All Questions
669 questions
13
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When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
13
votes
5
answers
3k
views
Example of a projective module which is not a direct sum of f.g. submodules?
This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...
- 65.4k
13
votes
1
answer
947
views
When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?
Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
- 11.9k
13
votes
1
answer
228
views
Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 85.6k
13
votes
2
answers
2k
views
Basics of classification of trilinear forms (when is it non-discrete)
Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,\dotsc,n_1$, $j=1,,n_2$, $k=1,\dotsc n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices.
...
- 2,232
13
votes
1
answer
7k
views
Chinese Remainder Theorem for rings: why not for modules?
This is a followup to Analog to the Chinese Remainder Theorem in groups other than Z_n. . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese ...
- 33.8k
12
votes
1
answer
989
views
Is an irreducible ideal in $R$ also irreducible in $R[x]$?
Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
- 1,961
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
12
votes
3
answers
3k
views
A remark on Cohen's theorem
It is well known as Cohen's theorem that a commutative ring is Noetherian if all its
prime ideals are finitely generated. Is this statement true or false when prime ideals are replaced by maximal ...
- 1,207
12
votes
1
answer
544
views
Square of primary ideals
Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
- 131
12
votes
1
answer
744
views
Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 38.6k
12
votes
1
answer
266
views
PID expressed as finite union of subrings
There is a classical theorem that no field can be expressed as finite union of proper subfields.
In contrast, there is an example of an integral domain that can be expressed as finite union of proper ...
- 365
12
votes
1
answer
574
views
Commutative algebras whose bidual is commutative
Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Call $D(A) := \mathrm{Hom}_k(A,k)$ the dual of $A$ as a $k$-module, and $DD(A) := \mathrm{Hom}_k(D(A),k)$ the dual of the latter. Let ...
- 32.5k
12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.3k
12
votes
2
answers
1k
views
Cohen-Macaulay domain with non-Cohen-Macaulay normalization?
Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
- 10.7k
11
votes
3
answers
1k
views
The concept "conjugate class" in monoids.
Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
- 6,201
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 ...
- 6,123
11
votes
1
answer
4k
views
henselization and completion
This might not be a question appropriate for this forum, I apologize in this case...
Is it true that any element of the completion of a valued ring $R$ that is algebraic over the field of fractions of ...
- 111
11
votes
3
answers
972
views
Is Krull dimension non-increasing along ring epimorphisms?
Let $f \colon R \to S$ be an epimorphism of commutative rings, where $R$ and $S$ are integral domains. Suppose that $\mathfrak{p} \subset S$ is a prime such that $f^{-1}(\mathfrak{p}) = 0$. Does it ...
- 7,338
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 ...
- 1,209
11
votes
2
answers
574
views
Identifying a group without 2-torsion
Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there ...
- 18.7k
11
votes
2
answers
2k
views
Is a scheme Noetherian if its topological space and its stalks are?
Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
- 564
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.
- 4,074
11
votes
1
answer
2k
views
geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings
Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...
- 1,355
11
votes
1
answer
4k
views
Does completion commute with localization?
Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
- 2,857
11
votes
2
answers
1k
views
Valuations on tensor products
Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equivalent valuations on ...
- 111
11
votes
1
answer
498
views
Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?
$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$Question 1: What is an example of a sequence $(X_\alpha)_{\alpha<\kappa}$ of ...
- 64k
11
votes
6
answers
33k
views
Are submodules of free modules free? [closed]
Are all submodules of free modules free? I would like a reference to a proof or counterexample please.
- 637
11
votes
3
answers
942
views
What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
10
votes
1
answer
388
views
Lifting $G$-invariants from characteristic $p\gg 0$ to characteristic 0 for a reductive algebraic group $G$
Let $S\subset \mathbb{C}$ be a finitely generated ring, let $R$ be a finitely generated commutative ring over $S$. Let $G$ be a linear algebraic group over $S$, such that $G_{\mathbb{C}}$ is reductive....
- 103
10
votes
2
answers
716
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 613
10
votes
1
answer
440
views
Reference for a generalization of Γ-spaces to monoidal model categories
Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
- 37.8k
10
votes
1
answer
316
views
Commutative algebras with modules of small complexity
Let $A$ be a finite dimensional commutative algebra. We can assume that it is local.
Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (...
- 26.5k
10
votes
2
answers
610
views
When is tensoring with a module representable by a scheme?
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...
- 5,548
10
votes
5
answers
1k
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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,...
- 10.5k
10
votes
1
answer
673
views
Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?
Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
- 2,506
10
votes
1
answer
2k
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Who invented Monoid?
I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...
- 203
10
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1
answer
1k
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Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence
I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
Formally etale iff $L_{B/A}$ (this denotes the cotangent ...
- 25.6k
10
votes
1
answer
818
views
Is $k(\!(x,y)\!)$ a topological field?
More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...
- 19.6k
10
votes
3
answers
2k
views
Irreducible/prime/indivisible elements
in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is irreducible if $\...
- 183
10
votes
2
answers
3k
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Generalization of finitely generated, finitely presented modules?
Let $R$ be a commutative ring and $M$ an $R$-module.
The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$.
Similarly, $M$ is finitely presented iff there is an ...
- 523
10
votes
1
answer
409
views
Does every set have a rigid self-map?
The question was asked on Mathematics Stackexchange
but has remained unanswered so far.
A self-map is a map $f:X\to X$ from a set $X$ to itself. There is an obvious notion of morphism, and thus of ...
- 4,416
10
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0
answers
454
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
10
votes
0
answers
1k
views
What is the etale fundamental group of Spec Z((x))?
I know the etale fundamental group of $\mathbb{Z}$ is trivial. For algebraically closed fields $K$, the etale fundamental group of $K((x))$ is $\hat{\mathbb{Z}}$, since all covers in this case are ...
- 10.7k
10
votes
3
answers
756
views
Degree of generators of irreducible components
Let $V$ be a Zariski-closed subset of $\mathbb{A}^n_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is ...
- 4,132
10
votes
3
answers
1k
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What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
user178596
10
votes
1
answer
439
views
Irreducibility of a class of polynomials
This question is directly inspired by this question. Consider polynomials of the form
$$p(x) = \prod_{i=1}^n(x-i)^2 - d.$$ For which values of $n$ and $d$ is $p(x)$ irreducible? There is a theorem of ...
- 96.4k
10
votes
2
answers
1k
views
Formal completion of the normal bundle
Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...
- 1,545
10
votes
0
answers
561
views
Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
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