All Questions
6,057 questions
4
votes
0
answers
560
views
Filtration over tensor product
Let
$$ M \supset M_1 \supset \ldots \supset M_n \supset \ldots \text{ and } N \supset N_1 \supset \ldots \supset N_n \supset \ldots$$
be exhaustive decreasing filtrations of modules over a commutative ...
- 385
3
votes
0
answers
336
views
Intersections of strict transform and strict transform of intersections
Let $Z_1,Z_2$ and $Y$ be subvarieties of a locally complete intersection variety $X$ over $\mathbb C$.
Consider the strict transforms of $Z_1$ and $Z_2$ in the blowup $Bl_YX$, the question is: when ...
- 31
3
votes
3
answers
681
views
on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 30.5k
51
votes
3
answers
3k
views
Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...
- 105k
4
votes
1
answer
220
views
In a commutative Koszul algebra, does every ideal generated by a subset of variables have linear resolution?
Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset ...
- 553
5
votes
1
answer
294
views
The Kronecker--Hurwitz property for rings of integers in global function fields
In Ireland and Rosen's book on number theory they give a proof of the finiteness of the class group of a number field which they attribute to Hurwitz, but which is essentially due to Kronecker (as I ...
- 3,823
2
votes
0
answers
95
views
Extending finitely many primitive elements of $\mathbb{Z}^{n+1}$ to bases with non-trivial intersection
Given a finite number of primitive elements $v_1,\dots,v_k\in\mathbb{Z}^{n+1}$ (i.e. the gcd of the entries of each $v_i$ is $\pm1$), is it always possible to find an element $v\in\mathbb{Z}^{n+1}$ ...
- 291
12
votes
2
answers
799
views
Normal Macaulayfications
Given any reduced excellent scheme $X$, there exists a Macaulayfication $\pi : Y \to X$. In other words, there exists a proper birational map $\pi$ from a Cohen-Macaulay scheme $Y$ to $X$. These ...
- 3,065
2
votes
0
answers
247
views
Ring isomorphism of multivariate polynomials/functions
It's well-known that over an infinite integral domain $R$, the ring of univariate polynomials $R\left[X_{1}\right]$ is isomorphic to a ring of one-argument "polynomial functions" (see, for ...
- 111
4
votes
1
answer
192
views
(Infinite) free resolution of $R/(x-z, y-w)$ for $R=\mathbb C[x,y,z,w]/(xy-zw)$
For a Noetherian local ring $R$, Koszul complex is a useful tool to construct finite free resolution of any quotient of $R$ by regular sequences $R/(x_1,\dots, x_r)$.
Let $R=\mathbb C[x,y,z,w]/(xy-zw)$...
- 125
5
votes
1
answer
385
views
Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
- 63.9k
21
votes
1
answer
2k
views
Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
- 35.5k
2
votes
1
answer
222
views
Wild ramification in composite fields
Let $ F $, $ F_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F_i \subseteq E $ and $ E = F_1 \cdot F_2 $ is the composite ...
- 3,065
5
votes
2
answers
261
views
Algebraically closed ring extension
Suppose that $B \rightarrow A$ is an extension of rings where $A$ and $B$ are integral $k$-algebras ($\mathrm{char}\,k = 0$) finitely generated over $k$.
It is well known that if $B \rightarrow A$ is ...
- 3,720
10
votes
0
answers
575
views
How general are Gröbner degenerations?
While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
- 3,513
12
votes
1
answer
863
views
Do commutative rings with "interesting" Jacobson radicals turn up "in nature"?
Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is uninteresting if
$J(R)$ coincides with the nilradical, or
$J(R)$ is the intersection of a finite number of maximal ...
CommunityBot
- 1
3
votes
2
answers
386
views
Lci local rings with isolated singularity are irreducible?
Let $R$ be a noetherian local ring; I say it has isolated singularity if its spectrum is regular outside the closed point. Such rings certainly don't need to be irreducible, for example the ...
- 7,300
0
votes
0
answers
105
views
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
0
votes
0
answers
222
views
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
4
votes
0
answers
141
views
Existence of a global completion functor
Is there any such thing as a ``global completion functor'' for commutative Noetherian rings?
More specifically, let $R$ be a commutative Noetherian ring. For each maximal ideal $m$ of $R$, let $W_m :=...
- 1,802
8
votes
1
answer
670
views
Derivation of formal power series
The basic idea of this question is to see if there is any other derivations than 'formal derivations'.
Let $\mathbb{K}$ be a field. Given a commutative $\mathbb{K}$-algebra $A$, a derivation of $A$ is ...
- 19.6k
2
votes
1
answer
262
views
How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?
I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...
- 53.3k
2
votes
0
answers
77
views
When are classes with prescribed reducts "pseudo"-elementary?
Let $\mathsf{Set}$ be the class of all sets and let $\mathcal{L}$ be a first-order language. Let $M \subseteq \mathsf{Set}$ be a set of $\mathcal{L}$-structures and let $$\mathfrak{Th}_{\in}(M) = \\ \{...
- 365
11
votes
5
answers
8k
views
An example of two elements without a greatest common divisor
Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.
"Easy" means that I can explain it to my ...
- 33.8k
11
votes
3
answers
1k
views
The concept "conjugate class" in monoids.
Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
- 12.9k
5
votes
1
answer
208
views
Conjugacy classes of monoids II: Abelianising a monoid, wrongly
$\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is
$$
G/\left(ab\sim ba\ \middle|\ a,b\in G\right)?
$$
Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$.
When ...
- 38.6k
0
votes
0
answers
137
views
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
2
votes
1
answer
85
views
Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$
The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...
- 19.6k
3
votes
1
answer
212
views
Is a tower of locally-free modules locally a tower of free modules?
Suppose we have a (commutative, unital) ring $R$ and a (commutative, unital) $R$-algebra $A$ such that $A$ is projective of constant rank $n$ as an $R$-module. This condition is equivalent to there ...
- 23.4k
5
votes
0
answers
324
views
Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
0
votes
1
answer
429
views
Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
- 28.7k
10
votes
1
answer
410
views
Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally ...
- 2,051
2
votes
2
answers
899
views
Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 35.5k
9
votes
0
answers
204
views
Reverse mathematics of Noetherian rings over $\mathbb{Q}$
Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic: For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
- 35.5k
8
votes
6
answers
2k
views
How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
- 4,862
5
votes
1
answer
252
views
Infinitely many initial ideals for non-Artinian monomial orders?
Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...
- 85.6k
6
votes
3
answers
596
views
What's the current state of one-rule semi-Thue system termination problem?
What's the current state of one-rule semi-Thue system termination problem? Search produces a lot of references, but it's hard to find out if decidability of this problem has been proven or not.
2
votes
0
answers
111
views
If $\mathfrak{p}\subset R$ is a minimal prime divisor of $\mathrm{Ann}_R(M)$, then $\mathrm{Ann}_R(M/\mathfrak{p}M)=\mathfrak{p}$
$\DeclareMathOperator\Ann{Ann}$Let $R=\mathbb{C}[x_1,\dots,x_n]$. I am looking for a reference of the following statement.
$(*)$ Let $M$ be an $R$-module, and let $\mathfrak{p}$ is a minimal prime ...
- 2,788
5
votes
1
answer
394
views
Kähler differentials on an Artinian local ring
Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...
- 63.9k
4
votes
1
answer
355
views
A noneffective descent datum: isomorphism not satisfying the cocycle condition
Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times_{S} S'$ and $S''' := S' \times_{S} S' \times_{S} S'$ with ...
- 984
4
votes
1
answer
731
views
matrix congruence and smith normal form
Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
- 121
8
votes
1
answer
493
views
General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
- 385
1
vote
1
answer
150
views
Algebraic structure of the space of multiaffine maps
Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number.
It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector ...
- 40.2k
9
votes
1
answer
1k
views
Zero scheme of global sections of vector bundles on affine varieties
I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties.
Let $\mathbb{K}$ be an algebraically closed field, $...
CommunityBot
- 1
1
vote
0
answers
87
views
Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations
Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary ...
- 225
1
vote
1
answer
344
views
Minimum number of generators of the product of ideals
Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two ...
- 30.5k
2
votes
1
answer
202
views
How to compute cup product of derived limits / presheaf cohomology
I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
- 37.8k
3
votes
1
answer
185
views
$K_0(\mathsf{Nil}(R))$ when $R$ is a field
$\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel. There I found a very interesting category $\Nil(R)$, which consists of ...
- 63.9k
0
votes
1
answer
81
views
Non-fractions in faithfully flat extension were non-fractions
All rings are commutative with unity.
Let $\phi:A \to B$ be a faithfully flat ring homomorphism. Let $f \in A$, $g = \phi(f) \in B$, and $\psi:A_f \to B_g$ the induced homomorphism on the ...
- 81
1
vote
1
answer
3k
views
Commutation of tensor products with inverse limits in a specific case
For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings)...
- 3,738