All Questions
6,057 questions
2
votes
1
answer
148
views
Terminology for a ring where every right cancellable element is cancellable
Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
- 218
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
2
votes
2
answers
400
views
What fraction of polynomials with integer coefficients are indecomposable?
It is well-known that "most" integers are composite: the Prime Number Theorem tells us that only about $1/\log(N)$ of the integers in the interval $1 \ldots N$ are prime. For polynomials, ...
- 2,051
17
votes
3
answers
3k
views
Ghost components of a Witt vector - Motivation
I'd like to know if anyone has a good explanation for where the ghost components that are used to define Witt vectors come from. A lot of sources I've read take the ghost components for their ...
- 323
2
votes
3
answers
225
views
Conditions for exact projective limits for some Mittag-Leffler systems?
Let $(M_i)_{i\in I}$ and $(N_i)_{i\in I}$ be Mittag-Leffler systems of $R$-modules. I have a map $(h_i)$ of projective systems such that every $h_i$ is surjective. I search for conditions for $\lim \...
CommunityBot
- 1
3
votes
0
answers
351
views
When is a maximal Cohen-Macaulay module of finite homological dimension?
Let $R$ be a local Cohen-Macaulay Noetherian ring. A maximal Cohen-Macaulay module or mCM-module is an $R$-module $M$ of finite type such that $\text{dim }M = \text{depth }M =d$
A module $M$ is of ...
- 26k
4
votes
1
answer
558
views
derived tensor product and finite projective dimension
Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.
Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...
- 361
5
votes
1
answer
298
views
Endomorphisms of Artinian modules
The following claim is from a paper [On the moduli spaces of bundles on K3 surfaces, I, p. 358] of Mukai. Consider an artinian module $\mathrm{M}$ over a local ring, and let $\mathrm{M}_0$ be the ...
- 2,808
9
votes
1
answer
224
views
What is the largest subcategory $C$ of a module category over an Artin algebra, such that $C$ is Krull-Schmidt (and abelian)? Does $C$ exist?
Let $A$ be an Artin algebra, $\text{Mod}\,A$ the category of $A$-modules and $\text{mod}\,A$ the category of finitely generated $A$-modules. It is well-known that $\text{mod}\,A$ is a Krull-Schmidt ...
- 35.2k
4
votes
1
answer
322
views
Understanding full set of sections as in Katz-Mazur
I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...
- 63.9k
4
votes
1
answer
669
views
When every proper submodule of a free module is contained in a maximal submodule
Let $(R, m)$ be a commutative local ring (it is not Noetherian in general) and $F$ be a free $R$-module. Under what conditions every proper submodule of $F$ is contained in a maximal submodule.
- 38.6k
2
votes
0
answers
173
views
finite type k-subalgebras of k[[t]]
in the middle of working on some deformation related questions, I bumped into a question regarding finite type subalgebras of some completed algebras. I hope someone may know the answer, or may ...
- 937
3
votes
2
answers
324
views
An integral transform and the Stone-Weierstrass theorem
For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
- 17.2k
6
votes
0
answers
230
views
Gelfand ring in Bourbaki's exercises
In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...
- 3,738
3
votes
1
answer
149
views
How to compute the intersection of an ideal with the maximal order of a subfield?
I asked this earlier on math.stackexchange but I think this is a better place for this question.
Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
- 37k
7
votes
1
answer
400
views
Does perfect fraction field imply perfect residue field?
Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect?
Thoughts:
If $A$ is ...
- 28.2k
10
votes
2
answers
706
views
Set of primes $p$ such that $\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset$
For which sets of primes $P$ is there a finite type $\mathbb{Z}$-algebra $A$ such that$$p\in P\iff\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset?$$Do all the finite $P$ arise this way?
$A=\mathbb{Z}/n$ works ...
- 28.7k
3
votes
1
answer
230
views
Why is this $ \mathbb{G}_{a} $ bundle trivial
Please tell me why the following example of a principal $ \mathbb{G}_{a} $-bundle over an affine ring is trivial. Let $ \{x_{1},x_{2},x_{3}\} $ a basis of $ \mathbf{V}^{\ast} $, $ c_{1}(t),c_{2}(t) $ ...
- 782
5
votes
1
answer
145
views
Example request: seriously deficient homogeneous spaces
In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
- 309
13
votes
1
answer
347
views
Existence of a translation-invariant basis of $\ell^2$
This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural).
I give four equivalent formulations of the same ...
16
votes
5
answers
5k
views
When are dual modules free?
Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes_A K$ is a finite dimensional vector space $V$. Let $T^*$ be ...
- 35.5k
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
- 18.7k
3
votes
0
answers
134
views
Partial orders on $\mathbb{N}^m$ compatible with addition
I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $\mathbb{N}^{m}$ that are compatible with addition. That is, partial orders $\leq$ satisfying two additional ...
- 645
4
votes
1
answer
153
views
Are finitely presented algebras over VNRs projective?
Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module?
In the ...
- 66.3k
4
votes
0
answers
402
views
Is every Dedekind domain the integral closure of some principal ideal domain?
I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...
- 1,053
3
votes
1
answer
129
views
Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient
Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we ...
CommunityBot
- 1
17
votes
4
answers
4k
views
Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
- 2,607
5
votes
1
answer
500
views
Question about a proof in Berthelot's crystalline book
Below is an excerpt from Berthelot's book on crystalline cohomology. I don't understand the last sentence, namely why it follows that $\sigma\circ \varepsilon$ is an isomorphism. For what it's worth, $...
- 38k
4
votes
0
answers
104
views
Ascent for projective modules
Let $A \subseteq B \subseteq C$ be commutative rings such that: $(1)$ $B$ is a projective $A$-module $(2)$ $C$ is a finitely presented $B$-module and $(3)$ $C$ is a flat $A$-module.
Does it follow ...
- 938
1
vote
1
answer
291
views
Systems of regular parameters of a regular local ring
Let $R$ be a Noetherian regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$. Given two systems of regular parameters $\vec{u}=\left<u_1, \dots, u_n\right>$ and $\vec{v}=\left&...
- 1,313
0
votes
0
answers
74
views
Sufficient conditions for $b\not\in I^2$ given that $b\in I$
Let $I$ be an $R$-ideal in a commutative algebra $B$ over a commutative ring $R.$ Given $b\in I$ I want to prove that $b\not \in I^2$.
Are there any sufficient conditions for showing that $b\not\in I^...
- 1,615
3
votes
0
answers
95
views
Growth of dimension of a good filtration on finitely generated modules over polynomial rings
Let $S=K[x_1,\cdots,x_s]$ be a polynomial ring over a characteristic zero field $K$.
For a partition $\{1, \cdots, s\} = I_1 \sqcup \cdots \sqcup I_r$ of the variable index, let $\mathbb{x}_i^{\mathbb{...
- 31
1
vote
0
answers
121
views
Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
- 59
3
votes
1
answer
263
views
Cohomology of commutative monoid acting on module
I have a some naive questions about how to define the cohomology of a commutative monoid.
One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i_{\...
- 38.6k
1
vote
0
answers
241
views
Smooth normalization and blow-up of the exceptional locus
Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\...
- 1,463
2
votes
0
answers
95
views
Socle of a quotient of the ring of differential operators of a polynomial ring
I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...
- 81
6
votes
2
answers
533
views
Künneth formula for de Rham cohomology with respect to an integrable connection
I am reading through https://stacks.math.columbia.edu/tag/0FM9 which proves that for $X,Y$ schemes over some base $S$ and $X \times _S Y \overset{p}{\rightarrow} X$ resp. $X \times _S Y \overset{q}{\...
- 1,144
2
votes
0
answers
90
views
On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik
Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...
- 81
2
votes
0
answers
227
views
Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal
Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
- 81
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
- 33.8k
1
vote
0
answers
365
views
Flatness over a local noetherian ring
Let $(R,\mathfrak m)$ be a local noetherian ring, and $M$ an arbitrary $R$-module. Suppose that $\mathrm{Tor}_1(M,R/\mathfrak m)=0$. Does it follow that $M$ is flat?
The answer is positive when $M$ ...
- 1,313
-8
votes
1
answer
351
views
Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
- 38.6k
2
votes
2
answers
589
views
Graded-irreducible ideals are irreducible?
One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?
Let $I$ be a graded ideal in a polynomial ring over a field....
- 1,313
3
votes
2
answers
249
views
For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay?
Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&...
- 1,214
15
votes
4
answers
3k
views
Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?
Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more ...
- 50.6k
8
votes
2
answers
475
views
Non-isomorphic smooth affine varieties dominating each other
Can non-isomorphic smooth affine varieties dominate each other?
In the projective case one can take isogenous abelian varieties.
- 7,722
3
votes
1
answer
404
views
How many monoids with $n$ arrows exist?
How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
- 38.6k
0
votes
0
answers
173
views
Can the notion of algebraic closedness be generalized to the rings with zero divisors?
Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero?
For instance, it seems that any polynomial of non-zero-divisor-...
- 63.9k
11
votes
3
answers
887
views
Are the rings $\mathbb{C}[X]/\langle X^2- c \mathrm{Tr}(X) X \rangle$ isomorphic when $c$ ranges over a neighborhood of 0?
Let $X=(X_{IJ})_{I,J=1,\ldots,N}$ be a family of $N^2$ indeterminates and consider the ring $$
R_{N,c}=\mathbb{C}[X] / J_c,\quad J_c=\langle X^2 -c \mathrm{Tr}(X) X \rangle .
$$
Here the notation ...
- 7,300