219,358pages on
this wiki

# 2009 October 6th - questions

#### 1 Sune

Is it possible to explicitly define (without AC) a total ordering on a set with cardinality greater than $\mathbb{R}$?

#### 2 Yuhao

Let $K$ be the function field of an algebraic curve $C$. Then by the birational nature of the genus, it determines the genus $g$ of the curve. How can we see $g$ directly from the field?

(One way is to PICK a function in $K$ and we get a ramified morphism $C \rightarrow P^1$, then use Riemann-Hurwitz to compute the genus; However, I think there should be a more "intrinsic" way to see that, i.e. without picking a function in an arbitrary manner. The genus is a well-defined invariant for any extension of the ground field of tr.deg. 1, right?)

#### 3 Anon

Determine explicitly a partition of the plane into two sets A and B such that neither A nor B contains (the image of) a non-constant continuous curve.

#### 4 An H

Let $K$ denote the cyclotomic field obtained by adjoining $p$th roots of unity to $Q$, where $p$ is prime. Let $q$ be a prime element in the ring of integers of $K$ which is coprime to $p$. Let $L$ denote the Kummer extension of $K$ by adjoining a $p$th root of $q$ to $K$.

Question: is it always true that $p$ divides the exponent of the prime ideal $(1-\zeta_p)$ in the relative discriminant $D_{L/K}$? (as usual, $\zeta_p$ denotes a primitive pth root of unity)

#### 5 Critch

In the categories Set and AbGrp (and I believe any topos), finte limits (e.g. kernels, products, equalizers...) commute with directed colimits (aka direct limits). Give an example category where this fails.

#### 6 Anton

Can a coequalizer in the category of Schemes fail to be surjective? (Note: it must hit all closed points in the target, because otherwise the closed point could be removed to make the coequalizer smaller.)

See the mathoverflow.net question: [1]

#### 7 Darsh

Let $f: \mathbb{C}\cup\infty \to \mathbb{C}\cup\infty$ be rational.

In what generalized notions of convergence does the sequence $x,f(x),f^2(x),\ldots$ converge?

#### 8 Harold

In the category of smooth manifolds, when does the fibre product of two maps exist?

A) Necessary conditions

B) Sufficient conditions (e.g. that each map is a fibration).

#### 9 James T

Which bounded linear maps on a Hilbert space are the exponentials of other maps? I.e., what is the image of the map $exp: B(H)\to B(H)$?

#### 10 Pablo

Give a simple example of a (necessarily infinte dimensional) Lie algebra that is not the lie algebra of any (necessarily infinite dimensional) lie group.