User randy brown - MathOverflow

How does one compute induced representations for modular representations?

2011-06-17T23:05:49Z

The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its character). I want to decompose $Ind^G_H(\rho)$ into irreducibles. I am given character tables of both $G$ and $H$.

If $K$ were $\mathbb{C}$, Frobenius reciprocity (http://planetmath.org/encyclopedia/FrobeniusReciprocity.html) will do the trick. However, I am in the modular case; meaning: $char(K)||H|$. I still have all the character tables, except now they are Brauer character tables for the correct characteristic.

Is there a method for decomposing $Ind^G_H(\rho)$ into irreducible (Brauer) characters?

Edit: I wanted to make clear that since in the modular case we don't have Maschke's theorem, the ``decomposition'' into irreducibles would be in the Grothendieck group of Brauer characters of $G$. (representations of $G$ wouldn't nec. be direct sums of irreducible representations)

Map of Number Theory

2010-01-27T07:11:25Z

I've attempted going past basic number theory several times, and always got lost in its vastness. Do any of you, perhaps, know a good review that pieces together the many concepts involved (Hecke algebras, SL₂(ℤ), Fuchsian groups, L-functions, Tate's thesis, Ray class groups, Langlands program, Fourier analysis on number fields, cohomological versions of CFT, Iwasawa theory, modular forms, ...)?

Thanks.

Versality in deformation theory vs. versality in moduli spaces

2010-03-08T19:25:59Z

As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:

Deformation theory

"Moduli of Curves" defines a versal deformation space as a deformation $\phi: X \rightarrow Y$ such that for any other deformation $\xi: X \rightarrow Z$ and for every point in $Z$ there exists an open set (in the complex topology) $U$ such that the pullback of $\phi$ via $f: U \rightarrow Y$ is $\xi$ restricted to $U$. (I imagine that in general instead of an open set one takes open etale covers - is this true?)

Moduli Spaces

In moduli spaces, versality has always meant a space such that instead of the geometric points being in 1-1 correspondence with the objects we're interested in (over the field of the geo. point), each object is going have several geometric points in the versal space corresponding to it.

Question

Are these two notions related? If so - how?

Degrees of etale covers of stacks

2010-02-19T18:53:06Z

This is probably pretty basic, but as I said before I'm just beginning my way in the language of stacks.

Say you have an etale cover X->Y of stacks (in the etale site). Is there a standard way to define the degree of this cover? Here's my intuition: if X and Y are schemes, we can look etale locally and then this cover is Yoneda-trivial in the sense of http://front.math.ucdavis.edu/0902.3464 , meaning that etale locally it is just a disjoint unions of (d many) pancakes. Can we do this generally? Is there some "connectivity" conditions on Y for this to work? Is there a different valid definition for degree of an etale cover of stacks?

Yoneda-Triviality

I figured since nobody answered so far, maybe I should write down what a possible Yoneda-triviality condition could mean for stacks:

Def: Call f:X->Y (stacks) Yoneda-trivial if there exists a set of sections of f, S, such that the natural map Y(Z)xS->X(Z) is an isomorphism (or maybe a bijection?) for any connected scheme Z.

But I'm still clinging to the hope that there's a completely different definition out there that I'm just not aware of.

Decomposition of primes, where the residue field extensions are allowed to be inseparable

2010-03-30T22:37:11Z

I've been dealing with the following situation:

Let $R\subseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L \subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $\mathfrak{p}$ be a prime of $R$, and $\mathfrak{q}$ a primes of $S$ above $\mathfrak{p}$. Let $D_{\mathfrak{q}}$ denote the decomposition group, and $I_{\mathfrak{q}}$ the inertia group, of $\mathfrak{q}$ over $\mathfrak{p}$.

However, unlike in the classic case, I allow the residue fields of $\mathfrak{p}$ to be infinite, with positive characteristic. So the extension of residue fields may be inseparable.

It seems that the paper I'm reading implicitly assumes:

$|I_{\mathfrak{q}}|=e[\kappa(\mathfrak{q}):\kappa(\mathfrak{p})]_ i $ (the ramification index times the inseparability degree of the residue extension)

$|D_{\mathfrak{q}}|=e[\kappa(\mathfrak{q}):\kappa(\mathfrak{p})]$

$|G|=re[\kappa(\mathfrak{q}):\kappa(\mathfrak{p})]$ (where $r$ is the number of primes above $\mathfrak{p}$)

Is that right? I keep hitting walls when I try to prove it.

Deformations of Tame Coverings

2010-03-06T18:39:25Z

To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to generalize it:

Theorem

Let $R$ be a complete DVR, $X$ a proper smooth curve over $R$, and $D$ a simple divisor on $X$. Let $\bar f:\bar Y\rightarrow \bar X$ be a tame covering of $\bar X$ with branch locus contained in Supp($\bar D$). Then there exists a unique lifting of $\bar f$ to a D-tame covering $f:Y\rightarrow X$. In addition, $f$ is a tame covering of $X$, and the branch locus of $f$ is contained in Supp($D$).

Notation

I'm not really sure how much of it is standard, so here's what I mean by the terms in the theorem:

A simple divisor on $X$ - a divisor that has no multiple components when base-changed to any geometric point of the base scheme.

Tame covering of integral varieties over a field ($f:\bar Y\rightarrow \bar X$ in our theorem) - what you think: the ramification indices in codim 1 are coprime to the characteristic.

D-tame covering of schemes over a DVR ($f:Y\rightarrow X$ in the theorem) - if for every (natural number) $k$, base change to $R/m^k$ is $D\times_{R}R/m^k$ -tame in the following sense:

D-tame covering of schemes over an Artin local ring, $\Lambda$ ($R/m^k$ in the previous notation) - a finite, flat morphism which is etale away from Supp($D$)., and such that if $x$ is in Supp($D$), and t is a local equation for D at $x$, then $f_\ast(O_Y)_x$ is a t-tame extension of $O_x(X)$. What's that? $A'$ in the following, plays the role of $O_x(X)$ here.

t-tame extensions of a ring $A'$ s.t. $A'/Nil(A')$ is a DVR with parameter $\check t$ (and assume (0) is primary in $Spec(A')$, meaning $Spec(A')$ has no imbedded components)- tame when base changed to $A'/tA'$ (where $t$ is some element going to $\check t$). $A'/tA'$ is an Artin ring. So what do I mean by tameness of extensions over Artin rings? Well:

A tame extension of an Artin ring $\Lambda$ - it is a (finite) product of $\Gamma_i$'s, such that for each $\Gamma_i$ it is free over $\Lambda$ of rank $e_if_i$; the extension of residue fields is $f_i$; $e_i$ is prime to the characterstic of the residue field of $\Lambda$; and $\Gamma_i$ contains a principal ideal $J$ such that $J^{e_i}$=0, and such that $\Gamma_i/J$ is free over $\Lambda$ of rank $f$.

Question

Phew, that was a lot of work... So - other than finding the whole thing very confusing, here is a concrete set of questions:

1. Can this be generalized to $X$ a higher dimensional scheme?

2. What would be a good reference to read for this type of deformation theory?

3. Well - just about any insight and context you can give would be great (whether deformation theoretic or about the various generalizations of tameness here.)

This is a bit long winded, but if nothing else - it was good exercise for me to write this.

Degrees of subvarieties of projective space

2010-03-26T03:55:42Z

I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some projective space?

Answer by Randy Brown for How do you motivate a precise definition to a student without much proof experience?

2010-03-20T03:50:41Z

I like explaining that without definitions, they don't even know what they think they already know. For example: What is the definition of $2^3$? $2$ times $2$ times $2$. Right. What is $2^{-3}$? $1/2^3$, right. What is $2^{1/3}$? $3^{rd}$ root of $2$, right. What is $2^{\sqrt{2}}$? Nobody? What does it even mean?

Fiddling with p-adics

2010-03-06T22:36:05Z

A paper I'm reading implicitly assumes the statement: Let $K_0$ be the completion of $\mathbb {Q}_ p^{un}$. Then any finite extension of $K_0$ is complete with residue field $\bar {\mathbb {F}} _p$. So here are a few questions:

1. Why is this true? Is it true in general that if you complete, and then take algebraic closure - then that is complete?

2. Is it true that any finite extension of $K _0$ comes from the completion of a finite extension of $\mathbb {Q} _p ^ {un}$?

Group cohomology vs. topological cohomology in the case of non-trivial action

2010-02-18T16:42:35Z

When A is an abelian group with trivial G-action (G being a discrete group) we get that Hⁿ(G,A)≅Hⁿ(BG,A). Is there a similar connection between group cohomology and topological cohomology if A is a non-trivial ℤG-module? What can we say in that case?

Different interpretations of moduli stacks

2010-02-13T18:29:23Z

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to classify (smooth curves of genus g over ℂ, for example). Fine moduli spaces are defined to be the objects that represent the functor that takes an object and gives you the [set, for schemes; groupoid, as I understand it, for stacks] of ways that that object parametrizes families of the object we want to classify. Now, for schemes - this makes sense in the following way:

Let that functor be F, and let it be represented by M. Then F(Spec ℂ) are the families of (desired) objects parametrized by Spec ℂ (a point), so it corresponds to all desired objects (the ones we want to classify). But F(Spec ℂ) is also Hom(Spec ℂ, M), and so corresponds to the closed points of M. Thus M really does, intuitively, have as points the objects it wishes to classify.

Does this idea go through to moduli stacks? Of course, it probably does, and this is all probably trivial - but I feel like I need someone to assure me that I'm not crazy. So let me put the question like this: Can you formulate how to think of a fine moduli stack (as an object that represents an F as above; also: how would you define this F in the category of stacks?) in a way that makes it clear that it parametrizes the desired objects?

A specific branched cover of S^2 as a subgroup of Pi_1

2010-02-04T19:53:27Z

This is a follow-up question to: http://mathoverflow.net/questions/14024/degree-2-branched-map-from-the-torus-to-the-sphere

This is a silly computation, but for whatever reason this is taking me much, much longer than it should. So hopefully more geometrically-oriented people can answer this easily.

Let's say we have a 2-cover of S² branched at 4 points. We can visualize this, as stankewicz aptly put it, as having a torus, putting it on a skewer (that meets it in four points), and quotienting by the action of turning the torus by a 180 degrees around that skewer. If you prefer to think of the torus as the plane quotiented by a lattice, it's the same as identifying vectors with their minuses (the ramification points being the 2-torsion).

As we know $\pi_1$(Torus - 4 points,basept)$\cong$< a,b,c,d,e,f| [a,b]cdef=1>, and $\pi_1$(S² - 4 points,basept)$\cong$< g,h,x,w| ghxw=1>. This 2-cover corresponds to an identification, therefore, of < a,b,c,d,e,f| [a,b]cdef=1> with an index 2 subgroup of < g,h,x,w| ghxw=1>. For the life of me, I can't figure out how this identification would go! I've been drawing tori to no avail for two days now.

Degree 2 branched map from the torus to the sphere

2010-02-03T19:06:41Z

Algebraic geometry predicts a degree 2 branched cover from an elliptic curve to the projective line. What does this map look like topologically?

Minimal size of an open affine cover

2010-01-30T16:25:33Z

This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?

Answer by Randy Brown for Good Algebraic Number Theory Books

2010-01-28T22:44:14Z

Look at: http://mathoverflow.net/questions/13106/map-of-number-theory

The book recommended there (Manin/Panchishkin's "Introduction to Modern Number Theory") does seem amazingly comprehensive, and very readable.

Nonsingular/Normal Schemes

2010-01-22T20:17:35Z

I always had trouble remembering this. Is it true that a curve over a non-algebraically-closed field is normal implies that it's non-singular? How about a 1 dimensional scheme? How about dimension 2? I think I heard once that surfaces over a non-algebraically closed field is normal does imply that it's non-singular. Is it true for2 dimensional schemes? What is the reason that these theorems are true for small dimensions, but fail for higher dimensions?

Points and DVR's

2010-01-23T03:08:03Z

In the familiar case of (smooth projective) curves over an algebraically closed fields, (closed) points correspond to DVR's.

What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points? Is there a characterization of the DVR's that aren't induced by closed points?

And how about for a general projective variety that is regular in codimension 1 (both for algebraically closed and non-algebraically closed)? Point of codimension 1 induce DVR's. Do they induce all of them? What is the characterization of the ones they do induce?

How about complete integral schemes that are regular in codimension 1?

Computing Integral Closures

2010-01-13T23:58:03Z

I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of ℤ[x] in Quot(ℚ[x,y]/fℚ[x,y]), for some specific f(x,y).

Comment by Randy Brown

2011-06-18T16:04:33Z

I'm not sure what you mean... I am given all the character tables. Here's what I mean: $Ind^G_H(\rho)=\sum a_i\chi_i$. $(Ind^G_H(\rho),\chi_i)=a_i$, and so if we can compute all the $(Ind^G_H(\rho),\chi_i)'s$ then we're done. Frobenius reciprocity does indeed allow to compute those. I know what all the $\chi_i$'s are because I'm given the character tables of $G$ and $H$. Am I missing something?

Comment by Randy Brown

2010-06-12T16:18:31Z

@Robin: that link was in the body of the question.

Comment by Randy Brown

2010-06-12T16:09:31Z

The same reason you "square" a number. When you square it, you multiply only 2 copies of itself. But a square has 4 sides. It's a very interesting etymological issue, but not a math question.

Comment by Randy Brown

2010-04-18T05:12:13Z

If you don't read French, the beginning of Milne's "Lectures on Etale Cohomology" gives a concise framework, relating algebraic stuff with topological stuff. The relevant material is just a couple of pages.

Comment by Randy Brown

2010-04-05T04:30:09Z

Aha, so Hilbert irreducibility only implies that there are infinitely many points where it's a field, but not an open affine?

Comment by Randy Brown

2010-03-06T22:51:54Z

Thank you! I was not aware of Krasner's lemma.

Comment by Randy Brown

2010-03-06T19:44:08Z

Whoops, no. I'll edit that.

Comment by Randy Brown

2010-03-06T19:34:54Z

Thanks! I'll give it a look-see.

Comment by Randy Brown

2010-02-04T20:40:49Z

Can you give an explicit example of such an assignment? I can't make it work.

Comment by Randy Brown

2010-02-04T20:39:20Z

To be precise: the question is what would be an assignment for A, b, d, e and f that would give an index 2 subgroup?

Comment by Randy Brown

2010-02-04T20:12:07Z

So what are you saying a, b, c, d, e and f go to?

Comment by Randy Brown

2010-02-03T22:45:11Z

Can you give us some motivation?

Comment by Randy Brown

2010-01-27T06:49:58Z

Is there a way to search for reviews directly? (rather than going through a paper that it reviews)