All Questions
6,057 questions
4
votes
1
answer
158
views
Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...
- 642
1
vote
0
answers
107
views
Topologically finitely generated non-abelian isomorphic absolute Galois groups
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are topologically finitely generated, non-abelian and ...
- 55
3
votes
1
answer
184
views
Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
- 2,051
1
vote
2
answers
132
views
Characteristic zero field with unique regular Noetherian dense unital subring
Let $F$ be a field. Call a unital subring $R\subset F$ dense if there is no subfield $K\subsetneq F$ such that $R\subset K$.
Is there a characteristic zero field $F$ such that the only regular ...
- 2,928
52
votes
1
answer
1k
views
Is there a notion of polynomial ring in "one half variable"?
Let $C$ be the category of commutative rings.
Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ?
(Here, we may assume those isomorphisms to be natural ...
- 64k
8
votes
1
answer
319
views
An invariance property of rational singularities
Let $X$ be a normal variety over a field of characteristic zero with rational singularities.
If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational ...
- 661
2
votes
0
answers
74
views
Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.9k
1
vote
0
answers
139
views
Free monoids on posets
I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying
if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
- 12.5k
1
vote
0
answers
74
views
Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?
The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
- 10.1k
4
votes
0
answers
72
views
When is the submonoid preserving a subspace finitely generated?
Let $T$ be a topological space with at least one open set whose closure is not open.
Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace.
Under what ...
- 51
19
votes
2
answers
1k
views
"Formally unramified iff trivial Kähler differentials" using only universal properties?
For convenience we work with commutative rings instead of commutative algebras.
Fix a commutative ring $R$. Consider the functor $\mathsf{Mod}\longrightarrow \mathsf{CRing}$ defined by taking an $R$-...
- 136
2
votes
1
answer
194
views
On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$
Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}_R(k, R)\cong k$, and so
$\text{Hom}_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes_R R^{\oplus n} , \forall n \ge ...
- 149k
1
vote
1
answer
235
views
Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
- 16.1k
2
votes
2
answers
2k
views
Projective modules over semi-local rings
Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.
- 1,313
9
votes
1
answer
1k
views
Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 12.9k
2
votes
1
answer
296
views
When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?
By a result of Klein-Nagata rings of the form $A_Q=K[x_1,...,x_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form.
Question 1: When is $A_Q$ a principal ...
- 156k
3
votes
0
answers
50
views
Rings with terminating division chains of a given length
Let $R$ be an integral domain. Given $a,b\in R$, then a division chain for $(a,b)$ is a sequence where we take $r_{-1}=a$, $r_0=b$, and for each $n>0$ we take $r_n=r_{n-1}s_n+r_{n-2}$ for some $...
- 18.7k
1
vote
1
answer
264
views
Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
- 2,837
4
votes
0
answers
158
views
Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?
Motivated by this question, it seems natural to ask the following:
Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
- 63.9k
0
votes
0
answers
177
views
Passing over $O_K \otimes_{\mathbb{Z}} A$ from $O_K$, how it affects the rank of a module?
This question was asked in MSE as well.
Let $K$ be a finite extension of the rationals $\mathbb{Q}$ with $O_K$ its the ring of integers.
Consider a $\mathbb{Z}$-algebra $A$ such that $|A|<\infty$.
...
- 930
3
votes
1
answer
776
views
On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
CommunityBot
- 1
9
votes
0
answers
441
views
Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)
I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
- 639
8
votes
2
answers
2k
views
The Mumford-Tate conjecture
The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...
- 66.3k
4
votes
1
answer
459
views
A similar construction to Ext, can we describe it better and does it have any use?
Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow ...
- 4,709
5
votes
1
answer
299
views
The universal multiset for a finite scheme - reference request
If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma_n$, which we can call the universal multiset for $X$.
Now let $X$ ...
- 8,972
3
votes
0
answers
542
views
Is the category of quasi-coherent sheaves not a topos?
There are two parts to my question:
Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?
My reasoning is thus:
It ...
- 318
9
votes
3
answers
2k
views
Is every additive, left exact functor isomorphic to a hom functor?
Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{...
- 4,709
6
votes
2
answers
781
views
What is the combinatorial data classifying non-normal affine toric varieties?
Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
- 21
17
votes
1
answer
754
views
Principal ideal domains with finitely many units
Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units?
Can such rings be classified?
(This is a more specialised version of the question in ...
- 156k
1
vote
0
answers
51
views
Betti numbers of a polynomial ideal after homogenization
Let $I \subset k[x_1,\ldots,x_n]$ be an ideal in a polinomial ring over a field $k$. Assume that $I$ is quasihomogeneous (that is, $I$ is not homogeneous with the usual grading, but it is homogeneous ...
3
votes
1
answer
238
views
Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS
I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...
- 12.4k
1
vote
1
answer
324
views
F-splitting and F-purity from commutative algebra viewpoint
First I define two terms:
Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...
- 30.5k
1
vote
1
answer
147
views
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...
- 2,928
5
votes
1
answer
292
views
Is Koszul homology of a monomial ideal always generated by the "obvious" things?
Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form
$$r e_{...
- 30.5k
0
votes
1
answer
248
views
Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called
We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?
- 4,991
8
votes
1
answer
265
views
Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
- 411
3
votes
1
answer
178
views
On the degree of the Hilbert polynomial of a graded module over the Rees algebra
If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
- 661
2
votes
0
answers
93
views
Can we write the isomorphism $F_i \otimes_R k \cong H_i (K_\bullet \otimes R/I)$ explicitly?
Let $R$ be a regular local ring (or polynomial ring over a field). Let $I$ be an ideal of $R$ and $F_\bullet$ a minimal free resolution of $R/I$. Let $K_\bullet$ denote the Koszul complex resolving ...
- 553
3
votes
1
answer
156
views
The cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term?
I recently studied the structure of the AR quiver of Dynkin type $\mathbb{A}_n/I$,
$\mathbb{D}_n/I$ with $I$ any admissable ideal, and found that the cokernel of an irreducible monomorphism is always ...
CommunityBot
- 1
4
votes
0
answers
169
views
Integral domains with finitely many units
Question: Is there a classification of (noetherian if needed) integral domains with finitely many units ? (of course we can exclude fields as trivial examples)
Probably there are many such domains ...
- 26.5k
2
votes
1
answer
218
views
Does a local ring with a separably closed residue field admit a universal homeomorphism from a local ring with algebraically closed residue field?
Let $A$ be a local ring with separably closed residue field. Under what conditions does there exist a local ring $B$ with an algebraically closed residue field and a local homomorphism $A\rightarrow B$...
- 28.7k
2
votes
1
answer
343
views
Quotient of ideal generated by regular sequence is a perfect module
I have recently started reading Bruns-Herzog's 'Cohen Macaulay rings' and this is problem 1.4.27 in it.
We say that a module $M$ over a Noetherian ring $R$ is perfect if the projective dimension of $M$...
- 661
1
vote
0
answers
80
views
Normal simple ring extensions
Let $R$ be a $k$-algebra, $k$ is an algebraically closed field of characteristic zero.
Assume that $R$ is an integral domain or a UFD (being a UFD makes things easier),
$a$ is an algebraic element ...
- 2,837
8
votes
0
answers
337
views
Passing to torsion of an exact sequence
If
$$
\Theta\colon\quad 0\to A\to B\to C\to 0
$$
is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$
0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\...
- 13k
0
votes
0
answers
195
views
Nice small resolution and normality of blow-up
Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is ...
3
votes
1
answer
553
views
Lax monoidal functor
Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal ...
- 15.9k
7
votes
1
answer
767
views
Is the Euler–Mascheroni constant an EL-number?
This question is based on Chow - What is a closed-form number?.
The author of the linked paper had proposed a plausible definition of "elementary numbers" (which he calls "EL-numbers&...
- 4,829
18
votes
1
answer
783
views
Are there any "simple" monoids with intermediate growth?
The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
- 1,947
2
votes
1
answer
195
views
Length of a module and Frobenius map
Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I ...
- 4,713
2
votes
0
answers
221
views
Field whose absolute Galois group is $\mathbb{Z}_p$
Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups.
Is there a ...
- 63.9k