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most recent 30 from 2019-04-17T18:38:31Z

A question about combinatorial model categories

Mon, 12/31/2018 - 01:35

I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)

He uses this result to prove Jeff Smith's theorem, but the proof on the n lab ( does not seem to use such a technical preliminary result.

So I am wondering why does Lurie's proof is this much more complicated? And, if the added complexity is somehow necessary for the proof to work, could someone point out the "mistake" in the n lab and give reference?

Conditions on a local ring for f.g modules to have full support

Mon, 12/31/2018 - 01:05

Let $A$ be a noetherian local ring of Krull dimension $d$, and let $M$ be a finitely generated $A$-module. Assume $M$ also has Krull dimension $d$. What are some conditions on the ring $A$ that will ensure that the support of $M$ is equal to $Spec(A)$?

Edit: it was pointed out in the comments that this happens exactly when $A$ is only one minimal prime ideal. I would then like to ask: given a noetherian ring, when is it the case that all of its localizations have this property that they have only one minimal prime?

An inequality involving $L^1$ and $L^\infty$ norms of a function of a real variable and its derivative

Sun, 12/30/2018 - 16:06

I got to the following inequality by a (hopefully correct) tortuous argument:

If $F:[a,b] \to \mathbb{R}$ is a absolutely continuous monotone function then: $$ \|F'\|_1^2 \leq 4 \|F\|_1 \, \|F'\|_\infty $$

Question 1: Does this inequality have a name?

Question 2: Is there a short proof?

Remark: Cases of equality should be with $\pm F$ like that:

(The plateau in the middle may vanish; in this case $F$ is affine with zero mean.)

Quaternion algebra-action on $\ell$-adic cohomology

Fri, 12/28/2018 - 08:15

Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).

For every prime $\ell\neq p$, there is a faithful $\ell$-adic algebra-representation:

$$\rho_{\ell} : H\to \text{End}_{\mathbf{Q}_{\ell}}(V_{\ell}(E))$$

where $V_{\ell}(E) := (\varprojlim_{n\ge 0} E_{\overline{\mathbf{F}}_p}({\overline{\mathbf{F}}_p})[\ell^n])\otimes_{\mathbf{Z}_{\ell}}\mathbf{Q}_{\ell}$ is the rational $\ell$-adic Tate module of $E$.

  • Using the natural isomorphism $H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell}) \simeq \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(E),\mathbf{Q}_{\ell})$ we have an $H\otimes_{\mathbf{Q}}\mathbf{Q}_{\ell}$-module structure on $H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$, as is classically known.

  • Now we make a more naive construction. Functoriality of the étale site of $E_{\overline{\mathbf{F}}_p}$ gives that for any endomorphism $f : E\to E$ there is a map $$H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},f^{-1}\mathbf{Q}_{\ell})$$ and since $\mathbf{Q}_{\ell}$ is constant (here I am being imprecise about the nature of $\mathbf{Q}_{\ell}$, which is not a constant sheaf on the étale site, but the meaning is clear from the context) we also have an isomorphism $H^i(E_{\overline{\mathbf{F}}_p},f^{-1}\mathbf{Q}_{\ell})\simeq H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$, and we call $$f^* : H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})\to H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$$ the composition of the two. In other words, every element $f\in\text{End}(E)$ has an effect $f^*$ on $H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$. On the other hand, the effect of each element of $\mathbf{Z}\subset\text{End}(E)$ on $H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ is invertible, and so the above construction defines, for every element $f\in H$, an effect $f^*$ on $\ell$-adic cohomology.

Does the construction in the second point, for $i=1$, give an action of $H$ on $H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ as an algebra? If so, does this action agree with the one constructed in the first point?

My expectation is that the answer to the first, and hence second, question is no.

The second construction should only define on $H^i(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$ the structure of representation of $H^{\times}$, and it should not be possible to upgrade this to an algebra action of $H$.

Existence of the $\ell$-adic Tate-module functor, and the fact that for abelian varieties $A,B$ the map $\text{Hom}_{\rm AV}(A,B)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}\to \text{Hom}_{\mathbf{Q}_{\ell}}(V_{\ell}(A),V_{\ell}(B))$ is injective, should be crucial to have an algebra action of $H$ on $H^1(E_{\overline{\mathbf{F}}_p},\mathbf{Q}_{\ell})$, and it feels it should not be possible to construct it just as a consequence of functoriality of the étale site, a much less deep fact.

Road map to learn Deformation theory

Fri, 12/28/2018 - 08:01

Question is as in the title.

What is the road map to learn Deformation theory?

What are the prerequisites for doing that?

I am familiar with scheme theory and very little Cohomology from Hartshorne’s Algebraic geometry book.

You can also see this question for Deformation theory tag wiki.

There seem to be more Deformation theory than that of Deformation theory of Schemes. I have seen questions under this tag and it has questions from category theoretic point of view, Lie algebra theoretic view and much more.

Characterizing the Graph of a Line as an Equivalence Class

Fri, 12/28/2018 - 07:49

I'm reflecting on something rather abstract, but which gave me a new insight.

I'm wondering if a line, (consider here in $ \mathbb{R}^2 $ Euclidean Space), represents in fact a coordinate-wise equivalence relation (given by parametric representation of the line) between any two distinct points in the plane?

The way I formalized this:

Let $ f: X \to Y $, and fix the unity point on the line: $ u = (1, f(1)) $.

Let's define the relation $ R $:

  • $ \forall (a,b) \in X^2 $ and pairwise distinct, we have, for some $ \lambda $ real:

$$ a R b \implies [ a_x = \lambda b_x \implies f(a_x) = \lambda f(b_x) ] $$

which is true on any Archimidean Field.

So, we can define a line $(L)$ as the set: $ \{ (a,f(a)) | aRu \} $, or just $ \{ \dot{f(1)} \} $.

This definition may imply other conclusions:

  1. I wanted to study if we could generalize this kind of approach for any curve, it seems it's not valid.

  2. But along the way, it gave me insights about a further study case: Could it be valid in $\mathbb{C}$? If yes, maybe it could allow to define a class of morphisms between $\mathbb{R}$ and $\mathbb{C}$

Thank you for any hint and comment for this approach.

How to count poles for rational functions in a compact Riemann surface?

Fri, 12/28/2018 - 07:34

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in a Riemann surface like that $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$.

I'm doing an extensive research in previous references looking for a formula that could give to me that number and I kind of feel that it is just a language problem between algebraic-geometry and the papers I'm dealing with. Am I Right?

Sometimes in my research, the calculation of allowed poles showed up like a simple statement that makes me feel that perhaps it could be a basic fact in algebraic geometry. I'll give an example simply copy pasted from one of papers I'm dealing with:

Proposition 1: Let $p(t)=\sum_{i\in\mathbb{Z}}a_it^i\in\mathbb{C}[t]$ and $R=\mathbb{C}[t,t^{-1},u]/\langle u^2-p(t)\rangle$.

The number $n$ where poles are allowed in $R$ depends on $p(t)$ according to the formula $n=4-r$ where $r$ is the number of ramified points in $\{0,\infty \}$: $0$ is ramified exactly when the constant term $a_0=0$, and $\infty$ is ramified exactly when the degree $d$ is odd.

Even opening all the references I cound't find some talking explicit about how to reach the Proposition 1 (that I call the hyperelliptic case) . My research colleges working in algebra also doesn't know how to find it.

I would like to know how to count the allowed poles in $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t) \rangle$ (that I call the superelliptic case). Is there any formula?

Extended Abel-Jacobi theorem

Fri, 12/28/2018 - 07:22

Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole or zeros on $\{P_1, ..., P_n\}$. Let $d\log(u)$ be the logarithmic differential form of $u$. Do we have the following equality in $\mathbf{C}^{\times}$: $$\exp(\int_{div(f)} d\log(u)) = \prod_{i=1}^n f(P_i)^{\text{ord}_{P_i}(u)}\text{ ?}$$ Here, $\text{div}(f)$ is the divisor of $f$, and the integral is well-defined (independant of the choice of the path) up to $2\pi i \mathbf{Z}$.

This identity is true if for all $i$ we have $f(P_i)=1$, as it relates to the analytic Abel-Jacobi description of the generalized Jacobian of $X$ with respect to the divisor $(P_1)+...+(P_n)$.

Influential results by Swinnerton-Dyer

Fri, 12/28/2018 - 06:07

The conjecture of Birch and Swinnerton-Dyer had a tremendous influence on the development of arithmetic geometry. Which other results of Swinnerton-Dyer have had a lasting influence?

Perpendicular bisector in trapeze

Fri, 12/28/2018 - 06:00

I am solving a task and I need your help.

ABCD - isosceles trapezium (AB || CD)

I have to prove that AB and CD have common perpendicular bisector.

Thank you in advance!

Eigenvalue and Eigenmatrix of a 3D Tensor - How to calculate it?

Fri, 12/28/2018 - 06:00

How to calculate easily the eigenmatrix of a 3D tensor.

I try immersing the tensor in a big matrix, in my case, the tensor is of nxnxn and I can build an n^2 x n^2 matrix that contains all the "coefficients" of my original tensor, but my calculations give me more eigenvalues that I have in my original problem (n^2).

There exist any numerical library for eigenvalues/eigenmatrices of a tensor?

Spectral theory without topology

Fri, 12/28/2018 - 05:38

How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ?

Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Fri, 12/28/2018 - 05:02

Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$.

  1. For each monic $\varphi \in A[x]$, there's an induced surjection (bijection) $$Z(A,\varphi)\longrightarrow Z(B,f(\varphi)).$$
  2. For each monic $\varphi\in A[x]$ the boolean algebra morphism $\mathrm{idemp}(A[x]/(\varphi))\to \mathrm{idemp}(B[x]/(f(\varphi)))$ is surjective (bijective).
  3. (Compare 09XI) For each integral base change the induced boolean algebra morphism between idempotents is surjective (bijective).

Are these properties equivalent? The motivation for my question comes from trying to get an overview of the notion of Henselian. There are many bits and pieces which I hope are underlain by equivalence of the above conditions. I am especially hopeful for "direct" proofs that go through as few other conditions as possible.

Forgive me if this question is too elementary for MO.

On different definitions of a prequantization space

Fri, 12/28/2018 - 04:32

Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).

Without talking about curvatures of connections and hermitian line bundles, one can also define a prequantization space over $(M,\omega)$ as a principal $S^1$-bundle $\pi : (V,\alpha) \to (M,\omega)$, where $\alpha$ is an $S^1$-invariant $1$-form on $V$ satisfying $d \alpha = \pi^* \omega$. These two conditions imply that $\alpha$ is a contact form.

I am looking for explanations or references regarding the following questions:

  1. it is said everywhere that the existence of such prequantization bundle is equivalent to the cohomology class of $\omega$ lying in the image of the natural homomorphism $$H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R}).$$ I know that it has to do with some identification involving the first Chern class, but I can't find any good reference or detailed proof about this anywhere;
  2. I would also like to know what classifies such bundles over a given symplectic manifold;
  3. I would like to find a detailed proof of the equivalence of these two definitions;
  4. Is there a physical meaning to the fact that $\alpha$ is a contact form ?

What minimal structure is required to define Clifford modules in a way as abstract as possible?

Fri, 12/28/2018 - 02:52

Start with a quadratic form $q$ on a vector space $V$. A module $M$ over the corresponding Clifford algebra is determined by a map $\cdot:V\otimes M\to M$ satisfying $v\cdot(v\cdot m)=-q(v)m$.

Now try to abstract this as follows. The bilinear form $B(x,y)=q(x)+q(y)-q(x+y)$ determines a natural transformation $\varepsilon:TT\to\text{identity}$, where $T$ is the endofunctor $T=V\otimes-$ on vector spaces. A Clifford module structure on $M$ in these terms is a morphism $\mu:TM\to M$, and - here starts my question - certain relationship between the composite $\mu\circ T\mu:TTM\to TM\to M$, and $\varepsilon_M:TTM\to M$.

The question is what minimal structure does one need to capture this relationship. Seemingly either some kind of nonadditive transformation $\delta:T\to TT$ is needed to express $v\mapsto v\otimes v$, or some kind of self-distributive law $\text{switch}:TT\to TT$. In the latter case however one seemingly needs the additive structure to express $x\cdot(y\cdot m)+y\cdot(x\cdot m)=B(x,y)m$.

Has any of this been carried out somewhere? Is it possible to avoid the additive structure, at least using some restrictions? For example, if $B$ is nondegenerate, the endofunctor $T$ will become self-adjoint, maybe one can use this somehow, I don't know how.

As Liviu Nicolaescu points out, this probably needs some motivation. My motivation is purely abstract-nonsensical in this case. It is known that the category of modules over any algebra can be uniquely (up to equivalence) determined by an abstract category-theoretic universal property. This is because for an algebra $A$ the functor $A\otimes-$ gets a monad structure, and the category of algebras over a monad is a lax limit in the well known way.

Now for a Clifford algebra, the monad is very special, so that the category of algebras over this monad is equivalent to another category with objects determined by more concise data. I have only described part of these data, but morally this looks like (lax (left)) categorification of the fixed point set of an involution. And the question can be reformulated as follows - given a natural transformation $\varepsilon:TT\to\text{identity}$, is there a category-theoretic universal construction (some sort of lax limit again, presumably) that would yield the category of Clifford modules in the particular case when $T=V\otimes-$ and $\varepsilon$ is induced by a $q$ as above?

Uniform inequality for an analytic perturbation

Fri, 12/28/2018 - 02:47

Let $T$ be a bounded linear operator acting on a complex Banach space. Suppose that $T$ has spectral radius strictly less than $1$. If we introduce an analytic perturbation to $T$, $s\mapsto T_s$ for $|s|<\epsilon$ (with $T_0 = T$), then by upper-semicontinuity of the spectrum, assuming that $\epsilon$ is sufficiently small, the spectral radius of each $T_s$ is less than $\rho$ for some $\rho <1$. My question is the following:

Is it possible to find $K>0$ and $\rho <\rho '<1$ such that $$\|T_s^n\|\le K (\rho ')^n,$$ for all $|s|<\epsilon$ and $n\in \mathbb{Z}_{\ge 0}$?

This certainly seems like it should be true but I can't find a proof - I think I'm missing something obvious. Any help would be greatly appreciated - cheers!

Finding a good transversal basis

Fri, 12/28/2018 - 02:46

A hypergraph $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A transversal of $H$ is a set $T\subseteq V$ such that $|T\cap e| = 1$ for all $e\in E$.

It is easy to see that transversals need not exist: Take $V = \{0,1,2\}$ and let $E$ be the collection of $2$-element subsets of $V$.

A transversal basis is a set $B\subseteq V$ such that $|B\cap e|\leq 1$ for all $e\in E$. Setting $I_B:=\{e\in E:B\cap e\neq \emptyset\}$, we say that that a transversal basis $B$ is good if for all transversal bases $B_1$ with $I_{B}\subseteq I_{B_1}$ we have $I_B=I_{B_1}$.

Question. Does every hypergraph $H=(V,E)$ have an good transversal basis?

Sets are a popular construction in programming languages. Is there a solid theory behind it?

Thu, 12/27/2018 - 23:27

What is known as Set Theory does not have much to do with "sets" as used in programming languages. In programming, one needs something like a pair: a predicate and, probably, a complete order, with an equivalence relation that only uses predicates.

Or there could be other approaches.

Is there a theory behind all this that programming languages model?

Swinnerton-Dyer passed away? [migrated]

Thu, 12/27/2018 - 23:02

His wiki article has been updated to say that he passed away recently. Are there any other sources confirming that? Thanks.

If it is more appropriate for another stackexchange site, please feel free to move the question there.

Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Thu, 12/27/2018 - 22:55

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following product evaluation.

If $T(n)=\frac{(3n-2)(n-1)}2$ and $i=\sqrt{-1}$ then $$\prod_{j<k}^{0,n-1}(\eta^k-\eta^j)=n^{\frac{n}2}i^{T(n)}.$$