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https://mathoverflow.net/questions/10413 | 26 | Denote Zermelo Fraenkel set theory without choice by ZF. Is the following true: In ZF, every definable non empty class A has a definable member; i.e. for every class $A = \lbrace x : \phi(x)\rbrace$ for which ZF proves "A is non empty", there is a class $a = \lbrace x : \psi(x)\rbrace$ such that ZF proves "a belongs to... | https://mathoverflow.net/users/2689 | Definable collections without definable members (in ZF) | **Update.** (June, 2017) François Dorais and I have completed a paper that ultimately grew out of this questions and several follow-up questions.
>
> F. G. Dorais and J. D. Hamkins, [When does every definable nonempty set have a definable element?](http://jdh.hamkins.org/when-does-every-definable-nonempty-set-have... | 30 | https://mathoverflow.net/users/1946 | 10415 | 7,108 |
https://mathoverflow.net/questions/10404 | 13 | Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C *does* have all finite colimits, the suspension functor on C is an equivalence, and C is enriched in Spectra in a way I don't want to make too precise (basically the Hom functor Cop × C → Spaces f... | https://mathoverflow.net/users/126667 | Stable ∞-categories as spectral categories | According to Corollary 8.28 in DAG I a pointed $\infty$-category is stable iff it has finite colimits and the suspension functor is an equivalence.
| 8 | https://mathoverflow.net/users/1100 | 10434 | 7,124 |
https://mathoverflow.net/questions/10433 | 6 | Inspired by the two solutions to Harry's question
[Can a topos ever be an abelian category?](https://mathoverflow.net/questions/10290/can-a-topos-ever-be-an-abelian-category)
I was wondering whether all coproducts of 1 in a topos are distinct up to isomorphism? That is $1 + 1 + \dots + 1 \cong 1 + 1 + \dots + 1$ if... | https://mathoverflow.net/users/1106 | Are all coproducts of 1 in a topos distinct ? | At least if you're talking about *finite* coproducts, then the answer is yes. If $n\le m$, then we have a canonical inclusion $\sum\_{i=1}^n 1 \hookrightarrow \sum\_{j=1}^m 1$, which is in fact a complemented subobject with complement $\sum\_{k=1}^{m-n} 1$. If this inclusion is an isomorphism, then its complement is in... | 8 | https://mathoverflow.net/users/49 | 10443 | 7,132 |
https://mathoverflow.net/questions/10408 | 28 | This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ exists for all $a<t<b$. Then there exist numbers $a < t\_1 < t\_2 < \dots < t\_n < b$ such that the vectors $F'(t\_1),\dots... | https://mathoverflow.net/users/2912 | Rolle's theorem in n dimensions | Here is a solution in the $C^1$ case [but see upd]. Suppose the vectors $F'(t\_1),\ldots, F'(t\_n)$ are linearly independent for all $0\leq t\_1< \cdots < t\_n\leq 1$. Let $L(t\_1,\ldots,t\_{n-1})$ be vector space spanned by the first $n-1$ of these. Let $$t^i=(t\_1^i,\ldots,t\_{n-1}^i),0\leq t^i\_1< \cdots < t^i\_{n-1... | 16 | https://mathoverflow.net/users/2349 | 10445 | 7,133 |
https://mathoverflow.net/questions/10409 | 23 | Does anyone know of an introduction and motivation for [W-algebras](https://en.wikipedia.org/wiki/W-algebra)?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice algebraic/Lie group. It acts on its Lie-algebra by the adjoint action. Fix a nil... | https://mathoverflow.net/users/2837 | Introduction to W-Algebras/Why W-algebras? | W-algebras appear in at least three interrelated contexts.
1. **Integrable hierarchies**, as in the article by Leonid Dickey that mathphysicist mentions in his/her answer. Integrable PDEs like the KdV equation are bihamiltonian, meaning that the equations of motion can be written in hamiltonian form with respect to t... | 22 | https://mathoverflow.net/users/394 | 10448 | 7,136 |
https://mathoverflow.net/questions/10400 | 3 | Although the question is easy to pose, I think some background will help to motivate it, so I'll start with it.
Consider variables $X=(X\_1, \ldots, X\_n)$ over a field $K$ and the elementary symmetric functions $T=(T\_1, \ldots, T\_n)$ in $X$. In other words $X$ are the roots of the polynomial $Y^n + T\_1 Y^{n-1} + ... | https://mathoverflow.net/users/2042 | Explicit expression of an alternating polynomial in characteristic $2$? | (Here is a more detailed version of Felipe's answer.)
In 1976 Elwyn Berlekamp defined characteristic 2 analogues of the discriminant and its square root, related to the expressions you wrote down. Later these were related to what you get if you lift the original polynomial to characteristic 0, compute the usual discr... | 6 | https://mathoverflow.net/users/2757 | 10451 | 7,138 |
https://mathoverflow.net/questions/10453 | 11 | Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}\_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A\_p$, under the reduction map?
| https://mathoverflow.net/users/2938 | Torsion of an abelian variety under reduction. | Let me try again at an alternate answer. If $A\_{\mathbb{Z}\_p}$ is an abelian scheme of dimension $g$ and $\ell \neq p$ is a prime, then for any positive integer $n$, the isogeny $[\ell^n]: A \rightarrow A$ is an etale map. [If I am not mistaken, the proof of this does not require formal groups!] Since the special fib... | 11 | https://mathoverflow.net/users/1149 | 10462 | 7,144 |
https://mathoverflow.net/questions/7679 | 7 | Let $K$ be a number field, and let $S\_x$ denote the set of primes of norm at most $x$. Is it possible to find a smaller set of places $T\_x\subset S\_x$ so that a lot of the solutions of the $S\_x$-unit equation $a+b=1$ for $a,b\in S\_x$ are solutions of the $T\_x$-unit equation?
Here's a possible precise statement ... | https://mathoverflow.net/users/2046 | S-unit equation and small sets of places | I think that the precise statement you ask for is false, but I have a feeling that current bounds on number of solutions to $S$-unit equations are not good enough to prove this, even for $K=\mathbf{Q}$.
What I can prove is that it is false if you require $T\_x$ to be the subset of the specified size consisting of th... | 6 | https://mathoverflow.net/users/2757 | 10464 | 7,145 |
https://mathoverflow.net/questions/10436 | 5 | Is there a theory of Newton-Puiseux type expansions
which works to parameterize singular surface germs $F(x,y,z) =0$?
Ideally, each branch would be the image of map of the form the
$x = u^m, y = v^n, z = \Sigma\_{i + j > n} a\_{i j} u^i v^j$
(after a linear change of the variables $x, y, z$).
| https://mathoverflow.net/users/2906 | Newton Puiseux expansions for singular surfaces? | I don't believe there is anything as general as that, but when your polynomial is over the complex numbers one can do the following. First shift and rotate coordinates so that you're working on a neighborhood of the origin where $F(0,0,0) = 0$ and $\partial\_z^n F(0,0,0) = 0$ for some $n$. Then you can use the Weierstr... | 6 | https://mathoverflow.net/users/2944 | 10472 | 7,151 |
https://mathoverflow.net/questions/10116 | 7 | **Background**:
I am focusing on $G=GL\_{2}(\overline{\mathbb{F\_q}})$ here. If you wonder why I am interested in this, I am trying a problem relating to the Deligne-Lusztig varieties defined over local rings by Stasinksi, and this background theory is relevant there. The definition I am using is this:
1. The first ... | https://mathoverflow.net/users/2623 | Explicit computations of small Deligne-Lusztig varieties (e.g. Drinfeld curve) | I found Teruyoshi Yoshida's exposition of the subject very helpful:
<http://www.dpmms.cam.ac.uk/~ty245/Yoshida_2003_introDL.pdf>
As JT commented, the curve you wrote down is really the Deligne-Lusztig variety for SL\_2, not GL\_2. Ben is also right about the curve being $\mathbf{P}^1 - \mathbf{P}^1(\mathbf{F}\_q)$,... | 6 | https://mathoverflow.net/users/271 | 10477 | 7,154 |
https://mathoverflow.net/questions/10481 | 30 | I hope someone can point me to a quick definition of the following terminology.
I keep coming across *wild* and *tame* in the context of classification problems, often adorned with quotes, leading me to believe that the terms are perhaps not being used in a formal sense. Yet I am sure that there is some formal defini... | https://mathoverflow.net/users/394 | When is a classification problem "wild"? | I am not an expert but in the algebra and representation theory the apparently standard definition is [as follows](https://encyclopediaofmath.org/wiki/Representation_of_an_associative_algebra) (see also [Drozd - Tame and wild matrix problems](https://doi.org/10.1007/BFb0088467 "Drozd, J.A. (1980). Tame and wild matrix ... | 26 | https://mathoverflow.net/users/2149 | 10484 | 7,158 |
https://mathoverflow.net/questions/10480 | 8 | In non-commutative geometry, Gelfand duality is the construction of multiplicative linear functionals of a commutative C\*-algebra, which can be viewed as the space of all its irreducible complex representations.
When encountered with a non-commutative C\*-algebra, we can speak of the space of pure states, which is ... | https://mathoverflow.net/users/2945 | Gelfand duality in NCG | On surjectivity: No, not every representation comes from a state; only the cyclic ones. Every nondegenerate representation of a C\*-algebra is a direct sum of cyclic representations (Zorn), and every cyclic representation comes from a GNS construction. But yes, every irreducible representation (which is also cyclic) co... | 7 | https://mathoverflow.net/users/1119 | 10485 | 7,159 |
https://mathoverflow.net/questions/10488 | 4 | There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n\_d = d + 1$. If $n = n\_d$ we obviously have $\binom{d+1}{2}$ distances which can be chosen (more or less) independently (restricted only by the triangle inequality).
I see two wa... | https://mathoverflow.net/users/2672 | Number of independent distances between n points in d-dimensional Euclidean space? | Here is a little generalisation of your observation.
Suppose we have a manfiold $M$ of dimension d with a metric. The isometry group $I(M)$ of the manifold has dimension at most $\frac{d(d+1)}{2}$. The maximal dimension of $I(M)$ is attained for $R^d$, $H^d$, $S^d$ and $RP^d$ with a constant curvature metric. Now we w... | 2 | https://mathoverflow.net/users/943 | 10490 | 7,162 |
https://mathoverflow.net/questions/10487 | 8 | Related to a previous [question](https://mathoverflow.net/questions/10126/reference-for-this-theorem-in-representation-theory) I am asking furthermore a proof
for the following:
Question 1: If $\chi$ is a faithful irreducible character of a
finite group $G$ then the regular character of $G$ is a polynomial
with integ... | https://mathoverflow.net/users/2805 | Faithful characters of finite groups | I assume that by "faithful irreducible character" you mean the character of a faithful (i.e., trivial kernel) irreducible representation. In this case, the answer to Question 2 is negative. For instance, the irreducible character $\chi$ of the symmetric group $S\_4$ indexed by the partition (3,1) is faithful and has th... | 12 | https://mathoverflow.net/users/2807 | 10498 | 7,167 |
https://mathoverflow.net/questions/10493 | 22 | I am interested in the general form of the Kirchoff Matrix Tree Theorem for weighted graphs, and in particular what interesting weightings one can choose.
Let $G = (V,E, \omega)$ be a weighted graph where $\omega: E \rightarrow K$, for a given field $K$; I assume that the graph is without loops.
For any spanning tr... | https://mathoverflow.net/users/2947 | The matrix tree theorem for weighted graphs | A very interesting weighting is obtained by just working with directed multigraphs (dimgraphs). 7 or 8 years ago, I applied the matrix-tree theorem applied to dimgraphs in conjunction with the BEST theorem to provide a structure theory for the equilibrium thermodynamics of hybridization for oligomeric (short) DNA stran... | 11 | https://mathoverflow.net/users/1847 | 10500 | 7,169 |
https://mathoverflow.net/questions/10514 | 25 | What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
| https://mathoverflow.net/users/2938 | Teichmuller Theory introduction | [The primer](http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf) on mapping class groups, by Farb and Margalit.
| 20 | https://mathoverflow.net/users/1650 | 10517 | 7,177 |
https://mathoverflow.net/questions/10502 | 8 | In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then X and Y have the same fundamental group, so the fundamental group isn't strong enough to distinguish them. We need to lo... | https://mathoverflow.net/users/343 | Examples of the varying strengths of topological invariants | To change up the nature of the responses some, IMO a good theorem to think about is the Kan-Thurston theorem. It states that given any space $X$ you can find a $K(\pi, 1)$ space $Y$ and a map $f : Y \to X$ inducing isomorphisms $f\_\* : H\_i Y \to H\_i X$, $f^\* : H^i X \to H^i Y$ for all coefficients (it can be souped... | 6 | https://mathoverflow.net/users/1465 | 10536 | 7,190 |
https://mathoverflow.net/questions/10266 | 5 | Imagine an n-simplex, the solution set for the expression: $a\_1$\*$x\_1$ + $a\_2$\*$x\_2$ + ... + $a\_n$\*$x\_n$ = S, where:
1. $a\_1$ through $a\_n$ are positive bounded integers
2. $x\_1$ through $x\_n$ are positive bounded real numbers
3. 'S' is the sum of the expression
This n-simplex therefore has a single v... | https://mathoverflow.net/users/2891 | Counting lattice points on an n-simplex | I am informed that you are "counting lattice points inside of a polyhedron."
[Here](http://www.math.ucdavis.edu/~deloera/.../manyaspectsofcountinglatpts.pdf) is a lecture on the subject - the picture on page six looks like the version of the problem you are interested in. To be honest, I found these notes by doing a ... | 5 | https://mathoverflow.net/users/1650 | 10551 | 7,203 |
https://mathoverflow.net/questions/10231 | 14 | Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements:
1. Every monomorphism is regular.
2. Every epimorphism in $C$ is surjective.
It is easy to see that 1. implies 2. what about the converse?
| https://mathoverflow.net/users/2841 | When are epimorphisms of algebraic objects surjective? | Update: the following exchange appeared on the categories mailing list several years ago: <http://article.gmane.org/gmane.science.mathematics.categories/3094>. Walter Tholen's response strongly suggests that the answer to Martin's question is that the converse does not hold, although I don't have access to the four-aut... | 13 | https://mathoverflow.net/users/2926 | 10553 | 7,205 |
https://mathoverflow.net/questions/10457 | 3 | Is a good characterization of Spec $\mathbb{Z}[\zeta\_n]$ known? Same question for its unit group.
| https://mathoverflow.net/users/434 | What are the prime ideals in rings of cyclotomic integers? | **Theorem:** Let $\alpha$ be an algebraic integer such that $\mathbb{Z}[\alpha]$ is integrally closed, and let its minimal polynomial be $f(x)$. Let $p$ be a prime, and let
$\displaystyle f(x) \equiv \prod\_{i=1}^{k} f\_i(x)^{e\_i} \bmod p$
in $\mathbb{F}\_p[x]$. Then the prime ideals lying above $p$ in $\mathbb{Z... | 12 | https://mathoverflow.net/users/290 | 10555 | 7,207 |
https://mathoverflow.net/questions/10509 | 15 | I'm not a number theorist, so apologies if this is trivial or obvious.
From what I understand of the results of Green-Tao-Ziegler on additive combinatorics in the primes, the main new technical tool is the "dense model theorem," which -- informally speaking -- is as follows:
>
> If a set of integers $S \subset N$... | https://mathoverflow.net/users/382 | Goldbach-type theorems from dense models? | Yes, this can be done, provided that k is at least 3. A typical example is given in this paper: <http://arxiv.org/abs/math/0701240> . (This uses a slightly older Fourier-based method of Ben that predates his work with me, but is definitely in the same spirit - Ben's paper was very inspirational for our joint work.)
... | 26 | https://mathoverflow.net/users/766 | 10558 | 7,209 |
https://mathoverflow.net/questions/10370 | 8 | Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about
$$c\_k\frac{N^2}{\log^kN}$$
for some constant $c\_k$ dependent on k. The paper of Green and Tao gives a similar lower bound for all k (with a much worse constant, but still), and recent work by Green, Tao and Ziegler have ... | https://mathoverflow.net/users/385 | Upper bound for number of k-term arithmetic progressions in the primes | Well, any standard upper bound sieve (e.g. Selberg sieve, combinatorial sieve, beta sieve, etc.) will give this type of result. I'm not sure where you can find an easily citeable formulation, though. One can get this bound from Theorem D.3 of [this paper of Ben and myself](http://arxiv.org/abs/math.NT/0606088) on page ... | 15 | https://mathoverflow.net/users/766 | 10559 | 7,210 |
https://mathoverflow.net/questions/10532 | 10 | Given a moduli problem, it appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space(is this strictly true?).
In any case, assuming the above, what additional condition on a moduli problem in algebraic geometry will make sure that a coarse moduli space is in fact a fine... | https://mathoverflow.net/users/2938 | When is a coarse moduli space also a fine moduli space? | I think this is an instructive question. Here are some partial answers.
A category fibered in groupoids whose fibers are sets (e.g. no automorphisms) is a presheaf. Strictly speaking, I mean equivalent to the fibered category associated to a presheaf. This truly follows from the definitions, and is a good exercise to... | 12 | https://mathoverflow.net/users/2 | 10562 | 7,212 |
https://mathoverflow.net/questions/10501 | 11 | Let $(R,m)$ be a local complete intersection of dimension $3$. Let $X=Spec(R)$ and $U=Spec(R) -\{m\}$ be the punctured spectrum of $R$. I am trying to understand the following comment by Gabber (see it [here](http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf) , page 1975-1976):
$Pic(U)$ is torsion-free is... | https://mathoverflow.net/users/2083 | Flat cohomology and Picard groups | This is not a complete answer by any means, but is intended to get the ball rolling.
First of all, it need not be the case that $H^1(X\_{fl},\mu\_n) = 0.$ Rather, what follows
from the vanishing of $H^1(X\_{fl},{\mathbb G}\_m)$ is that
$H^1(X\_{fl},\mu\_n) = R^{\times}/(R^{\times})^n.$
(This is not always trivial;... | 11 | https://mathoverflow.net/users/2874 | 10563 | 7,213 |
https://mathoverflow.net/questions/1554 | 21 | Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold structure extend across the point singularity?
(Penny Smith and I wrote a paper on this many years ago, but we had to a... | https://mathoverflow.net/users/613 | Point singularity of a Riemannian manifold with bounded curvature | Once we considered a similar problem but around infinity,
try to look in our paper "Asymptotical flatness and cone structure at infinity".
Let us denote by $r$ the distance to the singular point.
If dimensions $\not= 4$ then the same method shows that at singular point we have Euclidean tangent cone even if curvature... | 7 | https://mathoverflow.net/users/1441 | 10565 | 7,215 |
https://mathoverflow.net/questions/10567 | 15 | Let $N(x)$ be the number of uniform random variables (distributed in $[0,1]$) that one needs to add for the sum to cross $x$ ($x > 0$). The expected value of $N(x)$ can be calculated and it is a very cool result that $E(N(1)) = e$. The expression for general $x$ is
$E(N(x)) = \sum\_{k=0}^{[x]} (-1)^k \frac{(x-k)^k}{... | https://mathoverflow.net/users/2878 | Number of uniform rvs needed to cross a threshold | Another way to put it: the expected value of the sum right after it crosses x is x+1/3. If you simply conditioned the sum to be in (x,x+1) then the expectation would be about x+1/2, with the sum almost uniformly distributed. But you also condition on the overshoot being less than the last jump. The expectation of the s... | 14 | https://mathoverflow.net/users/2912 | 10568 | 7,217 |
https://mathoverflow.net/questions/10496 | 3 | Let [a,b] = {k integer | a < k <= b}. Further let
* Comp[a,b] = product\_{c in [a,b]} c composite;
* Fact[a,b] = product\_{k in [a,b]} k integer;
* Prim[a,b] = product\_{p in [a,b]} p prime.
*Question*: For n > 2 and n not in {10,15,27,39} is it true that
$$ \text{Comp}[{\left\lfloor n /2 \right\rfloor}, n] <... | https://mathoverflow.net/users/2797 | An inequality relating the factorial to the primorial. | This answer is just to point out that the result is true for large enough $n$. Let's rewrite it as $$\prod\_{n\le p\le 2n}p > \sqrt{\binom{2n}{n}}$$
Since $\binom{2n}{n}\approx \frac{4^n}{n}$ introducing Chebyshev's functions
$$\theta(x)=\sum\_{p\le x}\text{log}p\quad,\quad \psi(x)=\sum\_{p^{\alpha}\le x}\text{log}p$$
... | 9 | https://mathoverflow.net/users/2384 | 10573 | 7,220 |
https://mathoverflow.net/questions/10560 | 26 | Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with connected complement. Let $f:U \rightarrow\mathbb{C}$ be continuous and non-vanishing on $U$ and holomorphic on $U^{int}$. Then $\... | https://mathoverflow.net/users/1849 | Universality of zeta- and L-functions | Since, I believe, Jonas Meyer provided an answer to Q1, let me just say about the other questions: The concept of universality is much older. It was in fact introduced by Birkhoff, in the case for entire functions, in 1929 (and that is why universal functions are sometimes called Birkhoff functions) "Demonstration d'un... | 7 | https://mathoverflow.net/users/2384 | 10580 | 7,223 |
https://mathoverflow.net/questions/10594 | 9 | As a rule, the various groups and quotients of the divisor group on a variety have coefficients in $\mathbb{Z}$. That is, you take $\mathbb{Z}$-linear combinations of Weil divisors or Cartier divisors, and then to construct other groups you take quotients.
However, in some cases, people tensor with $\mathbb{Q}$ and $... | https://mathoverflow.net/users/622 | Is there any value in studying divisors with coefficients in a ring R? | The answer to the question is yes. For example, if $\omega$ is a meromorphic 1-form on a curve (smooth and projective, say) over a field $k$, then one can naturally form a degree zero divisor with coefficients in $k$, namely the residue divisor of $\omega$.
| 12 | https://mathoverflow.net/users/2874 | 10596 | 7,234 |
https://mathoverflow.net/questions/10576 | 6 | The question belongs to elementary category theory, so please forgive me if this is trivial. I think I even read a proof for this some weeks ago, but I can't find it.
In topology, you have the equation $\overline{\overline{A}}=\overline{A}$ for subsets $A$ of a topological space $X$. An analogous theorem in category ... | https://mathoverflow.net/users/2841 | completion of category is idempotent | The answer is no. Here is my counterexample:
**ARGUMENT SIMPLIFIED, THANKS TO SUGGESTIONS BY Reid, Scott Carnahan AND t3suji**
Let $X$ be the category of abelian groups. Let $A$ be the full subcategory on groups of the form $(\mbox{finite group}) \oplus (\mathbb{Q}-\mbox{vector space})$.
Let $D$ be any diagram in... | 8 | https://mathoverflow.net/users/297 | 10599 | 7,236 |
https://mathoverflow.net/questions/10578 | 20 | My question, roughly speaking is, what happened to the function fields Langlands conjecture? I understand around 2000 (or slightly earlier perhaps), Lafforgue proved the function fields Langlands correspondence for $GL(n)$ in full generality (proving all aspects of the conjectures). Since then, what's happened to the f... | https://mathoverflow.net/users/2623 | What is the current status of the function fields Langlands conjectures? | (1) Regarding the relationship between geometric Langlands and function field Langlands:
typically research in geometric Langlands takes place in the context of rather restricted ramification (everywhere unramified, or perhaps Iwahori level structure at a finite number of points). There are investigations in some circu... | 22 | https://mathoverflow.net/users/2874 | 10600 | 7,237 |
https://mathoverflow.net/questions/10607 | 7 | Let $E$ be an elliptic curve over $\mathbb Q\_p$. It is possible that $E$ has bad reduction but then when you see $E$ as a curve over a finite extension $K$ of $\mathbb Q\_p$, it obtains good reduction. Let $v$ be the valuation defined on $K$ and $R$ its valuation ring. I was interested in checking $E$ has good reducti... | https://mathoverflow.net/users/389 | Writing down minimal Weierstrass equations | A slightly more theoretical answer: there is an algorithm of Tate, called (unremarkably)
Tate's algorithm, which allows one to compute the minimal model over any local field. I have a vague memory that it's not proved that this algorithm terminates in general, although it is expected to. (Perhaps someone else can say s... | 6 | https://mathoverflow.net/users/2874 | 10614 | 7,246 |
https://mathoverflow.net/questions/10556 | 51 | Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over a sheaf of differential operators, but for spaces that aren't smooth in some sense, this definition doesn't work that we... | https://mathoverflow.net/users/66 | D-modules, deRham spaces and microlocalization | The first thing to say is that the abelian category of
sheaves on the de Rham space is only a good model
for D-modules if you're in the smooth setting, or very close to it (see for example arXiv:math/0212094 for a setting where all the different notions agree).. so unless you're fully derived you need to be careful w... | 41 | https://mathoverflow.net/users/582 | 10617 | 7,249 |
https://mathoverflow.net/questions/10635 | 48 | I remember one of my professors mentioning this fact during a class I took a while back, but when I searched my notes (and my textbook) I couldn't find any mention of it, let alone the proof.
My best guess is that it has something to do with Galois theory, since it's enough to prove that the characters are rational -... | https://mathoverflow.net/users/2363 | Why are the characters of the symmetric group integer-valued? | If $g$ is an element of order $m$ in a group $G$, and $V$ a complex representation of $G$, then $\chi\_V(g)$ lies in $F=\mathbb{Q}(\zeta\_m)$. Since the Galois group of $F/\mathbb{Q}$ is $(\mathbb{Z}/m)^\times$, for any $k$ relatively prime to $m$ the elements $\chi\_V(g)$ and $\chi\_V(g^k)$ differ by the action of the... | 77 | https://mathoverflow.net/users/437 | 10642 | 7,265 |
https://mathoverflow.net/questions/10627 | 18 | Excuse me for the specificity of this question, but this is a silly computation that's been giving me trouble for some time.
I want to explicitly realize the order 21 Frobenius group over ℂ(x), as ℂ(x,y,z) where y3=g(x) and z7=h(x,y). The order 21 Frobenius group is C7⋊C3, where the generator of C3 acts by taking the... | https://mathoverflow.net/users/2665 | A Galois Theory Computation | I'd like to change your numbers slightly. (EDIT: Slight adjustment to make the formula nicer and address the correction in the comments, nothing to see here)
One solution is to set $y^3 = x$, triply ramified only over $0$ and $\infty$, and if we want the 7-fold ramification over $x=1$ (which has solutions $y=1,\omega... | 8 | https://mathoverflow.net/users/360 | 10650 | 7,271 |
https://mathoverflow.net/questions/9874 | 5 | This is a revised version of [a question I already posted](https://mathoverflow.net/questions/9393/can-topologies-induce-a-metric), but which patently was ill posed. Please give me another try.
---
For comparison's sake, the axioms of a metric:
Axiom A1: $(\forall x)\ d(x,x) = 0$
Axiom A2: $(\forall x,y)\ d(x... | https://mathoverflow.net/users/2672 | Can topologies induce a metric? (revised) | I shall prove that there can be no characterization of metrizability along the lines that you seek. (This argument fleshes out and fulfills the expectation of Mariano in the comments.)
Your axioms are all stated in the language with two types of objects: points and basis elements. So let us be generous here, and ente... | 10 | https://mathoverflow.net/users/1946 | 10654 | 7,275 |
https://mathoverflow.net/questions/10630 | 58 | Suppose for some reason one would be expecting a formula of the kind
$$\mathop{\text{ch}}(f\_!\mathcal F)\ =\ f\_\*(\mathop{\text{ch}}(\mathcal F)\cdot t\_f)$$
valid in $H^\*(Y)$ where
* $f:X\to Y$ is a proper morphism with $X$ and $Y$ smooth and quasiprojective,
* $\mathcal F\in D^b(X)$ is a bounded complex of ... | https://mathoverflow.net/users/65 | Why do Todd classes appear in Grothendieck-Riemann-Roch formula? | You look at the case when $X=D$ is a Cartier divisor on $Y$ (so that the relative tangent bundle -- as an element of the K-group -- is the normal bundle $\mathcal N\_{D/X}=\mathcal O\_D(D)$ (conveniently a line bundle, so is its own Chern root), and $\mathcal F=\mathcal O\_D$. And the Todd class pops out right away.
... | 54 | https://mathoverflow.net/users/1784 | 10655 | 7,276 |
https://mathoverflow.net/questions/10631 | 3 | I had this simple question when formulating the [Todd class question](https://mathoverflow.net/questions/10630/why-todd-classes-appear-in-grothendieck-riemann-roch-formula).
>
> Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^\*(X)$ such that for all coheren... | https://mathoverflow.net/users/65 | Homology class orthogonal to image of Chern characters? | Why not just take a class that is orthogonal to all the algebraic classes on $X$? For instance you can take $Y$ to be a point, and take $X$ to be a generic abelian surface over $\mathbb{C}$, i.e. and abelian surface with $NS(X) = \mathbb{Z}$. The Chern character of any coherent sheaf on $X$ is contained in $H^{0}(X)\op... | 8 | https://mathoverflow.net/users/439 | 10660 | 7,281 |
https://mathoverflow.net/questions/10672 | 4 | Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of components of the critical set for F? Can we relax the Morse-Bott requirement? What if the critical set isn't smooth... c... | https://mathoverflow.net/users/492 | Morse theory and Euler characteristics | You could have a smooth function $f : \Bbb R \to \Bbb R$ whose critical point set is a Cantor set (minima) and the centres of the complementary intervals (local maxima) -- let $f$ be some suitable smoothing of the distance function from the Cantor set (or you could use the smooth Urysohn lemma to construct the function... | 3 | https://mathoverflow.net/users/1465 | 10676 | 7,288 |
https://mathoverflow.net/questions/10680 | 0 | Among all $n$-vertex graphs with $M$ edges and constant $k$,how to estimate the fraction of graphs of clique less than $k$? Thanks.
| https://mathoverflow.net/users/2976 | How to estimate the fraction of graphs with small clique among the graphs with certain edges | A little further elaboration on what you're looking for would be appreciated; as you've asked it, this could be anything from an elementary probability exercise to what I think might be an open problem!
So if we fix k and take n large, then the answer depends entirely on the size of M compared with n. The term of art... | 4 | https://mathoverflow.net/users/382 | 10681 | 7,289 |
https://mathoverflow.net/questions/10678 | 11 | Hi,
what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics?
Of course also mathematicians may answer from their perspective :)
Edit: Information on my background: My "primeval" motivatio... | https://mathoverflow.net/users/2837 | What are important examples of filtered/graded rings in physics? | It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the same language.
Having said, and coming from a Physics background, when I first learnt about filtered objects and associat... | 19 | https://mathoverflow.net/users/394 | 10683 | 7,291 |
https://mathoverflow.net/questions/10679 | 8 | To head off any confusion: I'm talking about the extremal-combinatorics [Sperner's theorem](http://en.wikipedia.org/wiki/Sperner_family), bounding the sizes of antichains in a Boolean lattice.
So the "canonical proof" of this theorem seems to be essentially Lubell's -- it's formulated in several different ways, but m... | https://mathoverflow.net/users/382 | Sperner's theorem and "pushing shadows around" | Let $\mathcal{B}$ be a collection of $k$-sets , subsets of an $n$-set $S$. For $k < n$ define the shade of $\mathcal{B}$ to be $$\nabla \mathcal{B}=\{ D\subset S : |D|=k+1,\exists B\in \mathcal{B},B\subset D\}$$ and the shadow of $\mathcal{B}$ to be
$$\Delta \mathcal{B}=\{ D\subset S : |D|=k-1,\exists B\in \mathcal{B},... | 10 | https://mathoverflow.net/users/2384 | 10689 | 7,294 |
https://mathoverflow.net/questions/10667 | 60 | Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = 1- e^{-D}$ or, by taking the inverses,
$$ \frac{1}{\bigtriangledown} = \frac{1}{1-e^{-D}} =
\frac{1}{D}\cdot \frac{D}{1-... | https://mathoverflow.net/users/1784 | Euler-Maclaurin formula and Riemann-Roch | As far as I understand this connection was observed (and generalised) by Khovanskii and Puhlikov in the article
A. G. Khovanskii and A. V. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasipolynomials over
virtual polytopes, Algebra and Analysis 4 (1992), 188–216, translation in St. Petersburg Math. J... | 27 | https://mathoverflow.net/users/943 | 10691 | 7,296 |
https://mathoverflow.net/questions/10701 | 8 | According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in H^2(X,T\_X \otimes I).
Can anyone give an example of being obstructed?
e.g. a smooth scheme over F\_p which does not lift s... | https://mathoverflow.net/users/1657 | obstruction to smooth lifting of smooth schemes | Ravi Vakil's paper *Murphy's Law in Algebraic Geometry …* gives many references of such things: see Section 2 of
<http://arxiv.org/abs/math/0411469>
The first example is due to Serre:
---
Serre, Jean-Pierre
[Exemples de variétés projectives en caractéristique $p$ non relevables en caractéristique zéro](http:/... | 12 | https://mathoverflow.net/users/1149 | 10704 | 7,304 |
https://mathoverflow.net/questions/10687 | 2 | Googling for "atomic morphism" gives me only 70 results. Is this concept so fruitless or does it have another standard name?
What I mean is a morphism $f: A \rightarrow B$ such that
$$(\forall g,h)\ f = g \circ h \rightarrow (g = f\ \wedge\ h = id\_A)\ \vee\ (g = id\_B\ \wedge\ h = f).$$
| https://mathoverflow.net/users/2672 | Standard name of "atomic morphisms"? | As mentioned in the comments, I would probably call such a morphism "irreducible" or "prime." A "less [evil](http://ncatlab.org/nlab/show/evil)," and perhaps more useful, version would be to ask that if $f = g \circ h$, then either $g$ or $h$ is an isomorphism. In this form, if you regard the multiplicative monoid of a... | 8 | https://mathoverflow.net/users/49 | 10711 | 7,310 |
https://mathoverflow.net/questions/10609 | 6 | Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields on $G$?
In the same vein I would like to know why the following construction works:
If one looks at the tangent v... | https://mathoverflow.net/users/2678 | Killing fields on homogeneous spaces | I think that if you generalize that statement a little it becomes clearer (also the proof).
Let $G$ be any Lie group (not necessarily compact) with a closed subgroup $H$ and a metric (not necessarily positive definite) on $G$ which is $G$-left-invariant and $H$-right-invariant (not necessarily bi-invariant).
Thes... | 9 | https://mathoverflow.net/users/2991 | 10713 | 7,311 |
https://mathoverflow.net/questions/10724 | 8 | I'm a little confused and in need of some clarification about the
relationship between algebraic and holomorphic differential forms:
(1) What is the exact definition of the module of differential
forms of a complex projective variety?
(2) What is the definition of its differential?
(3) Am I right in assuming that... | https://mathoverflow.net/users/1977 | Relationship between algebraic and holomorphic differential forms | Yes, every algebraic differential form is holomorphic and yes, the differential preserves the algebraic differential forms. If you are interested in projective smooth varieties then every holomorphic differential form is automatically algebraic thanks to Serre's GAGA. This answers (3).
Concerning (1) and (2) I sugges... | 9 | https://mathoverflow.net/users/605 | 10725 | 7,320 |
https://mathoverflow.net/questions/10736 | 15 | When I am testing conjectures I have about number fields, I usually want to control the ramification, especially minimize to a single prime with tame ramification. Hence, I usually look for fields of prime discriminant (sometimes positive, sometimes negative).
I get the feeling that I cannot be the only one who does ... | https://mathoverflow.net/users/2024 | Families of number fields of prime discriminant | [Klueners Malle online](http://www.math.uni-duesseldorf.de/~klueners/groups2.html) might be just the thing you're looking for. Make your own lists! And [here](http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html)'s some they made themselves, if you run out of ideas.
| 8 | https://mathoverflow.net/users/1384 | 10740 | 7,326 |
https://mathoverflow.net/questions/10718 | 0 | Are there non-regular strong monics in the category of locales?
| https://mathoverflow.net/users/2884 | Strong monics in the category of locales | No. The category of locales is dual to the category of frames, which is monadic (in fact, equationally presented) over Set. Any such category is Barr-exact, and in particular a regular category, which implies that every strong epic of frames is regular—hence every strong monic of locales is regular.
| 5 | https://mathoverflow.net/users/49 | 10745 | 7,330 |
https://mathoverflow.net/questions/10753 | 3 | I was reading [link text](https://mathoverflow.net/questions/10724/relationship-between-algebraic-and-holomorphic-differential-forms)
and these two much simpler questions occurred to me:
(i) What type of algebraic functions on complex *projective* varieties do the holomorphic functions correspond to? The rational func... | https://mathoverflow.net/users/2612 | Algebraic and Holomorphic Functions | Well, here's a few theorems that might help:
1: On a complex projective variety, a function that is meromorphic on the whole variety is a rational function. You can get this out of the embedding into projective space.
2: On a compact complex manifold, the only globally holomorphic functions are constant. This follo... | 8 | https://mathoverflow.net/users/622 | 10756 | 7,334 |
https://mathoverflow.net/questions/10743 | 11 | In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of smooth hypersurfaces of $\mathbb{P}^N$ are finite (with known exceptions such as quadrics, elliptic curves, K3 surfaces). ... | https://mathoverflow.net/users/2868 | Are automorphism groups of hypersurfaces reduced ? | If $X$ is a smooth hypersurface in $\mathbf{P}^{n+1}$ of degree $d$, where $n \ge 1$, $d \ge 3$, and $(n,d) \ne (1,3)$, then $H^0(X,T\_X)=0$ by Theorem 11.5.2 in Katz and Sarnak, *Random matrices, Frobenius eigenvalues, and monodromy*, AMS Colloquium Publications, vol. 45, 1999. So the connected component of the identi... | 17 | https://mathoverflow.net/users/2757 | 10759 | 7,336 |
https://mathoverflow.net/questions/10758 | 11 | For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a primitive polynomial, so the primitive polynomials form a multiplicative subset $S$ of $R[x]$ - hence we can form the r... | https://mathoverflow.net/users/1916 | Localizing at the primitive polynomials? | A prime ideal of $S^{-1}R[X]$ is the extension of a unique prime ideal of $R$, so that the morphism $Spec(S^{-1}R[X])\to Spec(R)$ is a bijection, and even an homeomorphism. All the extensions of residual fields induced are pure transcendental of transcendence degre $1$.
As an example, if you look at the case $R=\math... | 10 | https://mathoverflow.net/users/2868 | 10761 | 7,338 |
https://mathoverflow.net/questions/6253 | 4 | Can anything be said about the measure of the *topological* boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say anything about the topological boundary, for instance, points with density 0 or 1 can still be part of the boundary.
... | https://mathoverflow.net/users/1969 | Lebesgue measure of boundary of Caccioppoli set | The answer is no. Take countably many disjoint closed balls $B\_i$ contained in the square $Q=[0,1]\times [0,1]$ and such that:
(i) Sum of areas of $B\_i$ is less than 1
(ii) Sum of perimeters of $B\_i$ is finite
(iii) $\bigcup B\_i$ is dense in $Q$
Since the series $\sum \chi\_{B\_i}$ converges in BV norm,... | 6 | https://mathoverflow.net/users/2912 | 10766 | 7,340 |
https://mathoverflow.net/questions/10767 | 2 | Is the sum $$ S= \sum\_{n=2}^\infty \frac{1}{ \log^1n \log^2n \log^3n \cdots\log^{TL(n)}n} $$
convergent?
Here
$\log^i n$ denotes the $i$'th iterate of $\log$ (in base 2) of $n$, so $\log^2n$ means $\log\log n$, etc., and
$T(n)$ is the tower of $n$ (stack of $n$ 2's) defined by $T(1)=2$ $T(n+1)=2^{T(n)}$ for ... | https://mathoverflow.net/users/3005 | Convergence of a general Bertrand series | The sum diverges. This is [Putnam Problem A4, 2008](http://kskedlaya.org/putnam-archive/).
| 9 | https://mathoverflow.net/users/297 | 10769 | 7,341 |
https://mathoverflow.net/questions/10768 | 3 | Let $F=(F\\_n)\\_n$ be an $\ell$-adic sheaf on $X\\_{et}$, for a variety $X$ over an algebraically closed field $k$ of characteristic not equal to $\ell$. Does the presheaf sending $U$ to $H^i(U,F):=\lim\\_n H^i(U,F\\_n)$ sheafify to zero?
| https://mathoverflow.net/users/370 | sheafifying a projective limit of presheaves | CORRECTED ANSWER: I believe that the answer is no, at least in some contexts.
For example, suppose that
$X = $Spec $k$, with $k$ a field, and $F = {\mathbb Z}\\_{\ell}(1)$. Then
$U = $Spec $l$ for some finite separable extension $l$ of $k$, and $H^1(U,F)
= \ell$-adic completion of $l^{\times}$, which I will denote b... | 3 | https://mathoverflow.net/users/2874 | 10773 | 7,343 |
https://mathoverflow.net/questions/10726 | 4 | Luca Trevisan [here](http://lucatrevisan.wordpress.com/2008/02/16/approximate-counting/) gives a randomized polynomial-time approximation algorithm for #3-coloring given an NP oracle.
In a similar vein, I was wondering if there were any results on $BPP^{NP}\stackrel{?}{=}$ #P - i.e. outputting a correct count for a #... | https://mathoverflow.net/users/1612 | BPP being equal to #P under Oracle | First, let's be slightly pedantic and not make statements like P = #P, which cannot possibly be true just because P is a set of decision problems and #P is not. To get a decision version of #P, one can use PP, or something like P#P.
About your question, BPPNP is contained in PPP and P#P by Toda's theorem. On the othe... | 9 | https://mathoverflow.net/users/1042 | 10774 | 7,344 |
https://mathoverflow.net/questions/10778 | 1 | $\mathbb{R}^n$ admits a [tessellation by permutohedra](http://en.wikipedia.org/wiki/Permutohedron#Tessellation_of_the_space). The corresponding identification of facets of a permutohedron therefore gives a well-defined space: call it $X\_n$. For example, $X\_2$, the hexagon with opposite sides identified, can be shown ... | https://mathoverflow.net/users/1847 | The topology of periodic permutohedral boundary conditions | The quotient $X\_n$ is the same as the quotient of $\mathbb R^n$ by a subgroup of $\mathbb Z^n$ acting cocompactly through translations. Such a thing is always a torus.
| 6 | https://mathoverflow.net/users/1409 | 10780 | 7,347 |
https://mathoverflow.net/questions/10776 | 3 | In the category of normed vector spaces in which the morphisms are linear contractions, what do products look like?
| https://mathoverflow.net/users/3007 | What are the products in the category of normed vector spaces with linear contractions? | Two words: sup norm.
I.e., the product of a family is the uniformly bounded subset of the cartesian product of the family, and the norm is the smallest uniform bound.
Explicitly, if $\{X\_i\}\_{i\in I}$ is a family of normed vector spaces with all norms ambiguously denoted $\|\cdot\|$, then the product is $X=\{\{a\... | 6 | https://mathoverflow.net/users/1119 | 10784 | 7,350 |
https://mathoverflow.net/questions/10793 | 4 | There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. Is it true that the set of extreme points of a compact convex subset must be closed if the locally convex space in questio... | https://mathoverflow.net/users/792 | Compact Convex sets and Extreme Points | By definition, a non-extreme boundary point lies on an open line segment contained in the set, which happens to be an open subset of the boundary in two dimensions. Hence the set of extreme points is a closed subset of the boundary.
| 9 | https://mathoverflow.net/users/2912 | 10795 | 7,356 |
https://mathoverflow.net/questions/10319 | 4 | The yoneda lemma gives us a characterization of $Psh(\mathcal{C})$ that seems very similar to the theory of distributions. That is, we have a notion of representable presheaves, similar to representable distributions. The ability to talk about presheaves as colimits of representables correlates to the more complicated ... | https://mathoverflow.net/users/1353 | Distributions as presheaves? | While maybe not exactly what you were after, here is something that you might enjoy looking into, which relates presheaves and distributions.
There exists a category of sheaves on certain test objects, such that
* this category is a [smooth topos](http://ncatlab.org/nlab/show/smooth+topos) into which the category ... | 6 | https://mathoverflow.net/users/381 | 10811 | 7,364 |
https://mathoverflow.net/questions/10789 | 7 | Near the bottom of [the nlab page for Banach space](http://ncatlab.org/nlab/show/Banach+space) I see "To be described: duals (p+q=pq)".
Are $(\mathbb{R}^n)\_p$ and $(\mathbb{R}^n)\_q$ dual objects in the closed symmetric monoidal category of Banach spaces and linear contractions (with the tensor product described on ... | https://mathoverflow.net/users/126667 | Categorical duals in Banach spaces | My suspicion is "no", because if I recall correctly the map $I \to V \otimes V^\*$ naturally lands in the *injective tensor product*, not the *projective tensor product*, and it is the latter which appears as the ``correct'' tensor product for the SMC category of Banach spaces and linear contractions.
In the toy exam... | 2 | https://mathoverflow.net/users/763 | 10822 | 7,371 |
https://mathoverflow.net/questions/10819 | 3 | Related to this [question](https://mathoverflow.net/questions/3309/are-there-two-groups-which-are-categorically-morita-equivalent-but-only-one-of-wh) I also had some troubles to understand the classification of module categories over $Rep(G)$. Specifically, on page 12 of Ostrik's [paper](http://arxiv.org/PS_cache/math/... | https://mathoverflow.net/users/2805 | Module categories over $Rep(G)$. | Sebastian: your definition of Rep^1(\tilde H) is absolutely correct. If you have
a representation of G you can restrict it to H and consider it as a representation
of \tilde H (this operation is called inflation). Now you can tensor it with any
representation of \tilde H; this tensoring preserves Rep^1(\tilde H); this ... | 7 | https://mathoverflow.net/users/4158 | 10828 | 7,374 |
https://mathoverflow.net/questions/10827 | 9 | Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll give my definitions before concluding with my question.
Definitions
-----------
Let $C$ be a category with symmetric monoidal structure $\otimes$ ... | https://mathoverflow.net/users/78 | If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf? | There is a known theorem which, if I correctly understand your question, answers it (because the Hopf algebra antipode is defined as the $\ast$-inverse of $\mathrm{id}$):
**Theorem 1.** Let $A$ be an algebra and $\left(C,\left(C\_n\right)\_{n\geq 0}\right)$ a filtered coalgebra, i. e. a coalgebra $C$ and a sequence $... | 5 | https://mathoverflow.net/users/2530 | 10830 | 7,376 |
https://mathoverflow.net/questions/10831 | 19 | I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example would be the best but just a proof that some exist would also be nice. (In fact I am not sure they do.)
I know of exampl... | https://mathoverflow.net/users/2888 | Example of continuous function that is analytic on the interior but cannot be analytically continued? | Let $f(z) = \sum z^n/n^2$, which is continuous and bounded on the closed unit disc but not analytic near $1$. Then consider
$$\sum f(z^n)/n^2.$$
This should have a singularity at every root of unity; and should be analytic in the interior because it is uniformly convergent.
| 19 | https://mathoverflow.net/users/297 | 10837 | 7,379 |
https://mathoverflow.net/questions/10842 | 31 | The [Hawaiian Earring](http://en.wikipedia.org/wiki/Hawaiian_earring) is usually constructed as the union of circles of radius 1/n centered at (0,1/n): $\bigcup\_1^\infty \left[ (0, \frac{1}{n}) + \frac{1}{n}S^1 \right]$. However, nothing stops us from using the sequence of radii $1/n^2$ or any other sequence of number... | https://mathoverflow.net/users/1358 | Are all Hawaiian Earrings homeomorphic? | The Hawaian earring is the one-point compactification of a countable union of open intervals (with the coproduct or disjoint sum topology). This description is independent of the radii used to construct it.
A beautiful reference about this space is [Cannon, J. W.; Conner, G. R. The combinatorial structure of the Hawa... | 42 | https://mathoverflow.net/users/1409 | 10843 | 7,384 |
https://mathoverflow.net/questions/10848 | 5 | ### Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field, K. Then does this imply that cf is defined over K, for some c?
If so, do we have to assume that the zeros and poles are ind... | https://mathoverflow.net/users/2665 | Field of Definition of a Meromorphic Function | It is not sufficient that the subscheme of poles and zeroes is defined together over $K$, as the example of the function $(z+i)/(z-i):\mathbb P^1 \to \mathbb P^1$, defined only over $\mathbb Q(i)$, illustrates.
If poles of $f$, which form together a subscheme $D\_\infty$, are defined over $K$, then the line bundle $\... | 4 | https://mathoverflow.net/users/65 | 10850 | 7,389 |
https://mathoverflow.net/questions/245 | 7 | A pivotal monoidal category is called non-degenerate if the inner product $\left(x,y\right) = Tr\left(xy^{\*}\right)$ (where $y^{\*}$ is the dual map) is non-degenerate. As a rule of thumb non-degenerate is closely related to semisimplicity. For example, if a category is semisimple then it is automatically non-degenera... | https://mathoverflow.net/users/22 | Are abelian non-degenerate tensor categories semisimple? | I believe this is Proposition 5.7 in [Deligne's](http://www.math.ias.edu/files/deligne/Symetrique.pdf) “La Categorie des Representations du Groupe Symetrique S\_t, lorsque t n’est pas un Entier Naturel”. See also [this question](https://mathoverflow.net/questions/10857/is-tensor-product-exact-in-abelian-tensor-categori... | 2 | https://mathoverflow.net/users/297 | 10851 | 7,390 |
https://mathoverflow.net/questions/10832 | 3 | Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T\_K = \pi\_K T \pi\_K^\* :K \to K$, where $\pi\_K:H\to K$ is the orthogonal projection.
As an important special case, if $H=\mathbb{R}^n$ and $K$ is a coordinate subspace, then with respect to standard bases, $... | https://mathoverflow.net/users/1044 | Standard name for basis-independent submatrices? | The standard name in operator theory is "compression", and its partner in crime is "dilation". I.e., A is a compression of B if and only if B is a dilation of A (although sometimes "dilation" is reserved for cases where the compression respects powers). The Wikipedia entry is not proof, but [here it is](http://en.wikip... | 6 | https://mathoverflow.net/users/1119 | 10852 | 7,391 |
https://mathoverflow.net/questions/10857 | 8 | Suppose we are in an abelian tensor category with duals, where all objects have finite length. Let $0 \to A \to B \to C \to 0$ be a short exact sequence and $Z$ an object of the category. Is
$$0 \to Z \otimes A \to Z \otimes B \to Z \otimes C \to 0$$
exact?
**Motivation**: I am reading the proof of Proposition 5.7 i... | https://mathoverflow.net/users/297 | Is tensor product exact in abelian tensor categories with duals? | Yes, because if Z has a dual, then in particular Z ⊗ – has a left adjoint (Z\* ⊗ –) and hence commutes with limits (and similarly with colimits, but that's automatic if the category is closed monoidal).
| 11 | https://mathoverflow.net/users/126667 | 10859 | 7,396 |
https://mathoverflow.net/questions/10860 | 33 | ### Motivation
I learned about this question from a wonderful article [Rational points on curves](https://www.math.mcgill.ca/darmon/pub/Articles/Expository/12.Clay/paper.pdf) by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
* the set $\{$ objects $\dots$ over field $K... | https://mathoverflow.net/users/65 | Why no abelian varieties over Z? | It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then for any integer $n$ the $n$-torsion scheme $A[n]$ is a finite flat group scheme over $\mathbb Z$. Although this group sc... | 39 | https://mathoverflow.net/users/2874 | 10861 | 7,397 |
https://mathoverflow.net/questions/10853 | 4 | Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives of a local parameterization.
Alternatively, is there a "nice" variational characterization of surface curvature? (E.g., ... | https://mathoverflow.net/users/1557 | Variational characterization of curvature? | The curvature *is* a local invariant. There is such a thing as the curvature at a point. The curvature is described as a tensor, after all. It is different in, say, symplectic geometry, where because of the Darboux theorem all symplectic manifolds of the same dimension are locally symplectomorphic; a fact usually parap... | 13 | https://mathoverflow.net/users/394 | 10874 | 7,405 |
https://mathoverflow.net/questions/10870 | 46 | My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R\* in place of the usual R-valued metric.
That is, let us define that a topological space X is a *nonstandard metric space*, if there is a distance function, not into the reals R, but into some nonstanda... | https://mathoverflow.net/users/1946 | Which topological spaces admit a nonstandard metric? | The uniformity defined by a \*R-valued metric is of a special kind.
Let $(n\_i)\_{i<\kappa}$ be a cofinal sequence of positive elements in \*R. We may assume that $i < j$ implies that $n\_i/n\_j$ is infinitesimal.
Given a \*R-valued metric space $(X,d)$ we can define a family $(d\_i)\_{i<\kappa}$ of pseudometrics ... | 28 | https://mathoverflow.net/users/2000 | 10877 | 7,407 |
https://mathoverflow.net/questions/10881 | 1 | Let $E$ be a holomorphic vector bundle over $\mathbb{P}^n\setminus\begin{Bmatrix}[1,0,0,\cdots,0]\end{Bmatrix}$. Let $D$ be a connection on $E$. Let $\widetilde{E}$ be an extension of $E$. Since $\widetilde{E}$ is reflexive, i.e. double dual of $E$ is isomorphic to itself, then up to isomorphism, $\widetilde{E}$ is uni... | https://mathoverflow.net/users/2348 | Extension of a holomorphic vector bundle | Presumably, you assume $n\ge 2$.
1) Is it possible that $\tilde E$ is a vector bundle? Yes. Is it always a vector bundle for any $E$? No. Unless, of course, you assume that the connection is flat and holomorphic, then
it extends essentially for topological reasons.
2) Does the connection extends to $\tilde E$ if it... | 1 | https://mathoverflow.net/users/2653 | 10886 | 7,412 |
https://mathoverflow.net/questions/8497 | 7 | Given a $\sqrt{n}\times\sqrt{n}$ piece of the integer $\mathbb{Z}^2$ grid, define a graph by joining any two of these points at unit distance apart. How many spanning trees does this graph have (asymptotically as $n\to\infty$)?
Can you also say something about the triangular grid generated by $(1,0)$ and $(1/2,\sqrt{... | https://mathoverflow.net/users/932 | Number of spanning trees in a grid | I think the best way to deal with grids is to find the general eigenfunction of the infinite grid, and then apply appropriate boundary conditions. This is an idea of Kenyon, Propp and Wilson, you can find an outline in the very last section of my Diplomarbeit [link text](http://www.mat.univie.ac.at/~kratt/theses/rubey.... | 9 | https://mathoverflow.net/users/3032 | 10895 | 7,418 |
https://mathoverflow.net/questions/10868 | 12 | Let $G$ be a simple graph (undirected, no loops or parallel edges), with maximum degree $\Delta(G)$. I would like to add edges to the graph to make it regular, *without increasing the maximum degree*.
In general this is not possible. (For example, take the 5-vertex graph formed by taking a triangle ($K\_3$) and addin... | https://mathoverflow.net/users/1028 | Regularizing graphs | It is always enough to add k+2 more vertices where k denotes the maximum degree. This is sharp as shown by the graph which is a cycle of length 5 plus two independent edges.
The proof is the following. Add edges between non adjacent vertices whose degree is smaller than k until we can. After we cannot, we have some v... | 6 | https://mathoverflow.net/users/955 | 10898 | 7,420 |
https://mathoverflow.net/questions/10897 | 27 | There's a famous story about an exercise from Lang's Algebra that says something along the lines of *"pick up a homological algebra book and prove all of the theorems yourself"*. I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's only in the older editio... | https://mathoverflow.net/users/1353 | "Pick up a homological algebra book and prove all of the theorems yourself" (exercise from Lang's Algebra) | It's real, but only in the first and second editions. (I don't have any electronic proof, but I've seen it in my copy of the second edition and someone else's copy of the first edition.) It's the only exercise in the chapter.
The full quote in the second edition is:
>
> *Take any book on homological algebra,
> an... | 31 | https://mathoverflow.net/users/1119 | 10899 | 7,421 |
https://mathoverflow.net/questions/10914 | 8 | Is there some existing notation for
>
> `\[f(n)\leq g(n)\]` for sufficiently large n
>
>
>
Apart from just writing that itself?
I'm thinking of something compact like the Landau notation $f\ll g$.
(Apologies if this is too specific for MathOverflow - just close it if so. I was also unsure what tags to add, s... | https://mathoverflow.net/users/385 | Notation for eventually less than | In logic, this relation is called *almost* less than or equal, and is denoted with an asterisks on the relation symbol, like this: $f \leq^\* g$.
For example, the [bounding number](https://mathoverflow.net/questions/8972#9027) is the size of the smallest family of functions from N to N that is not bounded with respe... | 13 | https://mathoverflow.net/users/1946 | 10918 | 7,429 |
https://mathoverflow.net/questions/10731 | 44 | EGA IV 17.1.6(i) states that formal smoothness is a source-local property. In other words, a map $X\to Y$ of schemes is formally smooth if and only if there is an open cover $U\_i$ ($i\in I$) of $X$ such that each restriction $U\_i\to Y$ is formally smooth.
It seems however that there is a gap in the proof. The probl... | https://mathoverflow.net/users/1114 | Possible formal smoothness mistake in EGA | Let me point out the following remark made by Grothendieck in his book "Catégories Cofibreés Additives et Complexe Cotangent Relatif", 9.5.8
Please excuse my translation:
"Let $f:X \rightarrow Y$ be a morphism of schemes. We say that $f$ is "locally formally smooth" if X can be covered by opens $X\_i$ which are for... | 43 | https://mathoverflow.net/users/397 | 10923 | 7,433 |
https://mathoverflow.net/questions/2497 | 5 | The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal in $\mathbb{C}[X\_1,Y\_1..,X\_n,Y\_n]^{S\_n}$. By a theorem of Ginzburg, the zero-set of this ideal is the closure of the ... | https://mathoverflow.net/users/66 | What are the "special" strata of Sym^n(C^2)? | Just by coincidence, I was catching up on my arxiv reading and noticed that Losev's [paper](http://arxiv.org/abs/1001.0239) appears to answer your question. See part (3) of Theorem 4.3.1 there. He says it was "known previously".
### Edit
To be kinder to the reader: the Cherednik algebra is really a family of algebr... | 4 | https://mathoverflow.net/users/nan | 10924 | 7,434 |
https://mathoverflow.net/questions/10930 | 2 | is it possible to characterize the elements of a (special) direct limit only using the universal property? in detail:
let's first concentrate on the category of sets. by an element, I mean a morphism defined on the terminal object $\{\*\}$. let $A\_1 \to A\_2 \to ...$ be a sequence of sets, and $A$ their colimit whic... | https://mathoverflow.net/users/2841 | categorical description of elements in a direct limit | As with that [previous question](https://mathoverflow.net/questions/9921/equality-of-elements-in-localization-via-universal-property), I don't understand precisely what the rules of the game are when you say "without using an explicit construction". But maybe I can say something useful.
---
First I'll answer the ... | 4 | https://mathoverflow.net/users/586 | 10933 | 7,440 |
https://mathoverflow.net/questions/10934 | 27 | The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:
Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal ord... | https://mathoverflow.net/users/2024 | Class number measuring the failure of unique factorization | Theorem (Carlitz, 1960): The ring of integers $\mathbb{Z}\_F$ of an algebraic number field $F$ has class number at most $2$ iff for all nonzero nonunits $x \in \mathbb{Z}\_F$, any two factorizations of $x$ into irreducibles have the same number of factors.
A proof of this (and a 1990 generalization of Valenza) can be... | 26 | https://mathoverflow.net/users/1149 | 10939 | 7,443 |
https://mathoverflow.net/questions/10948 | 14 | This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum\_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum\_{i>0}i^{-2}$.
Some quick facts:
* Pretty much by construction, $\gamma$... | https://mathoverflow.net/users/750 | What is $\sum (x+\mathbb{Z})^{-2}$? | Use residues! For an entertaining narrative with the correct answer, see [here](https://mathoverflow.net/questions/8741/justifying-a-theory-by-a-seemingly-unrelated-example/8752#8752). For a derivation, see a complex analysis text.
### Added
Here's a link to an outline of the residue method of solving this: [Conway... | 17 | https://mathoverflow.net/users/1119 | 10949 | 7,449 |
https://mathoverflow.net/questions/10947 | 21 | There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and algebraic closures are analogous to universal covering spaces, hence the absolute Galois group is analogous to the fundamental g... | https://mathoverflow.net/users/290 | What's the analogue of the Hilbert class field in the following analogy? | This is a great question. Someone will come along with a better answer I'm sure, but here's a bit off the top of my head:
1) The Hilbert class field of a number field $K$ is the maximal everywhere unramified abelian extension of $K$. (Here when we say "$K$" we really mean "$\mathbb{Z}\_K$", the ring of integers. That... | 14 | https://mathoverflow.net/users/1149 | 10952 | 7,451 |
https://mathoverflow.net/questions/10966 | 38 | Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.
The first definition should coincide with what is given in most differential topology text books, for instance Warner's book.
>
> Orientability using differential forms: There exists a nowhere vanishing differen... | https://mathoverflow.net/users/2938 | Two kinds of orientability/orientation for a differentiable manifold | If $X$ is a differentiable manifold, so that both notions are defined, then they coincide.
The ``patching'' of local orientations that you describe can be expressed more formally as follows: there is a locally constant sheaf $\omega\_R$ of $R$-modules on $X$ whose stalk at a point is $H^n(X,X\setminus\{x\}; R).$ Of c... | 35 | https://mathoverflow.net/users/2874 | 10968 | 7,459 |
https://mathoverflow.net/questions/10960 | 9 | Suppose X is dimension two locally Noetherian scheme. Y is a closed subscheme of X, with codimension 1. Denote X' to be the blow up of X along Y. Prove that the structure morphism f:X'-->X is a finite morphism.
It suffices to show it's quasi-finite according to Zariski's main theorem. But I can't exclude the possibi... | https://mathoverflow.net/users/2008 | Blow up along codimension one closed subscheme | I think it's not true :
Let $X=Spec(A)$ with $A=k[x,y,z]/(x^2-y^2-z^2)$ be a quadratic cone. Let $Y$ be a line through the origin of the cone : its ideal is $I=(z,x-y)$. We calculate :
$$X'=Proj\_{A}A[t,u]/(zt-(x+y)u,(x-y)t-zu),$$ [**EDIT : THE FORMULA HAS BEEN CORRECTED**]
where, in the graded $A$-algebra $A+I... | 14 | https://mathoverflow.net/users/2868 | 10970 | 7,461 |
https://mathoverflow.net/questions/10971 | 17 | I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event $X$ and a model parameter $m$:
$P(X\mid m)$ is a probability density function
$P(m\mid X)$ is not
It feels like it should be, and I can't find a clear explanation of why it's not. ... | https://mathoverflow.net/users/3045 | Why isn't likelihood a probability density function? | If $X$ is data and $m$ are the parameters, then the likelihood function $l(m) = p(X | m)$. I.e. it's $p(X | m)$, considered as a function of $m$.
Both $p(X|m)$ and $p(m|X)$ are pdfs: $p(X|m)$ is a density on $X$ and $p(m|X)$ is a density on $m$. But the likelihood is $p(X|m)$, not as a function of $X$ (it would inde... | 9 | https://mathoverflow.net/users/3035 | 10978 | 7,465 |
https://mathoverflow.net/questions/10974 | 23 | Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
| https://mathoverflow.net/users/3046 | Does homology detect chain homotopy equivalence? | Yes, this is true. Suppose $C\_\*$ is such a chain complex of free abelian groups.
For each $n$, choose a splitting of the boundary map $C\_n \to B\_{n-1}$, so that $C\_n \cong Z\_n \oplus B\_{n-1}$. (You can do this because $B\_{n-1}$, as a subgroup of a free group, is free.) For all $n$, you then have a sub-chain-c... | 28 | https://mathoverflow.net/users/360 | 10983 | 7,469 |
https://mathoverflow.net/questions/10957 | 4 | This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question [here](https://mathoverflow.net/questions/10954/what-do-negative-knots-look-like), but Ilya N asked me to break that post up into several questions.
Consider the free commutative monoid $X$ on countably many generators. Let... | https://mathoverflow.net/users/78 | A hands-on description of a "completion" of the free commutative monoid on countably many generators | Preliminaries
-------------
First part of your question doesn't use the bialgebra structure. That is, you have a space of functions on countable many points which I'll denote $A = \mathbb C\_1\times \mathbb C\_2\times\cdots \times\mathbb C\_n\times\cdots$ equipped with $+$ and $\times$ pointwise. You would like to cl... | 4 | https://mathoverflow.net/users/65 | 10986 | 7,472 |
https://mathoverflow.net/questions/10954 | 18 | Background
----------
Recall that a (oriented) *knot* is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of oriented knots has a binary operation called *connected sum*: if $K\_1,K\_2$ are knots, then $K\_1 \# K\_2$ ... | https://mathoverflow.net/users/78 | What are the points of Spec(Vassiliev Invariants)? | Just some small comments here. Yes, the question of whether or not Vassiliev invariants separate knots is still open.
One broader context for the question is to consider how Vassiliev invariants were first observed -- via the Vassiliev spectral sequence for the space of "long knots". These are knots of the form $\mat... | 11 | https://mathoverflow.net/users/1465 | 10990 | 7,476 |
https://mathoverflow.net/questions/11001 | 3 | Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this generality there is no simple construction so feel free to restrict $m$ and $r$.
I'm reading some notes on group theory and... | https://mathoverflow.net/users/nan | non-abelian groups of prescribed order | I am also unsure of what "nontriviality" conditions you want to impose. Without any further conditions, the following answers your question:
Call a positive integer $n$ **nilpotent** if every group of order $n$ is nilpotent.
Call a positive integer $n$ **abelian** if every group of order $n$ is abelian.
Suppose t... | 12 | https://mathoverflow.net/users/1149 | 11006 | 7,484 |
https://mathoverflow.net/questions/11020 | 6 | Sorry if I'm missing something here, but what do we call $M$ if the functor $H\_M:N\mapsto Hom(M,N)$ is exact? Is this in fact equivalent to being flat through some adjointness properties?
| https://mathoverflow.net/users/1916 | Tensor product is to flat as Hom is to ? | We call such modules projective. If you take $N\mapsto Hom(N,M)$ then you get injective modules. This is fairly basic, and covered in any homological algebra book, and mentioned on [wikipedia](http://en.wikipedia.org/wiki/Exact_functor).
| 14 | https://mathoverflow.net/users/622 | 11022 | 7,495 |
https://mathoverflow.net/questions/10478 | 4 | Is there a variety $X$ over $\mathbb{Q}$ and a line bundle $L$ over $X$ (other than the trivial line bundle $\mathcal{O}\_X$ ) such that $L\_v$ is the trivial line bundle over $X\_v=X\times\_{\mathbb{Q}}\mathbb{Q}\_v$ for every place $v$ of $\mathbb{Q}$ ?
(Answer known. There is a pun on "locally trivial" in the titl... | https://mathoverflow.net/users/2821 | An everywhere locally trivial line bundle | The following example was provided to me by Colliot-Thélène some years ago : Let $X$ be the complement in $\mathbb{P}\_{1,\mathbb{Q}}$ of the three closed points defined by $x^2=13$, $x^2=17$, $x^2=221$. Then $\operatorname{Pic}(X)=\mathbb{Z}/2\mathbb{Z}$ but $\operatorname{Pic}(X\_v)=0$ for every place $v$ of $\mathbb... | 6 | https://mathoverflow.net/users/2821 | 11027 | 7,498 |
https://mathoverflow.net/questions/10993 | 14 | In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain properties. Examples include trying to classify (some class of) n-manifolds up to homeomorphism, or finite groups up to isomorph... | https://mathoverflow.net/users/2051 | Can you prove equivalence without being able to calculate it? | If you assume the axiom of choice, then every vector space has a basis, all bases of a given vector space have the same cardinality, and two vector spaces are isomorphic iff they have bases of the same cardinality. Now if the cardinality of the ground field is infinite, but smaller than the cardinality of the basis, th... | 20 | https://mathoverflow.net/users/1384 | 11028 | 7,499 |
https://mathoverflow.net/questions/10913 | 14 | Let $K$ be a finite extension of $\mathbf{Q}\_p$, with integer ring $R$ and residue field $k$. Say $G/R$ is a finite flat (commutative) group scheme of order $p^2$, killed by $p$. Say the special fibre of $G$ is isomorphic to the $p$-torsion in a supersingular elliptic curve over $k$. Is there some finite extension $L/... | https://mathoverflow.net/users/1384 | Lifting the p-torsion of a supersingular elliptic curve. | La réponse est oui. La raison est la suivante. Si $S$ est un schéma sur p est localement nilpotent, par définition (cf. thèse de Messing, chapitre I), un groupe de Barsotti-Tate tronqué d'échelon $1$ sur $S$ est un schéma en groupes fini localement libre sur $S$ annulé par $p$ tel que si $G\_0$ désigne la réduction de ... | 19 | https://mathoverflow.net/users/3052 | 11031 | 7,501 |
https://mathoverflow.net/questions/11025 | 11 | There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (and are both equivalent to X itself). I want to know why?
Background/Motivation
---------------------
Let $\Delta$ be... | https://mathoverflow.net/users/184 | Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set? | There are maps $|Sing(X)| \to |\underline{Sing}(X)| \to X$ which realize to weak homotopy equivalences. The inclusion of the n-skeleton $|Sing(X)|^{(n)}| \to |Sing(X)|$ is n-connected, because this is always true for CW-complexes, and so the map $|Sing(X)|^{(n)} \to X$ is n-connected. You can't really do any better tha... | 7 | https://mathoverflow.net/users/360 | 11034 | 7,503 |
https://mathoverflow.net/questions/11044 | 8 | Given 4 points ( not all on the same plane ), what is the probability that a hemisphere exists that passes through all four of them ?
| https://mathoverflow.net/users/3056 | What is the probability that 4 points determine a hemisphere ? | See [J. G. Wendell, "A problem in geometric probability", Math. Scand. 11 (1962) 109-111](http://www.mscand.dk/article/view/10655/8676). The probability that $N$ random points lie in some hemisphere of the unit sphere in $n$-space is
$$p\_{n,N} = 2^{-N+1} \sum\_{k=0}^{n-1} {N-1 \choose k}$$
and in particular you wa... | 9 | https://mathoverflow.net/users/143 | 11047 | 7,511 |
https://mathoverflow.net/questions/11045 | 19 | A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, which is Quillen equivalent to the usual model category structure on spaces.
The notion of test category was proposed by... | https://mathoverflow.net/users/437 | Are non-empty finite sets a Grothendieck test category? | That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left Kan extension/restriction along *both* adjunctions induced by your "forgetful functor" are Quillen equivalences (8.3.8).
... | 16 | https://mathoverflow.net/users/126667 | 11051 | 7,514 |
https://mathoverflow.net/questions/10997 | 14 | I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is called a projective resolution,
now pick one and use it to define the right derived functors of your
left exact functor". I w... | https://mathoverflow.net/users/391 | Founding of homological without quite involving derived categories | On rereading your question, Jacobson is not what you want after all.
Anton gave a very nice answer along these lines [here](https://mathoverflow.net/questions/1151/sheaf-cohomology-and-injective-resolutions). In comments to that post, Tyler Lawson recommends Cartan and Eilenberg.
| 2 | https://mathoverflow.net/users/297 | 11063 | 7,523 |
https://mathoverflow.net/questions/7775 | 23 | 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 through an "integral closure" that extends the notion of integral closure in degree zero for connective objects, and respects quasi-isomorphis... | https://mathoverflow.net/users/360 | Do DG-algebras have any sensible notion of integral closure? | I like this question a lot. It deserves an answer, and I really wish I had a good one. Instead, I offer the following idea. Maybe it has some merit?
### Background
Let me fix some terminology. Suppose $f:R\to S$ a homomorphism of (classical, commutative) rings.
1. An element $s\in S$ is said to be *integral over*... | 17 | https://mathoverflow.net/users/3049 | 11064 | 7,524 |
https://mathoverflow.net/questions/11059 | 12 | Let C be the category of topological monoids, that is, the category of [monoids](http://en.wikipedia.org/wiki/Monoid_%2528category_theory%2529) in (Top, $\times$).
1. Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy equivalence) be transferred to C along the free and forgetful p... | https://mathoverflow.net/users/2536 | Model Structure/Homotopy Pushouts in topological monoids? | The answer to question #1 is yes. You can use Kan's theorem on lifting model structures (11.3.2 in Hirschhorn's book) to obtain a model structure on $C$ such that the weak equivalences (resp., fibrations) are those morphisms of topological monoids that are weak equivalences (resp., Serre fibrations) on the underlying s... | 13 | https://mathoverflow.net/users/3049 | 11068 | 7,526 |
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