All Questions
6,055 questions
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Explicit representative for an extension class
Let $A$ be a regular local ring and $I\subset A$ a complete intersection ideal.
We have the natural map $\delta:Hom_A(I,A/I)\rightarrow Ext_A^1(A/I,A/I)$.
For a given $\alpha\in Hom_A(I,A/I)$ is there ...
- 1,463
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74
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Sufficient conditions for $b\not\in I^2$ given that $b\in I$
Let $I$ be an $R$-ideal in a commutative algebra $B$ over a commutative ring $R.$ Given $b\in I$ I want to prove that $b\not \in I^2$.
Are there any sufficient conditions for showing that $b\not\in I^...
- 1,615
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173
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Can the notion of algebraic closedness be generalized to the rings with zero divisors?
Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero?
For instance, it seems that any polynomial of non-zero-divisor-...
- 10.1k
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177
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Passing over $O_K \otimes_{\mathbb{Z}} A$ from $O_K$, how it affects the rank of a module?
This question was asked in MSE as well.
Let $K$ be a finite extension of the rationals $\mathbb{Q}$ with $O_K$ its the ring of integers.
Consider a $\mathbb{Z}$-algebra $A$ such that $|A|<\infty$.
...
- 930
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195
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Nice small resolution and normality of blow-up
Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is ...
- 1,463
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265
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Algebraic closure of field of fractions of multivariate polynomial ring over $\mathbb{R}$
I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(\mathbb{R}[x_{1},\dots,x_{n}])^{\operatorname{alg}}.$ I imagine that, when we try to ...
- 573
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293
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Quotient of monoids and monoid algebras
Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
- 327
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135
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On resolution of singularities over an Artin ring
For a locally noetherian scheme $X$, Grothendieck conjectured that if $X$ is quasi-excellent then there is a proper birational map $Y \to X$ s.t. $Y$ is regular.
We now fix an Artin ring $R$ whose ...
- 519
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105
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Flag variety as monoid and Schubert calculus
The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...
- 11
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222
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To show equivalence and full faithfulness of a functor PRESERVED under an action of a finite flat algebra
I have explained the two questions and then showed my effort on question $(1)$ as follows (Please at least check my effort below and suggest to make it perfect):
Let $R, S,T$ be three commutative ...
- 930
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137
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Ascend and descend properties for arithmetically Cohen-Macaulay/Gorenstein varieties
I had few questions regarding varieties admitting embeddings that make them arithmetically Cohen-Macaulay or Gorenstein varieties. A projective variety is called arithmatically Cohen-Macaulay/...
- 5,901
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110
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Decomposition an $A$-module to irreducible ones
Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra.
Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
- 4,058
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250
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Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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65
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Constant function on the generic fiber $f^{-1}(\eta)$ is contained in the function field $K(U)$
Let $U$ and $V$ be irreducible varieties and $f\colon V\rightarrow U$ be a proper surjective morphism.
Assume, $f^{-1}(\eta)$ is irreducible ($\eta$ is the generic point of $U$).
$\require{AMScd}$
\...
- 297
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137
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Elliptic units as Euler systems
I’m trying to understand elliptic units in order to work with the Euler systems of the abelian extensions of quadratic imaginary number fields. I’ve looked at few references about the topic, but they ...
- 99
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257
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How to prove the map of rings $\mathcal{R} \to \mathcal{R'}$ is flat?
We fix a finite extension $K$ of $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $\kappa$. Consider the ring of witt vectors $W(\kappa)$ over the residue field $\...
- 930
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182
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non constant regular function derivative is zero
Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.
More generally we could ask, Is it true that $...
- 629
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41
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Characterizing centralizer of nilpotent self-maps
Let $\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(...
- 109
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82
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Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
- 1,703
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147
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Is it possible to compute the basis of this module?
Let $A$ be a polynomial algebra in $n$ variables over field $\mathbb{F}$ of characteristic zero which is algebraically closed. Assume that $a_1,\ldots, a_n, b_1,\ldots, b_n\in A$ are such that $a_1b_1+...
- 291
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268
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Non-Noetherian local ring with nilpotent maximal ideal
Browsing through my notes on Artin rings, I have realized that I don't know an example for this and I wasn't able to google anything relevant.
What is an example for a commutative non-Noetherian ...
- 7,127
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191
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When $K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ is a flat morphism
Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial $K$-algebra morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,...
- 6,038
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93
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A semifield of characteristic zero may have a finite number of elements
A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.
I ...
- 203
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74
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Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples
$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
Is it true that there are ...
- 1,826
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413
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When are the cotangent and tangent sheaves isomorphic?
Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
- 327
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163
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Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
- 3,625
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124
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Increasing the number of ideals in an exact sequence
In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$
$...
- 183
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71
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Low rank approximation
Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
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114
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Compare degrees of a finite extension of domains and quotient domains
Let $A \subset B$ be a finite (finite type + integral) extension of integral domains and let $\mathfrak{p} \subset A, \mathfrak{q} \subset B$ be prime ideals such that $\mathfrak{q} \bigcap A =\...
- 393
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154
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Determinant of a special matrix in characteristic $p$
Let $K$ be a field of characteristic $p > 0$. Choose $p^i$ numbers of elements $c_1,\ldots,c_{p^i} \in K$ and consider the determinant $D$ of the following matrix$\colon$
\begin{pmatrix}\label{...
- 563
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103
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A certain property of integral domains $A \subseteq B$ with $Q(A) \cap B= A$
I have asked the following question in MSE:
Let $k$ be a field of characteristic zero. Let $A \subseteq B$ be $k$-algebras which are also (commutative) integral domains with fields of fractions $Q(A) \...
- 2,837
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113
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Special element of a commutative ring
Let $R$ be a commutative ring with $1$ and $S $ be a multiplicative subset of $R $. I am looking for an equivalence condition for the following property in $R $:
Property: There exists a fixed ...
- 1
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133
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Is the Hilbert series of an ideal related to the Hilbert series of its homogenization?
Suppose we have a field $k$ of characteristic 0, let $I$ be an ideal of $R=k[x_1,...,x_n]$, and let $H$ be the homogenization of $I$ in $S=R[z]$. Is there any relationship between the Hilbert series ...
- 573
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105
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An ideal invariant under an automorphism
The following question appears here; hopefully, it is appropriate for MO.
Let $k$ be a field of characteristic zero, and let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \...
- 2,837
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213
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Is the completion of an infinitely generated module, again infinitely generated
Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of ...
- 2,126
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94
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Tensor product of preordered rings
All rings in this post are commutative, unital, and contain $\frac{1}{2}$.
To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
- 277
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118
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Multiplicity of a polynomial in positive characteristic
Let $\mathbb K$ be a field of characteristic $p>0$.
Let $f\in\mathbb K[x_1,\dots,x_n]$ be a multivariate polynomial and let $q\in\mathbb K^n$. Is there a computational method to determine the ...
- 351
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215
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Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?
Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
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153
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Locus of trivialization of an extension of a vector bundle
Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
- 3,472
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112
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Homogeneous basis on a polynomial subalgebra
Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...
- 1
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287
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On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
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129
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Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$
Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $f: k[x,y] \to R_{-1}$ be ...
- 2,837
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558
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Intersection of an ideal and a subring
Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...
- 245
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89
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Structures like vector spaces but closed under heterogeneous products
The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...
- 93
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84
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Primage structures: induced domain partitioning by itterated inverse (reference request)
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example, the j-th such preimage list ...
- 23
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141
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Two commuting matrices over a commutative ring
I would like to know if there are results about the dimension of the Algebra generated by two commuting Matrices over a ring (as there are in the case of a Field).
The good news is that "my" ring is ...
- 337
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126
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Reference request: J-P Serre, "Groupes finis d'automorphismes d'anneaux locaux réguliers"
Does anyone have, or know a link to, a copy of the paper named in the title? It is published in Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11. I do not have ready access to Serre's ...
- 3,137
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68
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Normality of certain subrings of polynomial rings in characteristic p
Let $k$ be an algebraically closed field of characteristic p. Let
$Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a
polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in
$k[...
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90
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Do Plucker relation follow from a subsystem of equations?
The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$,
\begin{align*}
P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...
- 6,201
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448
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713