User yuhao huang - MathOverflow

Finitely generated monoids are finitely presented?

2010-08-25T04:27:14Z

I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and $P_0$ free commutative and finitely generated.

My question is:

Can we make $P_1$ maps to the kernel of $P_0\rightarrow M$? (to be more, precise, can we find a presentation of the form $P_1\rightarrow P_0\rightarrow M$ like the case for abelian groups?)

And if a monoid is finitely presented for one presentation, is it the same for other presentations? Thanks

What are p-adic period rings?

2012-03-15T00:20:46Z

I'm reading Illusie's survey on Crystalline cohomology, and I found him talking about those $p$-adic period rings like $B_{\text{dR}}, B_{\text{cris}}$. Can anybody explain what they are and give some heuristic on why they are what they are and what they are good for? So far I've only seen them appear in comparison isomorphisms. Are they just there to make the comparisons work?

I think I found a related question here: http://mathoverflow.net/questions/62454/fontaines-rings-of-periods but answers there simply didn't say much about it.

"Counter"-example for Gauss's Lemma on irreducible polynomials

2009-11-24T04:07:51Z

Gauss's Lemma on irred. polynomial says,

Let R be a UFD and F its field of fractions. If a polynomial f(x) in R[x] is reducible in F[x], then it is reducible in R[x].

In particular, an integral coefficient polynomial is irreducible in Z iff it is irreducible in Q. For me this tells me something on how the horizontal divisors in the fibration from the arithmetic plane SpecZ[x] to SpecZ intersects the generic fiber: a prime divisor (the divisor defined by the prime ideal (f(x)) in Z[x]) intersect the generic fiber exactly at one point (i.e. the prime ideal (f(x)) in Q[x]) with multiplicity one.

Now here is my question:

Give a ring R, with Frac(R)=F, and a polynomial f(x) in R[x] such that f(x) is reducible in F[x], but is irreducible in R[x].

Of course, R should not be a UFD.

I'd like to see an example for number fields as well as a geometric example (where R is the affine coordinate ring of an open curve or higher dimensional stuff). Thanks

Why does the (S2) property of a ring correspond to the Hartogs phenomenon?

2010-11-08T19:57:44Z

Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this is true. Can anybody explain this or give a heuristic argument?

Answer by Yuhao Huang for Topics for an Undergraduate Expository Paper in Number Theory

2012-09-25T20:27:59Z

How about Fibonacci numbers? Many of their mod $p$ properties can be easily deduced from arithmetic of finite fields. (e.g. $p|F_{p\pm 1}$, where the sign is determined by mod 5 class of $p$).

And similarly for Pascal's triangle. They have beautiful patterns mod $p$.

Pell's equation and continued fractions are also beautiful.

If the students know holomorphic functions, then Dirichlet's theorem on arithmetic progressions is a cool topic.

Characterization of schemes whose dualizing complex is perfect

2012-09-15T03:57:37Z

I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, the dualizing complex is a sheaf, when is this sheaf have a finite locally free resolution? (Or this never happens unless the scheme is Gorenstein?)

Recall a complex is called a perfect complex if it is locally quasi-isomorphic to a bounded complex of locally free sheaves.

How do you present a non-existence theorem?

2011-10-27T20:09:11Z

(This question might be too vague, feel free to edit or vote for closing.)

In math there are usually lots of non-existence theorems. When someone presents such a theorem, one natural response is "why shall I even care", or "why should such a thing be impressive".

The problem is, in the case of a non-existence theorem, usually all examples are trivial. If you tell some undergrad non-constant bounded entire functions don't exist, he/she will probably reply with a shrug. Similar thing happen to me and my friends when we talk about some fancier theorems (or when I see a paper stating such a non-existence theorem). I feel like it's really hard to convince people (or convince myself) "a priori this thing could exist, however by this awesome theorem it doesn't."

I think people would be impressed if all hypothesis look innocent, like the one in Liouville's theorem on entire functions, or maybe the fact that people have seen the existence of differentiable bounded non-constant functions helps. In general it is not necessary that all hypothesis look friendly. How would one figure out whether a non-existence theorem is a good one or is true just because one of the hypothesis is insanely strong?

Representation of Groupoids

2010-02-26T06:34:10Z

The title is vague, my actuall question is the following:

Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? (Either brand new things or pitfalls for those who too familiar with representations of groups). Does this point of view simplified any proof of theorems in representation of groups?

I've been only think of this for 15 mins. But I feel like it might be helpful to think of representation of groupoids for the following reasons:

When one talk about local systems, thinking of it as "representation of the fundamental groupoid" seems more natural than talking about "representation of the fundamental group".
When we talk about modules on stacks, if we choose a presentation of the stack (which is a groupoid), can we treat a given module as a representation of the groupoid? (Just as modules on BG gives representations of G.) Studying when two representations give the "same" module will be interesting.

What's the easiest example of a morphism of topoi that is not from that of a site?

2011-10-31T01:24:10Z

A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of sheaves. But I was told one of the cool thing about topos is that sometimes there are morphisms of topos that are not from morphisms of a site. When people talk about this they mention the word "crystalline"...

But is there a toy example I can play around with? What's the easiest example of this?

Why are finiteness conditions important (and how to recognize them)?

2011-05-02T18:09:49Z

I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a manifold to be compact, a sheaf to be coherent, and a complex to be bounded. And there are lots of good theorems once you assume some finiteness condition. (e.g. the Serre duality and Hodge decomposition for compact Kähler manifolds.) And removing those finiteness conditions seems to be non-trivial and interesting. (This applies to e.g. the two theorems mentioned above.)

So my question is,

why are finiteness conditions so important?

This question baffled me for a long time. I remember before I learnt compactness, when doing proofs in calculus, I felt I needed some finite covering of the closed unit interval, but I somehow thought I should avoid using that in my proof. Even after I learnt compactness for a while, the only thing I felt it gives me, was some "combinatorial advantage" ---- I mean, I didn't understand the necessity of assuming compactness in many theorems in elementary analysis, although I was sure I used it in the proofs and I can make some tricky counterexample if compactness wasn't assumed. [I don't feel I got a better understanding on that even now, I can only say I got used to it, i.e. assuming compactness then good things happen.]

The remarks on compactness also apply to me when I first learn the condition of a ring being Noetherian. Somehow the condition looks unnatural to me at the beginning, although after getting used to it I felt examples of non-noetherian rings are crazy.

And one more thing, I think one thing that Hartshorne/EGA make (early-level) readers confused is that they spent lots of time proving finiteness conditions, like proper pushforward of a coherent sheaf is coherent, or the cohomology of a coherent sheaf on a proper scheme over A is a coherent A-module. One can only appreciate them if he/she is sophisticated enough. (If you are about to prove these theorem in your algebraic geometry class, how do you motivate them and describe why people care about them?)

===============

A related question, maybe I should ask this in a separated thread, is, how do we recognize good finiteness conditions? Some are "easy", like compactness, and finite generation. But some are tricky, like the condition of a triangulated category being compactly generated. By recognizing good finiteness conditions one might hope to prove some good theorem, but how do we know whether the conditions are too restrictive or not? (I guess this requires hard work, but is there any convincing sign of a good condition before one dives into the details?) Anybody here knows the history of compactness (for topological spaces) and coherence (for sheave of modules)? [Judging by the name, coherent sheaves may come before quasi-coherent ones.]

Please re-tag it.

Can one find the hodge number by counting points over finite fields?

2012-04-03T03:15:59Z

Given a proper smooth variety $X$ of dimension $n$ over $\mathbb{C}$, assume it has a model over a DVR of mixed characteristic $(0,p)$ with residue field $\mathbb{F}_q$, and assume the closed fiber $X_0$ is smooth.

By the Weil conjecture, one can find the Betti number of the complex manifold $X^{an}$ by counting $\mathbb{F}_{q^r}$-points of $X_0$. If by counting points we find $|X_0(\mathbb{F}_{q^r})|=\sum\pm u_j^r$ for all $r$ and $b_i$ of the $u_j$'s has absolute value (in $\mathbb{C}$) equal to $\sqrt{q}^{i}$, then the $i$-th Betti number is equal to $b_i$.

My question is, can one find the Hodge number of $X$, which is $h^{ij}=\dim H^i(X,\Omega^j)$ by counting points of the closed fiber $X_0$? (The reason I'm asking this is, I guess both should connect to the theory of weights on the motive. So even if one cannot find the Hodge number this way, the reason must be interesting.)

Can one recover an A-algebra from its cotangent complex?

2011-11-03T19:48:19Z

Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor (forget, free-$A$-algebra).

As I understand it, the cotangent complex is a basic invariant of $A$-algebras --- like the complex of singular chains associated to a space. So I'm wondering if one can recover the original $A$-algebra from a cotangent complex (up to some equivalence).

In other words, if there is a map $B\rightarrow B'$ of $A$-algebras inducing a quasi-isomorphism of cotangent complex of $B'$ with the pull back of that of $B$, does this say something about the two $A$-algebras?

What I have in mind is something like the Whitehead theorem for CW complexes.

Is there a "knot theory" for graphs?

2011-10-30T00:46:32Z

I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) embeddings of graphs into R^3 or S^3.

If yes, can anyone show me any reference?

If the answer is basically no, then why? Is it just too hard, uninteresting, or can it be essentially reduced to the study of knots (and links)?

Can irreducibility of polynomials be figured out in polynomial time?

2011-08-17T04:47:05Z

I remember seeing somewhere "primarity test (of numbers) is harder than irreducibility test (of polynomials)", now as primarity test in polynomial time is known, can irreducibility test of polynomials over the integers be done in a fast way?

(I'm not sure if this is a well-defined question, as both the degree and coefficients can be large, maybe let me ask, can the primarity test of f(x) be done in $\text{O}(N^i(\log N)^j)$ operations, where N is f(m), with m = sum of absolute value of coefficients?)

Answer by Yuhao Huang for What elementary problems can you solve with schemes?

2011-05-04T04:43:46Z

I should mention the Hamilton-Caylay theorem for matrices: proof: base change to an algebraic closure of your underlying field, and use the fact that diagonalizable matrices are Zariski dense. (However, this doesn't used scheme that are glued from affines.)

What is known about the Picard scheme of a complete toric variety over C?

2011-04-22T06:16:55Z

Let $X$ be a complete toric variety over a field $k$. Its Picard scheme is defined to be the scheme representing the functor $Pic_{(X/k)(\text{fppf})}$, where $Pic_{X/k}: Sch_k^{op}\to Set$ sends a scheme $T$ over $k$ to the scheme $Pic(X\times T)/p_2^*Pic(T)$, and "(fppf)" denotes the sheafification of this functor w.r.t the fppf topology.

It is known that in this case the Picard scheme exists.

I want to know if there is any good combinatorial description of the Picard scheme. (We may assume $k=\mathbb{C}$ if this helps.)

Cohomological functor from triangulated category

2011-04-14T04:59:37Z

Say we have a cohomological functor F from a triangulated category $C$ to the category $Ab$ of abelian groups, e.g. $F=Hom(x,-)$, where x is an object in $C$. By definition, such a functor transform exact triangles into long exact sequence. And $F(y[i])$ is like the i-th "cohomology group" of the object $y$ w.r.t. the functor F. However, according to the philosophy of derived category, the right thing to look at is the complex, instead of the "cohomology groups". My question is, could there be a complex that "computes" these "cohomology groups"? e.g. Given two objects $x, y$ in $C$, can we find a complex that's like $RHom(x,y)$ whose cohomology computes $Ext_C^i(x,y):=Hom_C(x,y[i])$?

I would guess in full generality the answer is negative. (Otherwise it should have been done by Verdier.) But is there any mild or restrictive assumption that makes this true?

Or just for cohomological functors of the form $Hom_C(x,-)$, can we make the triangulated category $C$ enriched over $D^b(Ab)$ (such that the cohomology of the $Hom$ complex computes the old $Ext^i$'s?)

================

When I was writing this I feel like triangulated categories might not a good place to play game like this... If the above question isn't too interesting, could anyone just tell me what's the nature playground of questions like this? [Do dg-categories or $A_\infty$-category have an advantage on this?] Thanks.

Reverse Engineering to find deformation problem (from cohomology groups)?

2011-04-11T02:20:40Z

One of my favorite explanation of the cohomology groups of low degree is that they arise as the automorphism group, tangent space and obstruction space (or where the obstruction lives) of a certain deformation problem.

My question is, is it possible to reverse the process of going from deformation problem to cohomology groups? Say I have H^0, H^1 and H^2 of a certain sheaf, is it possible to find a deformation problem such that these groups control, i.e. such that they arises as the Automorphism-Tangent-Obstruction of that deformation problem?

To be more concrete, say we have a smooth scheme X over k, and our sheaf is the tangent sheaf T_X of X, how do I "find" the problem of "smooth deformation of X"?

(I know I have been pretty vague. I know how to interpret cohomology classes as gerbes/torsors, but I'm not quite satisfied (I like these interpretations though). I want a way to find deformation problem that produces gerbes as obstructions...)

Quote from wikipedia: Reverse engineering is the process of discovering the technological principles of a human made device, object or system through analysis of its structure, function and operation. [I hope my usage of this phrase is correct.]

Figure out the roots from the Dynkin diagram

2010-01-26T22:12:44Z

Just a d*mb question on Lie algebras:

Given a Dynkin diagram of a root system (or a Cartan Matrix), how do I know which combination of simple roots are roots?

Eg. Consider the root system of G_2, let a be the short root and b be the long one, it is clear that a, b, b+a, b+2a, b+3a are positive roots. But it is not clear to me that 2b+3a is a root just from the Dynkin diagram.

Does the derived category remember the homological dimension?

2011-03-05T06:48:16Z

Question:

Let $\mathcal{A}$ be an abelian category and $D^?(\mathcal{A})$ be its derived category, where ? could be empty, +, - or b (for boundedness). Is it possible to recover the homological dimension of $\mathcal{A}$ from the derived category?

Here I'm using the term homological dimension in the sense of Gelfand-Manin, i.e. if for all $X,Y\in\mathcal{A}$, $\text{Ext}_{\mathcal{A}}^i(X,Y):=\text{Hom}_{D(\mathcal{A})}(X[0],Y[i])=0$, then the homological dimension is said to be less than $i$. The homological dimension is the maximal $n$ such that there exists a non-vanishing $\text{Ext}^n(X,Y)$.

Note that in the derived category one could have all kinds of non-vanishing $\text{Ext}^n(X,Y)$, as $X,Y $ can be complexes shifted arbitrarily. Is it still possible to recover this information via other method?

Is it possible to do intersection theory in the derived category of a scheme?

2011-02-23T02:18:07Z

Let's say we are given a smooth scheme $X$ over $\mathbb{C}$, is it possible to do intersection theory in the (bounded) derived category of coherent sheaves?

I want to know if there is a way to extend the naive thought that for two subschemes $V$ and $Z$ of $X$, $\mathcal{O}_V \otimes^L \mathcal{O}_Z$ is the "intersection" of $V$ and $Z$. Reference guides are also welcome.

How many commuting nilpotent matrices are there?

2010-09-18T22:04:24Z

To be precise, fix $n$, fix a field $k$.

What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean all the products of matrices in this subspace are commutative.

(I feel like this can be formulated in terms of Lie algebras, but I don't find a good one. And I think the down-to-earth formulation might make it more accessible.)

Answer by Yuhao Huang for Generalize the Proj construction?

2010-12-03T07:50:57Z

My naive guess from the examples is the following:

Spec$k[P^{gp}]$ (which is just a product of $\mathbb{G}_m$'s) (somehow) acts on Spec$R$, and a GIT-quotient gives the construction you need, because the Proj construction is just a GIT-quotient of Spec$R$ by $\mathbb{G}_m$ w.r.t a certain linearization.

I'm not sure if the above construction generalizes directly, maybe some extra data is necessary.

Answer by Yuhao Huang for Global sections of $\Omega^{1} \otimes \mathcal{O} (2)$ over $\mathbb{CP}^{2}$

2010-11-29T05:10:21Z

Use the Euler sequence:

$0 \to \Omega^1_{\mathbb{P}^n_A/A} \to \mathcal{O}_{\mathbb{P}^n_A}(-1)^{\oplus n+1} \to \mathcal{O}_{\mathbb{P}^n_A} \to 0.$

Everything can be seen explicitly from here. Tensoring the sequence with $\mathcal{O}(2)$ gives an exact sequence, which is still exact if you take global sections because every monomial of degree 2 is a multiple of a monomial of degree 1.:) So the dim you want = 9-6 =3.

Is Mumford's GIT treated in SGA?

2010-11-29T03:31:55Z

I'm wondering to what extent is Mumford's Geometric Invariant Theory treated in the SGA volumes, and what's the Grothendieck point of view of GIT.

I looked at the TeXified and annotated version of SGA3 and found some reference to Mumford's GIT book in the modern annotation, especially in Exposé 5, quotient by groupoid. I think GIT is one of the major advances in Algebraic Geometry in Grothendieck's time, if the answer to my question is yes, then how (it is treated)? And if the answer to my question is no, then why (it is not treated)?

Also, has anyone seen Grothendieck's comments to GIT appeared somewhere?

How does one motivates the method of separation of variables when teaching PDE's?

2010-11-16T06:03:26Z

I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.

How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math behind it? It's not just because the guy who fancied it is very smart, right? (Although I feel like it does give students this impression...)

(Background: At Berkeley there is a course called Math 54, in which students learn linear algebra, linear ODEs and then 1 or 2 weeks of PDEs. Teaching Separation of variables is always my nightmare...)

What are unramified morphisms like?

2010-11-16T02:31:21Z

I'm wondering if finite unramified morphism between reduced schemes decomposes as closed immersions and etale morphisms. Suppose I have a morphism between reduced schemes which is finite, surjective and unramified, is it necessarily etale? I think this is certainly true if both source and target are curves, but I'm not sure about higher dimensional examples. Thanks

EDIT: to avoid trivial example let's assume the source and target are connected. What I'm wondering is precisely when one can deduce flatness from these conditions.

Calculate the zeta function of a scheme by from its étale covers?

2010-11-02T19:32:16Z

The title says it all. Given a (proper smooth) scheme $X$ over Spec$\mathbb{F}_q$, is it possible to calculate the zeta function of $X$ by from its étale covers?

Like for $\mathbb{P}^n$ you can cover it by $(n+1)$ copies of $\mathbb{A}^n$ which has as $k$-fold intersection of the form $\mathbb{G}_m^k \times \mathbb{A}^{n-k}$, and counting points are like doing the Inclusion-exclusion principle. Now I'm wondering if it can be done in the étale topology.

More concretely, given a scheme and a finite étale cover of it, how are their zeta functions related?

(BTW, I know there is an easier way to calculate the zeta function of $\mathbb{P}^n$ by using an affine stratification, the above example is just to illustrate my question.)

Why are the following varieties symplectomorphic?

2010-10-22T06:12:46Z

I saw a statement somewhere that for the Hirzebruch surfaces $F_n:=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(n))$, $F_n$ and $F_m$ are symplectormorphic when $m$ and $n$ have the same parity.

My question is: Why is this true?

I can see that they are diffeomorphic by Freedman's Theorem: computing the intersection pairing on $\text{Pic}(F_n)=H^2(F_n, \mathbb{Z})$, which is an even form when n is even, and an odd form when n is odd.

But this result is deep and abstract, is there any easy way to construct a symplectomorphism? Certainly it is not given by polynomial maps, I'm wondering what it will be.

What's a good dense open of $\bar{M}_g,n(X,\beta)$?

2010-10-22T07:28:11Z

The title says it all, what's a good dense open of $\bar{M}_g,n(X,\beta)$ which play the role of ${M}_g$ in $\bar{M}_g$?

My first (naive) guess is maps from a genus $g$ smooth curve to $X$ which represents the class $\beta$. But I'm a little bit concerned, is it dense for sure? Could it happen that in some cases one has singular curves only? Can a stable map from a singular curve always deform to a map from a smooth curve of genus $g$?

Comment by Yuhao Huang

2013-03-04T04:57:10Z

Normally the subscript should mean something the constant evolved in \ll depend on, but this case it looks weird.

Comment by Yuhao Huang

2012-10-05T02:10:12Z

Rank one and non-zero ==> generic isomorphism, and torsion-free ==> injective (The kernel is supported on a proper closed subset, thus torsion.)

Comment by Yuhao Huang

2012-09-15T06:47:16Z

Sure. I edited. What I wanted to ask is whether that $\omega_X$ being a perfect complex only happens in the Gorenstein case.

Comment by Yuhao Huang

2012-09-15T04:01:49Z

"...when the global space is normal." What does normal mean for analytic spaces?

Comment by Yuhao Huang

2012-09-15T03:50:04Z

What does G1 mean?

Comment by Yuhao Huang

2011-10-25T05:38:17Z

Being a bit picky, I think in this argument we need U to be quasi-compact. Otherwise I don't see why $$U\rightarrow X$$ is quasi-compact.

Comment by Yuhao Huang

2011-05-11T04:19:17Z

I don't think this is an interesting question, but I like the first picture. :)

Comment by Yuhao Huang

2011-05-04T03:31:21Z

That's an interesting point. I haven't thought about simplicial modules. Presumably the simplicial sets of a compact orientable manifold should be somehow distinguished from a random simplicial set (as the cohomology-ring functor factor thru taking simplicial set, one can expect some difference.)

Comment by Yuhao Huang

2011-05-02T23:21:54Z

And in what universe is compactness homotopy-invariant...?

Comment by Yuhao Huang

2011-05-02T23:16:44Z

@Shenghao, are you sure? I think it is only Quillen equivalent to the model category of CW complexes.

Comment by Yuhao Huang

2011-05-02T21:20:26Z

The Grothendieck ring of not-necessarily-finite-dimensional vector spaces is boring --------- this is just a fancy way of saying ∞ + 1 = ∞, right?

Comment by Yuhao Huang

2011-05-02T20:02:49Z

I really like the phrase "hygienic theorem". In fact I think in some sense "Property A implies property B" shouldn't be called a theorem, even if the proof is not trivial (maybe they can be called "Comparison", if one write down something showing A implies B and B doesn't imply A). But in this way we won't have many theorems...

Comment by Yuhao Huang

2011-05-02T18:13:15Z

After writing down the phrase "combinatorial advantage", I realized that I haven't seen any finiteness condition for simplicial sets, which capture combinatorial data. Has anybody here seen something like that (finiteness condition for simplicial sets)? I will open another thread for this if nobody answer it in comments.

Comment by Yuhao Huang

2011-03-30T08:02:13Z

People definitely use this (at least for holomorphic v.b. over algebraic varieties). And this gives the setup for Grothendieck Riemann-Roch. On a projective smooth variety K for holomorphic vector bundles is the same as K for the category of coherent sheaves.

Comment by Yuhao Huang

2011-03-08T06:04:12Z

Is this the first ad on mathoverflow?