All Questions
1,123 questions
2
votes
0
answers
122
views
First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
2
votes
0
answers
90
views
Poincaré Duality of a quasi-free algebra
I'm completely stumped on this one (yet I feel it is obviously true or obviously false)
If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality?
All i need to find is a protective ...
2
votes
0
answers
66
views
The norm of a polynomial f in a skew polynomial ring must be in the center
This is proved in Prop 1.7.1 in Jacobson's book ``Finite dimensional division algebras over fields". But I am not clear why the norm n(f), defined as the norm of the matrix representation of f by ...
- 43
2
votes
0
answers
124
views
Reasoning about "approximately" associative structures and "almost monoids".
If $(M,+)$ is a monoid then it obeys the laws:
$$m_1 + 0 = 0 + m_1 = m_1$$
$$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$
But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example,...
- 21
2
votes
0
answers
272
views
Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
2
votes
0
answers
91
views
Fat modules on some algebras.
Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely ...
- 21
2
votes
0
answers
63
views
Boundedness of modules on AS regular algebras
Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules ...
- 1,663
2
votes
0
answers
137
views
Noncommutative Castelnuovo-Mumford regularity
I am looking for noncommutative version of Castelnuovo-Mumford regularity. To be more precise, let $A=\oplus_{i=0}^{\infty}A_{i}$ be a $good$ (finite global dimension, connected etc) noncommutative ...
- 21
2
votes
0
answers
196
views
Recursion formula for odd holonomic function
Let $E$ be the space of maps
$f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$
satisfying $\forall i$, $f(i)=-f(-i)$.
The algebra
$$\mathcal{T}=\mathbb{C}[t^{\pm}]\langle L^{\pm},M^{\pm}\rangle/(LM-...
- 243
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
- 1,422
1
vote
3
answers
450
views
Smooth affine algebras are Calabi-Yau
Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
- 27
1
vote
1
answer
190
views
Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 87
1
vote
2
answers
367
views
"Exactness" of groupify functor
For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram ...
- 9,853
1
vote
2
answers
296
views
Are orbits of an affine algebraic monoid affine?
Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
- 4,902
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
1
vote
2
answers
375
views
What are the monoids in which every globally idempotent subsemigroup contains the identity element?
A semigroup is called globally idempotent when for any $x\in S$ there are $y,z\in S$ such that $x=yz$.
Is there a name for monoids whose every globally idempotent subsemigroup contains the identity ...
- 1,445
1
vote
2
answers
442
views
submonoids of Z_n
Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?
1
vote
1
answer
215
views
Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 313
1
vote
2
answers
339
views
Are monoids with zero and partial homomorphisms related?
Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U \...
- 2,464
1
vote
1
answer
434
views
A Version of Nullstellensatz for Rings of Dİfferential Operators
Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then $K[T_1,\ldots,T_n]/\mathfrak{...
- 355
1
vote
1
answer
182
views
A question about surjective maps between quadratic algebras
Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...
- 643
1
vote
1
answer
231
views
Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
1
vote
1
answer
203
views
Filtered Colimit of associative $k$-algebras that are domains
Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains.
Then the colimit of the obvious diagram is an integral domain.
...
user78817
1
vote
2
answers
378
views
Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 33.3k
1
vote
1
answer
106
views
Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
1
vote
1
answer
283
views
Ore Extensions and the Construction of the Quantum General Linear Group
In the usual (fomal) construction of the quantum general linear group $GL_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of ...
- 1,453
1
vote
3
answers
2k
views
Various Cartan's Lemmata
I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
- 2,275
1
vote
1
answer
142
views
Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 20.5k
1
vote
2
answers
333
views
Condition for equality of modules generated by columns of matrices
Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
- 431
1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 1,374
1
vote
1
answer
166
views
Reference for a proof of cancellation property of braid monoids
Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
...
- 6,201
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
1
vote
1
answer
410
views
Morphisms of a simple sheaf over an algebra to its double dual
Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...
- 1,391
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
1
vote
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,636
1
vote
2
answers
1k
views
An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
- 1,801
1
vote
1
answer
80
views
Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
- 155
1
vote
1
answer
110
views
Particular example of a quadratic extension of a nonunital ring
I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...
- 223
1
vote
1
answer
368
views
Wedderburn theorem for finite-dimensional algebras over the complex numbers
I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
1
vote
1
answer
165
views
Polynomial identities satisfied by the Weyl algebra in prime characteristic
The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...
- 3,318
1
vote
1
answer
402
views
Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
- 10.1k
1
vote
1
answer
77
views
What is the minimal possible size of a subset of this semigroup satisfying the following conditions?
Suppose $A$ is some set. Let's define a pair semigroup over $A$ as $P[A] = (A\times A \cup \{0\}, \circ)$, where the $\circ$ operation is defined by the following two identities:
$\forall a \in P[A]$ ...
- 2,618
1
vote
1
answer
163
views
Internal commutative monoid gives commutative monad
Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object.
The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
- 2,129
1
vote
1
answer
126
views
Do these sorts of submonoids go by a particular name?
Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows:
$$r(x)=\{y\in M:xy=x\}$$
$$l(x)=\{y\in M:yx=x\}$$
Do these sorts of sub-monoids go by a particular name?...
- 111
1
vote
1
answer
134
views
amalgamated sum of monoids
Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids.
Why does $v:= v_i \circ u_i$...
- 65
1
vote
1
answer
96
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
- 10.5k
1
vote
1
answer
68
views
Under what assumptions can endomorphisms of $M/IM$ be realized as a subquotient of endomorphisms of $M$?
Suppose we have an algebra $A$ (unital, associative), with an ideal $I \leq A$ and a finitely generated module $M$ over $A$.
It is possible to obtain both $\mathrm{End}_A(M)$ and $\mathrm{End}_A(M/IM)...
- 33
1
vote
1
answer
158
views
Graded category O for for rational Cherednik algebras, but at t=0
The paper [1] introduced the category $\mathcal{O}$ for rational Cherednik algebras $H_{t,c}(W)$. This construction is tailored for the $t=1$ case (equivalently, the $t\neq 0$ case). The general setup ...
- 1,066
1
vote
1
answer
274
views
Jacobi-Zariski exact sequence question
Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ $...
- 5,405
1
vote
1
answer
266
views
$Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$
I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
$$\mathbb{Z}^2\rtimes_n\mathbb{Z}=\...
- 1,528