All Questions
6,053 questions
3
votes
0
answers
181
views
Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
8
votes
1
answer
322
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
6
votes
3
answers
551
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 1,000
1
vote
0
answers
47
views
Examples of graded subrings of $\mathbb Q(T)$
The following question came up in some discussion on some very unrelated matters.
A graded algebra $A$ is an algebra $A$ with a decomposition $A = \oplus_{i \in \mathbb Z} A_i$ such that $A_i A_j \...
- 2,029
-2
votes
1
answer
77
views
integral ring extension implies algebraicity of their fraction fields extension?
$\DeclareMathOperator\Fr{Fr}$There is something I don't get about the following :
Start with the well known fact that if $A\subset B\subset C$ are rings with $B$ the integral closure of $A$ in $C$, ...
- 1,031
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
5
votes
0
answers
216
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
1
answer
223
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
1
vote
0
answers
110
views
How large can the Krull dimension of the Rees algebra be?
Let $d$ be a natural number. How large can the Krull dimension of the Rees algebra $A[It]$ be, where $A$ is a commutative ring of Krull dimension $d$, and $I$ is an ideal of $A$.
Currently, I know the ...
- 76
3
votes
0
answers
250
views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 321
3
votes
0
answers
91
views
Quillen-Suslin theorem for non-strict polydiscs in the sense of Berkovich
Let $K$ be a complete non-archimedean field of mixed characteristic $(0,p)$. Choose $\rho_1,\dots,\rho_n\in \mathbb{R}_{>0}$ and let $P$ be a finite projective module over $K\langle\rho_1^{-1}t_1,\...
- 63
2
votes
1
answer
327
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
11
votes
1
answer
513
views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
7
votes
0
answers
202
views
Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.1k
5
votes
0
answers
128
views
What are the conditions for the dual of the exterior algebra to be isomorphic to the exterior algebra of the dual?
The exterior algebra $\Lambda^*_kM$ can be defined for a $k$-module $M$, where $k$ is a commutative ring. A number of sources mention, without condition or proof, a (canonical) isomorphism $$(\Lambda^*...
- 171
4
votes
0
answers
176
views
What is the equivalent of Artin gluing for quasicoherent sheaves?
Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \...
- 15.9k
3
votes
0
answers
120
views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
3
votes
0
answers
95
views
History of the notion of integral ring extension?
[I asked that question in "history of science and mathematics" but didn't get any answer so I take my chance here. I hope this is not out of context]
Can anyone give me references about the ...
- 1,031
6
votes
1
answer
307
views
Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 985
3
votes
2
answers
255
views
Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
If we consider any differential graded algebra $A^\bullet$, then its homology is a graded algebra, since the tensor product interacts well with homology.
A sufficient condidtion for the homology to be ...
- 4,418
4
votes
1
answer
132
views
Zero dimensional complete intersection ring of length a power of $p$
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C$ be a $k$-algebra of finite dimension over $k$ such that $k[C^p]=k$. Under these hypothesis it is known by results of L. ...
- 657
1
vote
0
answers
207
views
When is the derived category of a ring generated by injective modules
Are there any equivalent conditions on a ring to the condition that the localizing subcategory of $D(R)$ generated by injective modules is the entire category? Are there any non-examples in Boolean ...
- 2,356
5
votes
0
answers
176
views
Example of a Boolean Ring with infinite injective dimension over itself
It is known that Boolean rings can have infinite global dimension (free Boolean algebra on a large enough number of generators) [ see The Global Dimension of Boolean Rings by Pierce]. Are there any ...
- 2,356
1
vote
0
answers
136
views
Local cohomology and image of $1$ under the canonical map from Ext to local cohomology
Let $R$ be a commutative Noetherian local ring, and $S$ be an $R$-algebra. Let $x_1,\dots,x_t$ be elements, in the maximal ideal of $R$, which is a regular sequence on both $R$ and $S$, and let $I$ be ...
- 412
3
votes
2
answers
271
views
Orbits under the automorphism group of projective space
Let $\mathbb{P}^d_K$ be projective space of dimension $d\geq 1$ over an infinite field $K$. Let $x\in\mathbb{P}^d_K$ with $\dim\overline{\lbrace x\rbrace}=n\leq d-1$.
My question: is the set $\lbrace ...
- 133
5
votes
1
answer
264
views
Non-existence of power divided structure on a maximal ideal of truncated polynomial rings (example from Koblitz)
In 3.3 of Berthelot-Ogus's book Notes on Crystalline Cohomology, an example from Koblitz is exhibited without proof.
Let $k$ be a field (or a commutative ring) with characteristic $p>0$, $$A:=k[x_1,...
- 441
2
votes
1
answer
159
views
Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
- 253
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
6
votes
0
answers
152
views
Can Harrison cohomology be written using Ext?
Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext'...
- 985
2
votes
1
answer
149
views
Finitely generated modules over completion
Let $k$ be a field, $A$ a finitely generated $k$-algebra and $I \subset A$ an ideal with $I$-adic completion $\hat{A} = \varprojlim A/I^n$. Is every finitely generated $\hat{A}$-module the completion ...
- 23
3
votes
1
answer
180
views
How to find equations of $\mathbb{C}^*$-curves
Fix positive integers $t_1,t_2,t_3$.
Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by
$$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{...
- 1,981
4
votes
1
answer
330
views
An assertion of Mahler
Let $\rho$ be an integer greater than $1$. In the article "Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen" https://link.springer.com/article/10.1007/...
- 3,998
5
votes
1
answer
151
views
Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 3,230
1
vote
0
answers
104
views
Is a normal domain a filtered colimit of Noetherian normal domains?
As described in the title, is any normal domain a filtered colimit of Noetherian normal domains? It will be great if one can show this, even with additional conditions, or if one can provide a ...
- 1,024
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
1
vote
0
answers
216
views
Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
0
votes
0
answers
114
views
Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
3
votes
0
answers
161
views
On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have
$$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$
So the ...
- 15.6k
3
votes
0
answers
92
views
Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
2
votes
1
answer
262
views
Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
- 341
5
votes
0
answers
211
views
On the natural map $\mathrm{Br}(R) \rightarrow \mathrm{Br}(S)$ of Brauer groups
$\DeclareMathOperator\Br{Br}$Let $R$ be a commutative ring, and let $\Br(R)$ be the Brauer group of $R$ as defined by Auslander and Goldman. Let $S$ be a commutative $R$-algebra, and consider the ...
- 181
3
votes
3
answers
387
views
Basic question about completion of local ring
Let $(A,m)$ be a Noetherian local ring and $\tilde{A}$ its completion by the maximal ideal $m$.
Are the following three statements true?
(i)
If $\tilde{A}$ is free over $A$, then $A\cong \tilde{A}$
(...
- 328
3
votes
0
answers
181
views
Conditions for an open mapping between spectra
Let $(A,\mathfrak{m})$ be a Locally Noether Ring, and $\hat{A} = \varprojlim A/ \mathfrak{m}^{n}$ .Furthermore, let $f : A \to \hat{A}$ be a canonical morphism, and consider the mapping $f^{*} : Spec(\...
- 41
9
votes
1
answer
326
views
Nonzero module with vanishing derived fibers
What's an example of a nonzero $R$-module with vanishing derived fibers at all points of $\mathrm{Spec}(R)$?
This was asked in When does a quasicoherent sheaf vanish?
but the answer there only says ...
- 2,356
1
vote
1
answer
67
views
An example of a commutative ring $R$ which has a proper right ideal which is not a right SIP $R$-module
Recall that a module $M_R$ ($R$ is a ring with unity) is called SIP if the intersection of any two summands of $M$ is also a summand. I asked before if there exists a commutative ring which is not an ...
- 324
2
votes
1
answer
232
views
Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 23
2
votes
1
answer
181
views
Idempotent algebras over absolutely flat ring
Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...
- 2,356