All Questions
6,057 questions
1
vote
0
answers
83
views
What is known about the algebraic completion of a monoid?
It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid:
Let $W$ be a monoid and let $p(x)=q(...
- 12.9k
1
vote
0
answers
109
views
Integer factorization given modular square root of 2
Let $N$ be composite. It is well-known that if $x^2 \equiv 1 \pmod N$, and $x \neq \pm 1 \pmod N$, then a factor of $N$ is easily found by computing gcd($N$, $x + 1$). I'm curious if there is a ...
- 1,703
0
votes
0
answers
110
views
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
7
votes
0
answers
235
views
Brauer group of the Henselization
Let $R$ be a Noetherian local ring and let $R^h$ be its Henselization. What can we say about the kernel and range of the map
$$
\operatorname{Br}(R) \rightarrow \operatorname{Br}(R^h)?
$$
Are there ...
- 12.9k
5
votes
2
answers
409
views
Extending monoids to a ring
I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid ...
- 38.6k
1
vote
2
answers
406
views
When splitting of short exact sequence preserves the kernels
This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...
- 35.2k
1
vote
0
answers
231
views
Has an "algebraic manifold" been defined before? Are there any non-trivial examples?
Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds:
for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $...
- 63.9k
1
vote
1
answer
282
views
Number of cluster variables
In the paper Hernandez and Leclerc - Cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster ...
- 12.9k
0
votes
0
answers
250
views
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] ...
- 63.9k
4
votes
1
answer
704
views
Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)
Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
- 10.5k
5
votes
2
answers
811
views
Cancellation property for commutative monoid
Let $(M,+,e)$ be a commutative monoid with unit $e$. An element $a\in M$ is called cancellative element if
for any $b,c \in M$ such that $a+b=a+c$ implies that $b=c$.
Let $(\mathbf{N},+,0)$ the ...
- 63.9k
1
vote
0
answers
102
views
What is the real name for the initial object in the category of "monoid-valued measures of intervals" on transitive relations?
(I'm not asking for a true/false answer; I have a true algebraic fact and I'm looking for a reference in the literature. By the way, there is a version of this theorem that replaces monoid with $R$-...
1
vote
0
answers
52
views
Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?
The terms are defined in a related question. [1]
Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
2
votes
0
answers
729
views
On Serre's "Local fields"
While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
- 63.9k
1
vote
1
answer
234
views
Krull dimension and elimination theory over the integers
Let $K:=\mathbb{C}$, and let $R:=K[x_1,\dots , x_n]$.
Then, a system of polynomial equations $p_1=0, p_2=0, \dots , p_r = 0$, where the $p_i$ are polynomials in the $x_j$, has finitely many solutions $...
- 311
3
votes
0
answers
53
views
Continuous differentiations of functional algebras
Let $A$ be some algebra (infinite-dimensional) of analytic functions on $\mathbb{C}^n$, and $D$ be some derivation of $A$, i.e. $D(fg)=Df \cdot g + f \cdot Dg)$ (so A may be considered as a ...
- 4,713
4
votes
1
answer
169
views
Is every invertible-free cancellative monoid action represented by "shifting" certain maps?
[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments]
Let $W,X$ be ...
1
vote
2
answers
256
views
Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$
I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its ...
- 10.8k
3
votes
1
answer
161
views
Are cofibrations in topological monoids preserved by forming the product with the identity?
Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
CommunityBot
- 1
0
votes
2
answers
285
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
0
votes
0
answers
65
views
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
12
votes
2
answers
820
views
Size of largest square divisor of a random integer
Let $x$ be an integer picked uniformly at random from $1 \ldots N$. Write $x = r^2 t$ where $t$ is square-free. How does the expected value of $r$ scale with $N$? Is anything known about the variance ...
- 151k
0
votes
1
answer
364
views
Koszul complex of $xy$, $yz$ and $xz$
Has anyone computed the homology of the sequence $xy$, $yz$ and $xz$ in $\mathbb{C}[x,y,z]$?
CommunityBot
- 1
1
vote
0
answers
94
views
Irreducibility of $\frac{x^{n+1}-(n+1) x+n}{(x-1)^2}$ [duplicate]
The question is motivated by this question.
Consider the polynomials
$$\dfrac{x^{n+1}-(n+1) x+n}{(x-1)^2} = \displaystyle \sum _{k=0}^n (n-k) x^k, n=1,2,3,\dots,$$
Are they all irreducible (over $\...
- 63.9k
8
votes
2
answers
2k
views
Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
Given two rings $R_1$ and $R_2$ (with or without identity: it's not specified). If $R_1[x]$ is isomorphic to $R_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant ...
- 63.9k
5
votes
1
answer
126
views
Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
- 2,356
2
votes
1
answer
1k
views
Pushout and pullback in the category of rings
Consider the following pushout diagram
$\require{AMScd}$
\begin{CD}
A @>f>> B\\
@V g V V @VV V\\
C @>> > D
\end{CD}
in the category of $\textbf{Rings}$ where $f,g$ both are flat ...
- 4,709
2
votes
1
answer
244
views
If a morphism from a commutative absolutely flat ring has integral fibers, does it induce an embedding of spectra?
All rings are commutative and unital.
Let $A$ be an absolutely flat ring and $A \rightarrow B$ a ring monomorphism with integral fibers (i.e. for each $\mathfrak{p} \in \operatorname{Spec}(A), B \...
- 1,388
6
votes
2
answers
2k
views
Online video of some courses
Who knows online video of Riemannian Geometry and Commutative Algebra? If you know, please recommend them to me. I am really eager to learn these courses.
- 30.3k
1
vote
0
answers
64
views
Continous morphisms of a local field with conditions in positive characteristic
Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
- 3,065
13
votes
1
answer
228
views
Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 7,875
1
vote
0
answers
71
views
terminology for a kind of two-sided module over a monoid
If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are ...
- 12.5k
4
votes
0
answers
265
views
Sections of smooth morphisms over henselian rings
Let $(A,\mathfrak m)$ be a henselian local ring. Let $R$ and $S$ be $A$-algebras of finite type and $f\colon R\to S$ be a smooth morphism. Assume that the induced morphism $R/\mathfrak m R\to S/\...
- 1,590
0
votes
0
answers
137
views
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 ...
- 63.9k
9
votes
1
answer
780
views
Is every field the residue field of a discretely valued field of characteristic 0?
Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?
- 9,263
3
votes
1
answer
607
views
Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$
I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
CommunityBot
- 1
1
vote
1
answer
158
views
Semigroups admitting commutative group actions
Let $(S,*)$ be a semigroup admitting a distinguished element $0$ such that $z*s = s*z = z$, for all $s \in S$. Moreover, let $(\mathbb{G},\cdot)$ be a commutative group. Consider an action
$$
\mathbb{...
- 63.9k
5
votes
1
answer
428
views
Analytic functions in arbitrary rings?
We have developed a rich theory of analytic functions over $\mathbb{R}^n$ and $\mathbb{C}^n$. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to ...
- 2,519
8
votes
1
answer
857
views
What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 321
10
votes
1
answer
422
views
Generalized cancelation properties ensuring a monoid embeds into a group
Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules:
$$xy=xz \quad\Longrightarrow y=z;$$
$$yx=zx \quad\...
- 38.6k
1
vote
0
answers
71
views
Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,230
7
votes
2
answers
544
views
A linearly orderable monoid which does not embed into a linearly orderable group
It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...
- 63.9k
5
votes
1
answer
598
views
Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
- 1,832
5
votes
2
answers
770
views
Integrally closed factor rings and projective modules
I have a weird vision that comes from reading a paper by Raphael and Desrochers..
Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is ...
- 938
2
votes
1
answer
168
views
Adding first generator to Cohen-Macaulay monomial ideal
Let $I$ be a Cohen-Macaulay monomial ideal of $R=K[x_1,...,x_n]$, where $K$ is a field. Can we say the ideal $(x_1)+I$ is Cohen Macaulay?
- 63.9k
1
vote
1
answer
299
views
Cohen-Macaulay monomial ideal
Let $R=K[x_1,...,x_n]$ be the polynomial ring over a field $K$ and $I[x_1,...,x_n]=(u_1,...,u_t)$ be a Cohen-Macaulay monomial ideal of $R$. If $m<n$, could we say that $I[x_1,...,x_m,0,0,...,0]$ ...
- 1,313
7
votes
1
answer
430
views
Is the Pierce spectrum useful elsewhere in Mathematics?
In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in ...
- 2,369
1
vote
0
answers
83
views
Characteristic of ring completions
This may be a completely trivial question, but I haven’t seen it stated in any of the references I checked. Is the characteristic of a ring $R$ equal to that of its completions? This is true for the ...
- 11
3
votes
0
answers
107
views
Do Frobenius algebras have a lattice basis and what lattices do appear?
Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients).
A (commutative) ...
- 63.9k
30
votes
2
answers
2k
views
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
- 7,893