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5
votes
1answer
388 views

Base change of trace for Gorenstein or Cohen-Macaulay morphisms

This is basically a question of functoriality for base change of CM morphisms. EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. ...
8
votes
0answers
260 views

Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples? Definition. Let $\mathcal{C}$ be a ...
2
votes
1answer
259 views

What is the dual of a pre-injective map?

In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...
4
votes
0answers
264 views

Dual of a weighted projective space

I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...
1
vote
3answers
333 views

Why the preimage rather than image in Stone-type dualities.

I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras. Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ ...
5
votes
3answers
291 views

Why are possibility and necessity dual?

Hello, Recently, I'm studying modal logic for my master's thesis, and my research background is category theory. So, I naturally have a question that why it is said that necessity (box) and ...
2
votes
2answers
526 views

How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ ...
2
votes
1answer
192 views

What is a de Vries algebra?

I've come across a set of slides by Guram Bezhanishvili where he claims the category of compact hausdorff spaces is related by duality to de Vries algebras. What are they?
1
vote
1answer
510 views

A bijective correspondence induced by Fourier Transform

Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection ...
2
votes
2answers
1k views

Dual Norm For Sum of 2-Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$ $\|\mathbf{x}\| = ...
5
votes
1answer
732 views

connections between Grothendieck's and Serre's duality

Hi, I would like to show that if $f: X \rightarrow Y=Spec \, \mathbb{C}$, where $X$ is a nonsingular complex projective variety, is the projection to a point, then the complex $f^! \mathcal{O}_Y$, ...
3
votes
2answers
561 views

Does equality of Hodge star and symplectic star imply Kähler structure?

Question The question asked is: On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does ...
5
votes
2answers
386 views

A name for a weak topology

Let $V$ be a real vector space and let $V'$ be the algebraic dual of $V$, i.e. the space of all the linear functionals $V\to\mathbb{R}$. Then there exists the weakest topology $\tau$ which makes all ...
1
vote
1answer
388 views

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows: ...
1
vote
1answer
203 views

In what sense is a generically submersive morphism of varieties subermersive over singular points?

Background/Motivation I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety. Let $V$ be a ...
7
votes
5answers
458 views

Dualizable classifying spaces

If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable. Are ...
0
votes
1answer
631 views

Proof that the Pontryagin dual of a topological group is a topological group

I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group. It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* ...
4
votes
2answers
934 views

Examples for “nice” Boolean algebras that are not complete or not atomic

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms). Boolean Algebras that are complete as ...
4
votes
0answers
165 views

Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem $$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$ where the functions $f_1$ and $f_2$ are concave but ...
3
votes
2answers
506 views

Alexander duality theorem for CW-complexes and stable homotopy theory

In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$ for finite CW-complexes with a "nice ...
1
vote
0answers
276 views

Generalized fourier transform and convolution?

Let $a(t)$ and $b(t)$ be two equal length sequences indexed by time index $t$. We know that $a(t) * b(t)$ corresponds to $A(\omega) \odot B(\omega)$ in the frequency domain where $A(\omega)$ and ...
2
votes
1answer
400 views

A category being self-dual vs. it being a groupoid

What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference? To go from a self-duality $F$ on a ...
38
votes
24answers
7k views

The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come. ...
1
vote
1answer
577 views

a “self-dual” adjunction

Is there a name for $(U,\eta)$ such that $(\eta, \eta^{op}):U^{op}\dashv U$ (is an adjunction). To clarify — $C:category$, $(I,I^{op})$ is the contravariant isomorphism with $I:C^{op}\to C$, ...
3
votes
0answers
536 views

Infinite Linear Programming

I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
4
votes
2answers
437 views

Continuous Transportation Problem

Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the ...
1
vote
1answer
211 views

“L^2_loc mod constants” as a reflexive space

In an article of Sten Kaijser ("A note on dual Banach spaces") I find the assertion that $E = L^2_{\text{loc}}({\mathbb R})$ modulo constants is a reflexive space. Question 1: which is the ...
4
votes
2answers
457 views

Tensor and Hom objects for finite flat group schemes

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to ...
10
votes
1answer
882 views

Questions about spectra of rings of continuous functions

I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
0
votes
2answers
628 views

Is division considered the mathematical dual of multiplication? [closed]

I'm doing a bit of research for a tech presentation that touches on the subject of mathematical duality. (To be clear, my presentation is not on mathematics or duality, but mentions duality in ...
12
votes
0answers
692 views

Frobenius upper shriek/flat of a dualizing complex

Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite. ...
5
votes
0answers
433 views

Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s?

Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so ...