All Questions
612 questions
27
votes
5
answers
14k
views
Flat module and torsion-free module
All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. ...
- 951
27
votes
2
answers
2k
views
Is every commutative ring a limit of noetherian rings?
Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...
- 4,416
27
votes
5
answers
3k
views
Algebraic description of compact smooth manifolds?
Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
- 6,546
26
votes
5
answers
3k
views
Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?
The question
Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...
- 27.7k
26
votes
2
answers
4k
views
Why are injective modules more complicated than projective modules?
For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...
- 2,040
26
votes
4
answers
3k
views
Is a domain all of whose localizations are noetherian itself noetherian ?
Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ?
The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory:
Let ...
- 558
25
votes
7
answers
3k
views
When can we prove constructively that a ring with unity has a maximal ideal?
Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
- 118k
25
votes
1
answer
5k
views
The Rabinowitz Trick
The recent question about problems which are solved by generalizations got me thinking about the Rabinowitz trick, which is used to prove a statement of Hilbert's Nullstellensatz, specifically, the ...
- 640
24
votes
6
answers
5k
views
Pythagorean 5-tuples
What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...
- 1,488
24
votes
3
answers
1k
views
Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$?
Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-...
- 7,527
23
votes
1
answer
3k
views
Modules and Square Zero Extensions
Let $R$ be a commutative ring, $RMod$ its category of modules and $CRing$ the category of commutative rings.
There's an embedding $RMod \rightarrow CRing/R$ that sends an $R$-module $M$ to the ring ...
- 1,484
23
votes
3
answers
4k
views
What are the units in the ring of Laurent polynomials?
What are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for ...
- 323
23
votes
1
answer
2k
views
Examples of Noetherian overkill
I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...
- 10.5k
23
votes
2
answers
909
views
How slowly can a power of an ideal grow?
For a polynomial ideal $I\subset \mathbb{C}[x_1,x_2]$, let $D(I)$ be the smallest degree of any polynomial in $I$.
How slowly can $D(I^n)$ grow as a function of $n$? For example, if $D(I^n)\leq 1....
- 7,836
22
votes
2
answers
5k
views
affine open subset of affine scheme
Let $X=Spec(A)$ be an affine scheme and $U=Spec(R)$ be an affine open subset of $X$. Is it true that $R$ is an localization of $A$, i.e. $R=S^{-1}A$ for some closed multiplication subset $S\subset A$ ?...
- 427
22
votes
6
answers
6k
views
When is a blow-up non-singular?
Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-...
- 3,284
22
votes
3
answers
2k
views
Nice algebraic statements independent from ZF + V=L (constructibility)
Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z}...
user1437
22
votes
3
answers
3k
views
Which rings are subrings of matrix rings?
In this question, all rings are commutative with a $1$, unless we explicitly say
so, and all morphisms of rings send $1$ to $1$.
Let $A$ be a Noetherian local integral domain. Let $T$ be a non-zero $...
- 41.4k
22
votes
2
answers
2k
views
Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?
QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...
- 7,500
21
votes
2
answers
2k
views
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
2
answers
2k
views
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
2
answers
1k
views
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
4
answers
4k
views
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
1
answer
2k
views
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
1
answer
584
views
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
2k
views
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
2
answers
3k
views
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
votes
6
answers
3k
views
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
20
votes
3
answers
2k
views
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
1
answer
2k
views
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
1
answer
2k
views
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
5
answers
2k
views
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
votes
1
answer
1k
views
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
3
answers
2k
views
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
19
votes
3
answers
3k
views
Total ring of fractions vs. Localization
Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a ...
- 63.1k
19
votes
3
answers
1k
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How to construct a constructive proof from a non-constructive proof using prime ideals?
The sum of two nilpotent elements of a commutative ring is nilpotent. This can be checked by a direct calculation using the binomial theorem. In fact, this calculation shows the stronger statement $x^...
- 5,482
19
votes
3
answers
4k
views
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
19
votes
5
answers
7k
views
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
742
views
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
19
votes
2
answers
5k
views
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
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 997
19
votes
2
answers
5k
views
Characterizations of UFD and Euclidean domain by ideal-theoretic conditions
This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ...
- 7,207
19
votes
2
answers
564
views
Ostrowski's Theorem for topological rings?
Ostrowski's theorem classifies all absolute values on a number field $K$.
Questions:
More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?
In ...
- 63.9k
18
votes
5
answers
8k
views
Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
- 17.9k
18
votes
4
answers
4k
views
Flatness of normalization
Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5).
What happens if we ...
- 3,704
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
18
votes
9
answers
2k
views
What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 118k
18
votes
3
answers
1k
views
The isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$
In a recent conversation with a colleague, the following question arose:
What is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$? That is to say, what is $...
- 323
18
votes
5
answers
2k
views
Is a complete homogeneous symmetric polynomial irreducible?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree }...
- 446
18
votes
3
answers
2k
views
Alternate proofs of Hilberts Basis Theorem
I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.
If $R$ is a noetherian ring, then so is $R[X]$.
or its sister version
If $R$ is a noetherian ...