# All Questions

100,865 questions

**6**

votes

**2**answers

522 views

### Naive Z/2-spectrum structure on E smash E?

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...

**7**

votes

**1**answer

2k views

### finite surjective l.c.i morphism is flat

Let $X,Y$ be locally Noetherian schemes. Let $f:X\to Y$ be a finite, surjective, and locally complete intersection morphism, i.e., locally it can be decomposed as regular immersion followed by a ...

**14**

votes

**2**answers

536 views

### When is a commutative ring the limit of its local rings?

Let $A$ be a commutative ring. Then we get local rings $A_p$ by localizing at each prime ideal $p$. Moreover, we get $A_p \rightarrow A_q$ when $p$ contains $q$. So we get a big diagram indexed by the ...

**6**

votes

**1**answer

533 views

### Some examples of depth

This is related to the question I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.
I am still not quite comfortable with the concept of depth, and ...

**15**

votes

**7**answers

846 views

### Extremal question on matrices

The following question was posed to me a while ago. No one I know has a given a satisfactory (or even a complete) proof:
Suppose that $M$ is an $n$ x $n$ matrix of non-negative integers. Additionally,...

**2**

votes

**3**answers

546 views

### Springer corresponding for nullcones other than the standard nilpotent cone

I understand the ordinary Springer correspondence gives a bijection between orbits in the nilpotent cone for the adjoint representation and irreducible representations of the Weyl group, through ...

**18**

votes

**0**answers

1k views

### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**0**

votes

**2**answers

290 views

### Paritioning a set of numbers A into two sets B,C so that abs(prod(B) - prod(C)) is minimal

Let A = $\{a_1,...,a_n\}$ be a set of numbers. We can assume all elements of A are integers.
Is there any efficient way to partition A into two sets B = $\{b_1,...,b_k\}$ and C = $\{c_1,...,c_l\}$ ...

**0**

votes

**1**answer

1k views

### Inversion of Laurent series

For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...

**15**

votes

**1**answer

1k views

### Is every finite-dimensional Lie algebra the Lie algebra of a closed linear Lie group?

This question is closely related to this one.
Ado's theorem states that given a finite-dimensional Lie algebra $\mathfrak g$, there exists a faithful representation $\rho\colon\mathfrak g \to \...

**0**

votes

**3**answers

701 views

### A website linking to most major math journals? [closed]

What's your usual action online in order to browse math journals? Like check Arxiv or MathSciNet. Any other good link directs you to most updated articles in major math journals. Or the traditional ...

**16**

votes

**4**answers

3k views

### Reference for Deligne's construction of Galois representations attached to modular forms

I was wondering if anyone can suggest some good reference for learning more about Deligne's construction of Galois representations attached to modular forms. I find Deligne's original paper hard to ...

**21**

votes

**3**answers

703 views

### A hypersurface with many points

Ok, it's time for me to ask my first question on MO.
Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...

**7**

votes

**2**answers

1k views

### When is every submodule pure?

Recall that a module is called
semisimple if every submodule is a direct summand
pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure ...

**17**

votes

**2**answers

1k views

### Are Jacobians principally polarized over non-algebraically closed fields?

How does one define the Torelli map $M_g \to A_g$ of moduli stacks? On geometric points a curve maps to its principally polarized Jacobian.
So what I am asking is: if I have a curve $C$ over a non-...

**16**

votes

**3**answers

2k views

### A riddle about zeros, ones and minus-ones

I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces ...

**16**

votes

**3**answers

2k views

### Algebraic varieties which are topological manifolds

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...

**5**

votes

**6**answers

3k views

### Differences between reflexives and projectives modules

Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make ...

**30**

votes

**2**answers

2k views

### Non-integral scheme having integral local rings

I can show that if $X$ is a scheme such that all local rings $\mathcal{O}_{X,x}$ are integral and such that the underlying topological space is connected and Noetherian, then $X$ is itself integral.
...

**6**

votes

**1**answer

3k views

### Tschirnhaus Transformation

Recently in my Intro to Proofs class, we've been talking about the fundamental theorem of algebra, which states that all polynomials of degree n always have n, not necessarily distinct, not ...

**22**

votes

**2**answers

2k views

### Is there a neat formula for the volume of a tetrahedron on $S^3$?

There is a nice formula for the area of a triangle on the 2-dimensional sphere;
If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we ...

**3**

votes

**3**answers

596 views

### Is (relatively) algebraically closed stable under finite field extensions?

Let $F\subset F'$ be a field extension such that $F$ is algebraically closed inside $F'$, i.e. if $x\in F'$ is algebraic over $F$ then $x$ belongs to $F$ itself.
Let now $F\subset L$ be a finite field ...

**12**

votes

**3**answers

2k views

### What are CR manifolds like?

The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR ...

**0**

votes

**1**answer

179 views

### Difference Equations & Possible Limits

The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...

**14**

votes

**3**answers

3k views

### Intuition about schemes over a fixed scheme

I am taking a first course on Algebraic Geometry, and I am a little confused at the intuition behind looking at schemes over a fixed scheme. Categorically, I have all the motivation in the world for ...

**6**

votes

**4**answers

2k views

### Choice of adviser

Not sure how to tag this one so feel free to edit and add tags.
When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that ...

**2**

votes

**3**answers

2k views

### Algebraic Varieties which are also Manifolds

Any non-singular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties - one would not expect varieties with singular ...

**1**

vote

**1**answer

600 views

### What do we know about the space of finite order distributions ?

Hi,
(Question updated)
My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ?
From in the information I gathered, it ...

**12**

votes

**5**answers

3k views

### Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension

Let $(A,m_A)$ and $(B,m_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}...

**14**

votes

**1**answer

897 views

### Tropical mathematics and enriched category theory

Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...

**12**

votes

**3**answers

711 views

### What is a monoidal metric space?

At time of writing, the highest rated answer to my question What is a metric space? is Tom Leinster's account of Lawvere's description of a metric space as an enriched category. This prompted my ...

**7**

votes

**2**answers

924 views

### Difference between Alexander polynomial and Blanchfield pairing

For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. ...

**14**

votes

**2**answers

1k views

### What is a projective space?

Is there a "recognition principle" for projective spaces?
What categories are there with projective spaces for objects?
Background: Although the title is a nod to What is a metric space?, this ...

**37**

votes

**33**answers

11k views

### What are the most overloaded words in mathematics? [closed]

This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or ...

**8**

votes

**6**answers

3k views

### Exact short sequences of vector spaces

If possible, how could one prove that every short exact sequence $0 \to A \xrightarrow f B \xrightarrow g C \to 0$ of vector spaces (here $A$, $B$ and $C$) splits without using any basis of $A$, $B$ ...

**30**

votes

**2**answers

8k views

### Explanation for the Thom-Pontryagin construction (and its generalisations)

In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...

**27**

votes

**4**answers

4k views

### The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...

**11**

votes

**5**answers

2k views

### Geometry Vs Arithmetic of schemes

Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind,
Where do we find some ...

**5**

votes

**2**answers

739 views

### What is $TC(\Sigma^\infty \Omega X)$?

I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and ...

**10**

votes

**1**answer

814 views

### Cyclic spaces and S^1-equivariant homotopy theory

I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...

**5**

votes

**3**answers

4k views

### photon propagator

I am reading Zee's book "QFT in a nutshell". I have a question on the photon propagator computation. For a massive photon, consider the Lagrangian
$L = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{...

**20**

votes

**1**answer

961 views

### Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...

**2**

votes

**2**answers

230 views

### morphisms from abelian varieties to rational curves.

Let $A$ be an abelian variety and and $\sigma$ an automorphism of $A$. Suppose $f:A\rightarrow P^1$ is a morphism. Is it true that $\sigma$ descends to an automorphism of $P^1$? I seem to remember ...

**155**

votes

**78**answers

32k views

### Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...

**14**

votes

**13**answers

15k views

### Math journal for high school students?

I recently discovered The College Mathematics Journal and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a ...

**3**

votes

**1**answer

903 views

### Interpolation of sequences by analytic functions

Given a sequence of complex numbers $a_n$ with $n\in\mathbb{N}$, is it possible to find an analytic (or meromorphic) function that interpolates this sequence in the sense that $f(n)=a_n$?
If this is ...

**12**

votes

**1**answer

1k views

### When is an Albanese variety principally polarized?

Let (X,x) be a pointed projective variety. Then there exists an abelian variety V which is universal for maps of pointed varieties $(X,x) \to (A,e_A)$, called the albanese variety. When X is a curve, ...

**4**

votes

**2**answers

329 views

### A graph connectivity problem (restated)

Given an undirected connected graph, our goal is to remove some edges to make the graph disconnected. The constraint is that each node of the graph can not lose more than $m$ edges incident to it. I ...

**34**

votes

**5**answers

2k views

### Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...

**21**

votes

**4**answers

5k views

### Etale cohomology and l-adic Tate modules

$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...