All Questions
6,056 questions
2
votes
0
answers
64
views
A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
14
votes
2
answers
3k
views
Maximal ideal and Zorn's lemma
It is known that any nonzero ring A (say commutative with 1) has a maximal ideal. The proof uses Zorn's lemma.
Now I heard some people saying that if we assume A to be noetherian, then we don't need ...
- 50.6k
8
votes
1
answer
626
views
Example of a connected finite group scheme which is not solvable
What would be an example of a connected finite group scheme over a field $k$ that is not solvable? Here $k$ is algebraically closed.
Let $\operatorname{GL}_n$ be the general linear group scheme over ...
- 12.9k
0
votes
0
answers
267
views
completion and tensor product
Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...
- 3,472
0
votes
0
answers
133
views
Example of a periodic free resolution over a hypersurface
I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION,
WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud
I'm wondering what would be a nice example illustrating Theorem 6....
- 839
8
votes
2
answers
2k
views
What generalizes symmetric polynomials to other finite groups?
Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
- 12.9k
2
votes
0
answers
44
views
Derivation of positively graded domain
We know that if $f\in k[X_1,{...},X_n]$ is quasi-homogeneous polynomial and $R :=k[X_1,{...},X_n]/(f)$, then any minimal generating set of $\operatorname{Der}_k(R)$ contains the Euler derivation. Is ...
- 63.9k
5
votes
0
answers
219
views
Cyclic homology can be recovered from topological cyclic homology?
Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type.
By an equivalence of ring spectra
$$
\operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR,
$$
...
- 12.9k
7
votes
1
answer
1k
views
How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?
(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...
- 42.9k
0
votes
1
answer
144
views
Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 63.9k
13
votes
1
answer
868
views
Is an ordinary scheme in Borger's Absolute Geometry the same as a "scheme over 𝔽₁" with a map to Spec(ℤ)?
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}_1}
\newcommand{\spec}{\operatorname{Spec}}$If I understand correctly, in Borger's paper $\Lambda$-rings and the field with one element about the ...
- 4,713
3
votes
0
answers
157
views
Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
- 85
8
votes
1
answer
448
views
Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
- 42k
4
votes
3
answers
904
views
Examples of (non-normal) unibranched rings?
For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...
- 253
3
votes
0
answers
235
views
Frobenius and mixed characteristic valuation rings
Let $R$ be an $\mathbf{F}_p$-algebra. Kunz's theorem says that if $R$ is Noetherian, then the Frobenius of $R$ is flat iff $R$ is regular. Following the philosophy that valuation rings often behave ...
- 12.9k
1
vote
2
answers
194
views
When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product?
In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le n$ imply $\prod_{1\le i\le n} r_i\mid r$ (when $r_i$ are fixed)?
Does there exist any criterion for this implication that ...
- 6,262
4
votes
1
answer
435
views
Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?
Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?
...
- 869
4
votes
0
answers
108
views
Quantum version of Kostant's basis of ℤ-form of U(𝔤)
Kostant showed that the subring of $\mathcal U(\mathfrak{sl}_2)$ generated by the divided powers $e^c/c!$ and $f^a/a!$ has a $\mathbb Z$-basis given by the elements $\frac{f^a}{a!}\binom hb \frac{e^c}{...
- 12.9k
4
votes
1
answer
224
views
For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements
Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...
- 4,111
4
votes
2
answers
586
views
Brauer group of $\mathbb{Z}_{(p)}$
This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
- 4,713
7
votes
1
answer
351
views
Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?
Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...
- 6,262
3
votes
1
answer
378
views
Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
$\...
- 25.8k
12
votes
1
answer
417
views
Are algebras of smooth functions formally smooth?
Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
(This is not a joke ...
- 10.8k
4
votes
1
answer
2k
views
Localisation of two rings which is an integral extension, then integral extension still holds?
Question seems simple, but I just can't find the solution.
Let A/B be an integral ring extension and let P be a prime ideal of B. By going-up theorem, there is Q, a prime ideal of A, lying over P. ...
- 173
1
vote
1
answer
106
views
On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings
Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
- 6,262
1
vote
1
answer
97
views
Projectivity of the fundamental ideal of Witt groups
Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a ...
- 1,766
10
votes
1
answer
826
views
Triangulations of polytopes and tilings of zonotopes
Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...
- 2,821
2
votes
0
answers
196
views
Commutative local rings which satisfy Krull-Remak-Schmidt
Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...
- 26.5k
10
votes
1
answer
3k
views
Latest "A Term of Commutative Algebra" by Altman and Kleiman? [closed]
Where can I find the latest revision of A term of Commutative Algebra by Allen B. ALTMAN and Steven L. KLEIMAN? Is my 2013 version ok?
It is hard to locate the latest one; many old revisions and ...
- 89
3
votes
0
answers
82
views
Example of a secondary representation of a module that is not a direct sum
Let $A$ be a commutative ring. An $A$-module $M$ is said to be
secondary if $M\neq 0$ and for each $a\in A $, the endomorphism
$\phi_a:M\to M$ defined by $\phi_a(m)=am$ for $m\in M$ is either
...
- 12.9k
2
votes
1
answer
237
views
Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?
Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{...
- 23
6
votes
2
answers
226
views
For which fields $k$ with $k^\times$ not $p$-divisible, does there exist finite $l/k$ such that $l^\times$ is $p$-divisible?
Is there a prime $p$ and a field $k$, not real closed, with $k^\times$ not $p$-divisible, such that there exists a finite extension $l/k$ such that $l^\times$ is $p$-divisible?
This question came up ...
- 337
1
vote
0
answers
192
views
Smooth surjective morphism to integral scheme
Suppose that $f : X \rightarrow Y$ is a smooth (or even étale) surjective morphism over a field $k$ to a scheme $Y$ of finite type over $k$.
I want to show that $X$ is locally integral, i.e. (in the ...
- 635
3
votes
0
answers
76
views
The existence of an idempotent in some special semigroups
Problem. Does a semigroup $S$ have an idempotent, if there exist elements $b\in S$ and $a_1,\dots,a_n\in S$ such that $b\in \bigcup_{i=1}^na_ixSxa_i$ for every $x\in S$?
What is the answer to this ...
- 63.9k
6
votes
1
answer
182
views
Commutative Frobenius algebra with non-invertible window element, but not square zero
For any commutative Frobenius algebra $A$ there is an associated window element $\omega \in A$. If $\mu: A \otimes A \to A$ denotes the multiplication, $1 \in A$ the unit, $b: A \otimes A \to k$ the ...
- 26.5k
10
votes
5
answers
1k
views
On the notion of partial semigroup
A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M,...
- 305
1
vote
0
answers
275
views
Does analytic isomorphism imply local isomorphism?
If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
- 912
3
votes
1
answer
447
views
About a corollary of the Briançon-Skoda theorem
The following is a corollary of the Briançon-Skoda theorem:
If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f_1,f_2,...,f_{d+1}\in R$. Then, $f_1^df_2^d...f_{d+1}^d \in (f_1^{d+1},f_2^...
- 2,821
13
votes
1
answer
637
views
Can a Dedekind domain have a power basis over a ring that isn't a Dedekind domain?
This aim of this question is to determine whether there exists a proof (or some counterexample) to the following statement : "If $R$ is a subring of Dedekind domain $S$, such that $S$ has a power ...
- 4,713
1
vote
0
answers
66
views
Existence of a minimal ideal with a specific property
Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...
- 329
1
vote
0
answers
121
views
Situations where extension of contraction is well-behaved?
It is known that one cannot expect the extension of the contraction of an ideal under a ring homomorphism $f:A\to B$ to be the original ideal, except in very special scenarios like surjections or ...
- 11
1
vote
0
answers
151
views
$K_1(k[x]/(x^2))$ for a field $k$
$\DeclareMathOperator\GL{GL}$The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $\GL(R)$, where $\GL(R)$ is the union of the sequence $R^{ \times} = \GL_1(R)...
- 63.9k
3
votes
1
answer
229
views
A pexiderization of the sine addition law on semigroups
Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
- 12.9k
8
votes
0
answers
185
views
Ring of invariants for graph automorphism
$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite simple graph with nodes numbered $1$ to $n$. Attach variables $x_1, ..., x_n$ to nodes. The graph automorphism group $\Aut G$ acts on nodes by ...
- 645
8
votes
1
answer
339
views
On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 28.2k
1
vote
0
answers
70
views
Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 883
11
votes
1
answer
722
views
A closed formula for $A_n(X)=\sum\limits_{i=0}^n X^{i^2}$
I want to know if there exists a closed formula for sum $A_n(X)=\sum \limits_{i=0}^n X^{i^2}$.
I have found if $n$ is odd then $(X^n+1)\text{ | } A_n(X)$, but I don't have found a closed formula.
- 6,367
13
votes
2
answers
446
views
Tensor product of finite type UFD algebras over an algebraically closed field is again UFD?
Let $K$ be an algebraically closed field, $A$ and $B$ two finite type $K$-algebras which are assumed to be UFD. Is $A \otimes_K B$ again a UFD?
This question has been already asked here and here, but ...
- 28.7k
3
votes
0
answers
182
views
Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?
Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the ``Euclidean'' plane $F^2$ endowed with the ...
- 63.9k
3
votes
1
answer
421
views
What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)
I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
- 11.2k