All Questions
669 questions
6
votes
1
answer
530
views
Does a variety contain a cartesian product of two curves?
We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
- 7,836
6
votes
1
answer
725
views
Who defined and who coined "module"?
The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...
- 7,500
6
votes
1
answer
269
views
Algebraization of Bayesian networks?
The algebraization of classical propositional logic is Boolean algebra.
Bayesian networks are a generalization of classical propositional logic with probability truth-values.
What is the ...
- 558
6
votes
3
answers
393
views
Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)
Question. Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons?
A monoid with ...
- 10.5k
6
votes
0
answers
618
views
Duality between conjugacy classes and irreducible characters for finite monoids?
Qiaochu's answer to this question suggests that the proper way to view the bijection between conjugacy classes and irreducible complex representations of a finite group is via a duality. My question ...
- 38.6k
6
votes
3
answers
2k
views
Dimension of polynomial algebras
Let $R$ be a commutative ring of Krull dimension $d$, let $n\in\mathbb{N}$, and let $R[X_1,\ldots,X_n]$ denote the polynomial algebra in $n$ indeterminates over $R$. One can show that then we have $\...
- 6,700
6
votes
1
answer
216
views
Coloring summands of given n-partition with given weights of colors
Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$
Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...
- 443
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
2
answers
457
views
Integer Gelfand-Kirillov dimension
Let $R$ be a (noncommutative) Noetherian affine $K$-algebra. The Gelfand-Kirillov dimension is known to be an integer for many classes of affine Noetherian algebras. I wonder, if this is true for any ...
user91931
6
votes
2
answers
529
views
A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$
Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$.
For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote $w(f)=f(x_{w(1)},x_{w(2)},\ldots,...
- 1,997
6
votes
1
answer
623
views
When is the cofibrant replacement of a product the product of the cofibrant replacements?
I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...
- 30.3k
6
votes
2
answers
547
views
Properties of ring epimorphisms that are true only over commutative rings
I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case.
Example: I learned from ...
- 2,160
6
votes
3
answers
472
views
Spaces with unique endomorphism monoids
If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$.
We ...
- 48.9k
5
votes
1
answer
362
views
On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
- 541
5
votes
1
answer
506
views
Injective homomorphism of modules and tensor product
Let $R$ be a commutative ring; let $M$, $N$ be $R$-modules and let $f\colon M\to N$ be an injective homomorphism of $R$-modules. Is $f\otimes {\rm id}\colon M\otimes_RN\to N\otimes_RN$ injective?
- 351
5
votes
1
answer
656
views
Uncountable chain of prime ideals in an arbitrary direct product of rings
I am only considering commutative rings with $1$. Dimension refers to Krull dimension.
In the paper "Products of commutative rings and zero-dimensionality", Gilmer and Heinzer give necessary and ...
- 415
5
votes
1
answer
345
views
Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective
Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?
I'm looking for conditions ...
- 6,406
5
votes
1
answer
182
views
Antisymmetrization of the Hochschild cocycle
Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define $$(b\varphi)(a_0,a_1,...,a_{n+1}):=\sum_{j=0}^{n}(-1)^j\...
- 9,330
5
votes
0
answers
144
views
Is there a good notion of higher-rank archimedean norm?
Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
- 64k
5
votes
2
answers
434
views
Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?
Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the ...
5
votes
1
answer
697
views
Does perfection of rings commute with products?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius, so being a filtered colimit in Rings, the perfection ...
- 533
5
votes
2
answers
933
views
Is every commutative group structure underlying at least one (unitary, commutative) ring structure
From the theorem of classification of finitely generated abelian groups, we can see that every finitely generated commutative group can be considered as the additive structure underlying (at least) ...
- 2,655
5
votes
0
answers
200
views
A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
- 2,837
5
votes
1
answer
545
views
For what varieties do we have results on the category of singularities?
Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...
- 1,423
5
votes
2
answers
332
views
Questions on weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called weakly symmetric in case $soc(P)=top(P)$ for every indecomposable projective module $P$ and it is called symmetric in case $D(A) \cong A$ as ...
- 26.5k
5
votes
0
answers
561
views
Brauer groups of a local ring and of its residue field
This is a question of DeMeyer (see the last paragraph of [1]):
What's an example of a local ring $A$ with residue field $k$ such that the restriction map on Brauer groups $\varphi : \operatorname{...
- 2,017
5
votes
2
answers
364
views
Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
- 505
5
votes
1
answer
342
views
Can the trivial module be stably free for a monoid ring?
Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ ...
- 38.6k
5
votes
1
answer
265
views
Is every matrix involution over a UFD diagonalisable?
Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...
- 7,722
5
votes
1
answer
759
views
On the annihilator of a module
Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such
that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$?
Remark. The annihilator of a ...
- 303
5
votes
2
answers
838
views
Epimorphisms and free submodules
By inspecting the accepted answer to this question
Are epimorphisms from a division ring isomorphisms ?
one obtains the following necessary condition for epimorphisms:
Let $R \le S$ be rings ...
- 2,160
5
votes
2
answers
754
views
A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 128k
5
votes
0
answers
187
views
Isbell duality for monoids and groups
Isbell Duality
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\B}{\...
- 11.8k
5
votes
2
answers
788
views
When does End(M) consist entirely of zero, zero divisors, and units?
Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...
- 1,197
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
5
votes
1
answer
306
views
Generators of an ideal with small degree
Let $P_1,\ldots, P_d, Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots, x_n]$ be homogenous polynomials of degree at most $r$.
Assume that $P_1 \cdot P_2 \cdots P_{d-1} \cdot P_d \in \langle Q_1, \ldots, ...
- 2,576
5
votes
0
answers
99
views
Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
5
votes
1
answer
504
views
Classification of the quotients of the ring Z/4 [X]
Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
- 63.1k
5
votes
3
answers
2k
views
Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
- 2,399
5
votes
1
answer
322
views
Non-negative coefficients polynomials
Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$.
Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ?
I have asked, this question here (*), two weeks ago, but no answers.
(*) ...
- 4,074
5
votes
1
answer
924
views
Is a base-change of an integral domain by an extension of its base field without algebraic elements still a domain?
Let $K$ be a field, and let $L/K$ be an algebraically closed field extension (i.e. the only elements of $L$ that are algebraic over $K$ are already in $K$). Let $R$ be a $K$-algebra that is an ...
- 1,802
5
votes
4
answers
6k
views
Are quotients of polynomial rings almost UFDs?
If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
- 1,412
5
votes
2
answers
402
views
Maximal commuting subsets of $\text{End}(X)$
Let $X$ be a set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. We say that $f, g\in \text{End}(X)$ commute if $g\circ f = f\circ g$, and $S\subseteq \text{End}(X)$ is a commuting ...
- 48.9k
5
votes
1
answer
408
views
On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
user315884
5
votes
2
answers
236
views
Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
- 6,700
5
votes
0
answers
283
views
irreducibility of a scheme; a "continuous" property or a "limit" property?
I bumped into the following question while working on a research problem on closed subschemes over a completed DVR. I hope someone could possibly answer it, or give a hint toward its resolution. To ...
- 937
5
votes
2
answers
689
views
Irreducible components of quotients of Cohen-Macaulay rings of the "correct" dimension
Suppose $R$ is a Cohen-Macaulay ring. It is well known that if $I$ is an ideal of $R$ generated by $n$ elements, and $I$ has codimension $n$, then $R/I$ is also Cohen-Macaulay.
Now suppose that $I$ ...
- 3,093
5
votes
0
answers
184
views
Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms
Let $K$ be an algebraically closed field of characteristic $p>0$
and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by
(homogeneous) polynomials of degree $d$. Assume that $I$ is reduced,...
- 3,813
5
votes
2
answers
441
views
Name and notation for a binary operation
Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore ...
- 19.7k
5
votes
1
answer
2k
views
Generalizing Dedekind's Factorization Theorem
A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...
- 14.6k