All Questions
6,053 questions
21
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Using a known result without a specific reference
This is a question of mathematical writing. Let me know if it would be better suited to academia.SE.
I am writing a paper in invariant theory. It uses some slightly heavy commutative algebra. There ...
21
votes
4
answers
2k
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The first female algebraist in US/Britain?
Recently I dug up some biographical details of Lindsay Burch, of Hilbert-Burch Theorem fame, whose few papers have had quite an impact on commutative algebra. This made me curious about the first ...
- 30.5k
21
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4
answers
4k
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Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?
I am looking for a proof of the following fact:
If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules.
(Apparently such rings $...
- 3,380
21
votes
2
answers
2k
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What is the dimension of the product ring $\prod \mathbb Z/2^n\mathbb Z$ ?
In an anwswer to a question on our sister site here I mentioned that a reduced commutative ring $R$ has zero Krull dimension if and only if it is von Neumann regular i.e. if and only if for any $r\...
- 47.5k
21
votes
6
answers
3k
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A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
- 30.5k
21
votes
1
answer
2k
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Does formally etale imply flat for noetherian schemes?
This is a followup to an earlier question I asked: Does formally etale imply flat? After some remarks I received on MO I noticed that this was answered to the negative by an answer to an earlier ...
- 705
21
votes
2
answers
3k
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Standard reduction to the artinian local case?
Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold?
...
- 411
21
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2
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3k
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Integer matrices with no integer eigenvalues
Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
- 1,045
21
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3
answers
3k
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Prime ideals in the ring of germs of continuous functions
We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.
Can anyone think of a single ...
- 13.5k
21
votes
5
answers
1k
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Computation of fraction field of formal series over the integers
What is the fraction field $K$ of the domain $\mathbb Z[[X]]$?
It is strictly smaller than the field of Laurent series $L=\operatorname {Frac}\mathbb Q[[X]]$, since $\sum_{i\geq 0}\frac {X^i}{i!}\in ...
- 47.5k
21
votes
1
answer
1k
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A Krull-like Theorem and its possible equivalence to AC
A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is ...
- 528
21
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2
answers
1k
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If a polynomial ring is finite free over a subring, is the subring polynomial?
Let $R = k[x_1, \ldots, x_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x_i = 1$) such that $R$ is a free $S$-module of ...
- 156k
21
votes
1
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When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 27.8k
21
votes
2
answers
2k
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Stability of real polynomials with positive coefficients
Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive.
For $f$ a ...
user2529
21
votes
5
answers
6k
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Fast computation of a Groebner basis. What is possible?
I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
- 1,256
21
votes
2
answers
1k
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What properties define open loci in excellent schemes?
Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open ...
- 30.5k
21
votes
1
answer
759
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Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure
Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
- 13.6k
21
votes
1
answer
584
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Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors
For all primes up to $p=89$ there exists a product $Q=\prod_{j=1}^d(x-a_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a_j$ in $\mathbb F_p[x]$ such that $Q'$ has all its roots in $\mathbb ...
- 17.9k
21
votes
1
answer
638
views
Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 1,543
21
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1
answer
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Two conjectures by Gabber on Brauer and Picard groups
In a paper I need to make reference to two conjectures by Gabber, from
Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37
...
- 30.5k
21
votes
1
answer
757
views
Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?
Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
user111524
21
votes
4
answers
5k
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The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
- 1,881
20
votes
4
answers
9k
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For which $c$ is $\mathbb{Z}[\sqrt{c}]$ a unique factorization domain? a Euclidean domain?
Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$
which form a subring of the ...
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20
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5
answers
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Constructively, is the unit of the “free abelian group” monad on sets injective?
Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...
- 21.3k
20
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1
answer
2k
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Joyal's construction of the spectrum of a commutative ring
I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well.
I know this is a lot to ask, but basically, I ...
- 10.5k
20
votes
10
answers
7k
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Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
20
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4
answers
4k
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What is interesting/useful about Castelnuovo-Mumford regularity?
What is interesting/useful about Castelnuovo-Mumford regularity?
- 10.5k
20
votes
2
answers
4k
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Ideals of the ring of smooth functions
The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
- 101
20
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3
answers
2k
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Simple example of a ring which is normal but not CM
I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
- 156k
20
votes
2
answers
15k
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Maximal Ideals in the ring k[x1,...,xn ]
Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field, are of the form $(x_1 - ...
20
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2
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Is there a choice-free proof that a Euclidean domain is a UFD?
I asked this question about a week ago on math.SE, without any answers. My motivation is pedagogical, but maybe the question comes closer to research-level than I thought.
The proof (at least the ...
- 11.4k
20
votes
2
answers
683
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non-isomorphic stably isomorphic fields
Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)?
Q2: Do we have a sufficient criterion for ...
- 7,609
20
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2
answers
1k
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Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups
A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...
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1
answer
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How is a descent datum the same as a comodule structure?
For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\...
- 10.4k
20
votes
2
answers
1k
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a determinantal identity
Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...
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20
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3
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Is every integral epimorphism of commutative rings surjective?
That's the question. Recall that a morphism $f\colon A\to B$ of commutative rings is integral if every element in $B$ is the root of a monic polynomial with coefficients in the image of $A$ and that $...
- 9,408
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1
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Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?
Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
- 7,893
20
votes
1
answer
518
views
Concept associated to the Eudoxus reals
I am aware of three different constructions of the field of real numbers :
The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the ...
- 201
20
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1
answer
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On a theorem of Jacobson
In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...
- 47.3k
20
votes
1
answer
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Tropical homological algebra
Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
- 6,229
19
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2
answers
5k
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Are morphisms from affine schemes to arbitrary schemes affine morphisms?
To put this question in precise language, let $X$ be an affine scheme, and $Y$ be an arbitrary scheme, and $f : X \rightarrow Y$ a morphism from $X$ to $Y$. Does it follow that $f$ is an affine ...
- 387
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4
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What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
- 1,961
19
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5
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7k
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When a formal power series is a rational function in disguise
Given a formal power series $f \in k[[X]]$, where $k$ is a commutative field, is there any good way to tell whether or not $f\in k(X)$?
Edit: To clarify, "good way to tell" means "computable ...
19
votes
2
answers
836
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Does the rational power series ring $\mathbb{Q}[[X]]$ embed as a ring into the field of real numbers?
The title says it all. I'm wondering if the power series ring $\mathbb{Q}[[X]]$ (with rational coefficients) embeds as a ring into the field of real numbers. There are various topologies one might ...
- 209
19
votes
6
answers
2k
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Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
- 65.4k
19
votes
4
answers
6k
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A $0$-dimensional ring that is not noetherian
$\DeclareMathOperator{\Spec}{Spec}$
[Edit] Martin pointed out that $\dim A = 0$ does not imply that $\Spec A$ is discrete. Therefore I changed the wording of question 2.[/Edit]
With dimension of a ...
- 5,504
19
votes
3
answers
794
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What is the shortest polynomial divisible by $(x-1)(y-1)(x^2y-1)$
I am interested in polynomials with few terms ("short polynomials", "fewnomials") in ideals. A simple to state question is
Given an ideal $I\subset k[x_1,\dots,x_n]$, what is the shortest polynomial ...
- 1,961
19
votes
2
answers
742
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Do all subtraction-free identities tropicalize?
If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+...
- 19.7k
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votes
1
answer
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Examples of solid abelian groups
I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying ...
- 455
19
votes
3
answers
4k
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Generalized Euler phi function
Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\...
- 337