All Questions
6,053 questions
26
votes
1
answer
4k
views
Underlying structure behind the infamous IMO 1988 Problem 6
This is the infamous Problem 6 from the 1988 IMO which has recently been popularised by the YouTube channel Numberphile:
Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{...
user117786
26
votes
2
answers
2k
views
Uniqueness of the "algebraic closure" of a commutative ring
There are several ways to generalize the notion of "algebraic closure" from fields to arbitrary commutative rings. A good overview is On algebraic closures by R. Raphael. I am more ...
- 63.1k
26
votes
1
answer
2k
views
Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?
I am working on some combinatorics problems. One of my problems leads to the following question:
Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \...
- 431
25
votes
4
answers
4k
views
Serre's theorem about regularity and homological dimension
One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional.
I'd like to ask a somewhat ...
- 47.3k
25
votes
5
answers
2k
views
Exotic principal ideal domains
Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that ...
25
votes
5
answers
3k
views
is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?
This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
- 1,162
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
25
votes
6
answers
2k
views
What does the semiring of ideals of a ring R tell us about R?
Here is something I've wondered about since I was an undergraduate. Let $R$ be a ring (commutative, let's say, although the generalization to noncommutative rings is obvious). Ideals of $R$ can be ...
- 82.7k
25
votes
3
answers
1k
views
What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?
This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
- 48.9k
25
votes
3
answers
2k
views
product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 5,163
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
2
answers
1k
views
The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 41.9k
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
5
answers
6k
views
To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field?
In his answer to a question about simple proofs of the
Nullstellensatz
(Elementary / Interesting proofs of the Nullstellensatz),
Qiaochu Yuan referred to a really simple proof for the case of an
...
- 1,411
24
votes
3
answers
3k
views
Non-abelian Grothendieck group
By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
- 63.1k
24
votes
4
answers
4k
views
Is there a Galois correspondence for ring extensions?
Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
- 1,415
24
votes
2
answers
4k
views
To what extent can fields be classified?
The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...
- 3,761
24
votes
3
answers
4k
views
Commutative algebra with a view toward algebraic _number theory_
Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...
24
votes
3
answers
3k
views
Automorphisms of a weighted projective space
What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},...,a_{n})$ ?
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any ...
- 8,998
24
votes
2
answers
2k
views
Can one prove the elementary divisor theorem for PIDs by elementary matrix operations?
The elementary divisor theorem was originally proved by a calculation on integer matrices, using elementary (invertible) row and column operations to put the matrix into Smith normal form. That is ...
- 11.1k
24
votes
2
answers
3k
views
Who named it the Snake Lemma?
What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.
- 2,550
24
votes
3
answers
3k
views
Origin of the term "localization" for the localization of a ring
I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...
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
24
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,099
24
votes
2
answers
3k
views
Discriminant and Different
First some context. In most algebraic number theory textbooks, the notion of
discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their ...
- 26k
24
votes
1
answer
4k
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Minimal number of generators of a homogeneous ideal (exercise in Hartshorne)
In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
Show that a strict complete intersection is a set theoretic complete intersection.
Here are ...
- 14.7k
23
votes
6
answers
7k
views
Noether's normalization lemma over a ring A
Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, ...
- 1,320
23
votes
2
answers
3k
views
Criteria for irreducibility of polynomial
If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
- 669
23
votes
3
answers
2k
views
What are some toy models for the stable homotopy groups of spheres?
The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...
- 63.9k
23
votes
6
answers
4k
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Is projectiveness a Zariski-local property of modules? (Answered: Yes!)
I know that for a finitely presented $A$-module $M$ ($A$ a commutative ring), TFAE:
$M$ is projective;
$M$ is max-locally free, meaning that $M_{\mathfrak m}$ is free for every maximal ideal $\...
- 11.3k
23
votes
3
answers
6k
views
Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 425
23
votes
6
answers
2k
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Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
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
2
answers
910
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 $D(I^n)\leq 1....
- 7,836
23
votes
1
answer
966
views
Do DG-algebras have any sensible notion of integral closure?
Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
- 52.6k
23
votes
1
answer
2k
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Examples of Noetherian overkill
I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...
- 10.5k
23
votes
0
answers
682
views
CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
- 17.7k
22
votes
6
answers
8k
views
A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite
I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
- 14.3k
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
8
answers
5k
views
Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
- 897
22
votes
4
answers
2k
views
Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
- 65.4k
22
votes
2
answers
977
views
Mapping from a finite index subgroup onto the whole group
Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...
- 1,437
22
votes
3
answers
2k
views
Discriminant of characteristic polynomial as sum of squares
The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity.
Therefore the discriminant $D(H)$ of this polynomial is zero or positive.
It is ...
- 8,086
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
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
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
2
answers
1k
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Toposes (topoi) as classifying toposes of groupoids
A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
- 38.6k
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
1
answer
948
views
Alternating forms on abelian groups
Let $G$ and $H$ be abelian groups. By an alternating form, I mean a bilinear function $A\colon G\times G\to H$ such that $A(x,x)=0$ for all $x\in G$.
Question. If $A\colon G\times G\to H$ is an ...
- 27.2k