://arxiv.org/abs/math/0201175), Ofer Gabber and Lorenzo Ramero define a valued field $K$ to be "deeply ramified" if the module of Kähler differentials $\Omega_{...
1
vote
0
answers
54
views
Elimination theory for variables packaged in a matrix
I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices.
For instance, consider the following:
...
- 367
1
vote
0
answers
82
views
If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?
Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...
- 1,479
1
vote
0
answers
110
views
Grobner basis for a general algebra
Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
- 2,384
1
vote
0
answers
156
views
A family of maximal ideals
Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} m_i\...
- 49
1
vote
0
answers
280
views
Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
- 11
1
vote
0
answers
172
views
Local cohomology commuting with fiber
Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
- 5,562
1
vote
0
answers
35
views
Projectivity of a faithfully balanced self-orthogonal bimodule
Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right $S$...
- 1,207
1
vote
0
answers
106
views
Upper bound for the minimum number of generators of the canonical module
Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...
1
vote
0
answers
125
views
A question from Hilbert's Nullstellensatz [closed]
From the Hilbert's Nullstellensatz, we know that for any algebraic closed field $K$ and any prime ideal $p$ of $K[X_1,X_2,\cdots,X_n]$, the intersection of all the maximal ideals containing $p$ is $p$....
- 1,997
1
vote
0
answers
135
views
Relation of primary decomposition of two ideals
I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...
1
vote
0
answers
895
views
Does the functor of taking invariants commute with tensor products? [closed]
Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the $G$-...
- 27