All Questions

6
votes
2answers
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
1answer
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
2answers
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
1answer
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
7answers
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
3answers
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
0answers
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
2answers
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
1answer
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
1answer
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
3answers
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
4answers
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
3answers
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
2answers
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
2answers
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
3answers
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
3answers
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
6answers
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
2answers
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
1answer
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
2answers
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
3answers
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
3answers
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
1answer
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
3answers
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
4answers
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
3answers
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
1answer
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
5answers
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
1answer
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
3answers
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
2answers
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
2answers
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
33answers
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
6answers
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
2answers
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
4answers
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
5answers
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
2answers
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
1answer
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
3answers
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
1answer
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
2answers
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
78answers
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
13answers
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
1answer
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
1answer
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
2answers
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
5answers
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
4answers
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 ...

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