All Questions
6,056 questions
4
votes
0
answers
149
views
Coherence of the I-adic completion of a local ring of a formal scheme
Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...
- 119
2
votes
1
answer
167
views
DCC on the powers of ideals
My apologies if this question is below the level of MO. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring. $R$ is called strongly $\pi$-...
- 5,407
7
votes
0
answers
138
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
- 42k
29
votes
5
answers
9k
views
Local complete intersections which are not complete intersections
The following definitions are standard:
An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X_1,\ldots, X_n]$. ...
- 2,821
0
votes
1
answer
265
views
Quiver representations over any commutative ring
I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this ...
- 356
4
votes
0
answers
216
views
Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
2
votes
1
answer
627
views
Rank 2 vector bundle with trivial first chern class is self-dual
I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...
- 631
42
votes
5
answers
4k
views
What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
- 10.5k
21
votes
4
answers
4k
views
Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?
I am looking for a proof of the following fact:
If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules.
(Apparently such rings $...
- 2,821
11
votes
1
answer
1k
views
A local ring not a quotient of a regular local ring
In his book Commutative Ring Theory, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional.
What is an example of a ...
- 2,821
3
votes
1
answer
257
views
Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $L(R)$ ...
- 14.6k
4
votes
0
answers
265
views
Completion of ring as direct limit
If $X$ is a variety and $x \in X$, there are several ways to look locally around the point $x$:
Localisation: taking the direct limit over open immersions around $x$.
Henselisation: taking the direct ...
- 635
6
votes
1
answer
135
views
Automorphisms of special egg-box diagrams
By a egg-box diagram I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (...
- 18.7k
0
votes
0
answers
132
views
Which consequences can be deduced from this peculiar property of tetration?
Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
- 1,451
0
votes
1
answer
127
views
Software to compute generators of a module over polynomial ring
Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.
Consider the subset
$$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}...
- 708
1
vote
1
answer
91
views
Duogenic quartic rings
Recall that a commutative, unital ring $R$ of finite rank which is isomorphic to $\mathbb{Z}^n$ for some $n \geq 1$ as $\mathbb{Z}$-module is said to be monogenic if there exists an element $\gamma \...
- 12.9k
5
votes
0
answers
210
views
Dependence of completion on the base ring
Let $M$ be a module over an $E_\infty$ ring, $A$. Let $I$ be an $A$-non unital commutative algebra together with an associative map $I \wedge_A M \to M$.
Define ${_A}(M/I^n)$ as the cofiber of $I^{\...
- 579
4
votes
1
answer
333
views
Flat essential ring extensions
We call a ring extension (where $R$ and $S$ are commutative) $R \subset S$ essential if for every ideal $I$ of $S$ we have
that $I \cap S \neq 0 \implies I \cap R \neq 0$.
Suppose now that $R \subset ...
- 1,388
3
votes
1
answer
147
views
Bounded torsion of quotients of affine formal models
$\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
2
votes
0
answers
75
views
Smoothness of homomorphisms between graded algebras
Let $A$ be a finitely generated $\mathbb{C}$-algebra. Let $S^{\bullet}=\bigoplus_{i\geq 0} S^i$ and $T^{\bullet}=\bigoplus_{i\geq 0} T^i$ be two graded $A$-algebras such that $S^0=T^0=A$ and $S^1, T^1$...
- 565
6
votes
0
answers
194
views
"Cluster algebra" structure for finite distributive lattices
Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).
For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
- 24.2k
3
votes
2
answers
172
views
On the Hilbert function of a numerical semigroup
Recall that a numerical semigroup $S$ is a submonoid of the non-negative integers $\mathbb Z_{\geq 0}$ whose relative complement $\mathbb Z_{\geq 0} \setminus S$ is finite. Observe that the collection ...
- 183
0
votes
0
answers
162
views
Finite-exponent abelian groups
Let $G$ be an abelian group and $G=\bigoplus_{i=1}^t{{\Bbb{Z}}_{p_i}^{n_i}}^{(\Lambda_i)}$ where each $\Lambda_i$ is a set (at least one of $\Lambda_i$ is infinite). Since $G_{\Bbb{Z}}$ is a finite-...
- 12.9k
1
vote
0
answers
186
views
How to calculate the periodic cyclic homology group of $\overline{\mathbb{Z}}/\mathbb{Z}$
$\newcommand{\ur}{\mathrm{ur}}$Fix a prime number $p$. We let $\overline{\mathbb{Z}}$ denote the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Z}}_p$ denote the ...
7
votes
2
answers
2k
views
Ideals in the ring of single-variable Laurent polynomials with integer coefficients
I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...
- 1,446
1
vote
0
answers
77
views
Subrings of rings of integers of quartic fields having prime index and a specific property
Let $L$ be a $S_4$ or $A_4$ quartic field, and $\mathcal{O}_L$ its ring of integers. Let $K$ be its cubic resolvent field, which necessarily has the same discriminant as $L$. If $\mathcal{O}_L$ has a ...
- 26.9k
2
votes
0
answers
101
views
Regularity before and after completion - reference request
Put $R=\mathbb{Z}[x_1,\dotsc,x_n]$ and $I=(x_1,\dotsc,x_n)$. Let $M$ be an $R$-module that is probably not finitely generated. Suppose that the sequence $x_1,\dotsc,x_n$ is regular on $M$; I believe ...
- 56.9k
5
votes
1
answer
179
views
An example where the non-Archimedean tensor product of normed modules is only seminormed?
Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies
$$ | 0_R| = ...
- 4,132
1
vote
0
answers
173
views
The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
3
votes
1
answer
267
views
Singularities of contractions of extremal faces
Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-...
- 2,427
5
votes
2
answers
236
views
An example of a local integral domain with special spectrum
I am looking for a local integral domain $(D, m)$ with $Spec(D)=\{0,m\}\cup\{ P_i\}_i$ such that $P_i's$ are incomparable (that is, $P_i\not\subseteq P_j$ and $P_j\not\subseteq P_i$ for $i\not= j$) ...
- 1,388
2
votes
1
answer
94
views
What are the n-ary subsemigroups of $\mathbb{N}$?
There is a well-known result about the subsemigroups of $\mathbb{N}$ stating that the additive subsemigroup generated by a (finite) set $A$ of $\mathbb{N}$ is cofinite in $\mathbb{N}$ if and only if $\...
- 63.9k
11
votes
1
answer
915
views
Triviality of vector bundles on affine open subsets of affine space
Let $k$ be a field. By the Quillen-Suslin theorem, all vector bundles on $\mathbb{A}^n_k$ are trivial for all $n \geq 0$. If $U \subset \mathbb{A}^n_k$ is an affine open subset, then vector bundles ...
- 23k
19
votes
3
answers
2k
views
Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
- 10.5k
1
vote
0
answers
89
views
Combinatorial models of the refined inverse Eulerian numbers
If I evaluate substitution of an infinite set of indeterminates $(c_1,c_2,c_3,\cdots)$ into the infinite set of refined Eulerian polynomials $[E]$ of OEIS A145271, I obtain the Taylor series ...
- 10.5k
0
votes
0
answers
72
views
countable direct sum of cyclic abelian $p^{2}$ groups
Let $G={{\Bbb{Z}}_{p^{2}}}^{(\aleph)}$ (countable direct sum of copies of ${\Bbb{Z}}_{p^2}$). It is clear that every subgroup of $G$ is a homomorphic image of $G$. Now this is my question:
Is it true ...
- 56.9k
5
votes
1
answer
664
views
Eudoxus real numbers
I recently remembered the eudoxus construction of the real numbers.
Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction?
Clearification: The usual ...
- 2,624
5
votes
0
answers
204
views
What are all of the topological (commutative) monoid structures on a closed interval?
Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.
Question 1: What are all of the topological commutative ...
- 64k
1
vote
0
answers
54
views
Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
...
- 2,605
1
vote
0
answers
66
views
Counting commutative rngs (rings without identity)
A037289 counts the number of commutative rngs (rings without identity). It is complete up to 31, that is, the number of commutative rngs with 32 elements is not known. Is this in the literature?
...
- 9,114
0
votes
0
answers
39
views
Countably infinite monoids with minimal right ideals
Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
- 237
10
votes
2
answers
1k
views
When is every submodule pure?
Recall that a module is called
semisimple if every submodule is a direct summand
pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure ...
- 50.6k
0
votes
0
answers
180
views
Proof of Co-H map the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$
How to show the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$ is Co-H-map?
45
votes
5
answers
4k
views
How to think about CM rings?
There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me....
- 12.9k
0
votes
1
answer
216
views
A Newton identity and the primes--the Faber partition polynomials and modular arithmetic
[Edit, July 6, 2022: Removed erroneous characterization of Faber polynomials as an Appell sequence.]
Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I ...
- 10.5k
41
votes
4
answers
2k
views
What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 7,545
27
votes
5
answers
14k
views
Flat module and torsion-free module
All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. ...
- 63.9k
9
votes
2
answers
790
views
Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
14
votes
0
answers
432
views
Surprisingly only real points on intersection of certains quadrics
Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by
\begin{align}
X_e &= 0\\
X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\
X_g &...
- 22.5k
7
votes
4
answers
798
views
A lost lemma about periodicity in a grid of long exact sequences?
This is a question about finding references and hopefully a larger
context for a lemma in homological algebra I proved recently.
The motivation is to understand properties of characteristic
classes of ...
- 9,263