# Questions tagged [duality]

Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

301
questions

0
votes

0
answers

74
views

### Local freeness of dualizing sheaf

I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...

2
votes

1
answer

138
views

### Tempered distributions at non-coinciding points and density of Schwartz functions

In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.
Let us consider the Schwartz space $\mathcal{...

4
votes

2
answers

412
views

### Is every closed subspace of the Schwartz space densely embedded into its dual space?

My original question is from this ME post but I think I need a broader understanding for this.
The Schwartz space $\mathcal{S}$ and its subspaces are examples of nuclear spaces. In fact, any closed ...

4
votes

1
answer

93
views

### Dual spaces of Banach-valued $L^{p}$-spaces

Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...

0
votes

1
answer

98
views

### Kernel of restriction in étale cohomology of curves over number fields

Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...

2
votes

1
answer

137
views

### In a monoidal category with duals is the coevaluation map determined by the evaluation?

For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...

2
votes

0
answers

236
views

### Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group.
Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence.
...

1
vote

0
answers

121
views

### Duality in Spc with ∧ and [-,-]

I am thinking about two duality theorems for H-spaces and their actions.
By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...

5
votes

1
answer

232
views

### Double hom with $\mathbb{CP}^\infty$

Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy.
$\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\...

4
votes

0
answers

112
views

### Length of dual module

It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...

4
votes

1
answer

174
views

### Lattice description of matroid duality

Apologies for this very basic question in matroid theory, but I could not find anything about it online after a bit of searching.
There is a well-known bijective correspondence ("cryptomorphism&...

4
votes

0
answers

138
views

### Examples of $\ast$-autonomous $(\infty,1)$-categories

A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...

3
votes

0
answers

74
views

### Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?

Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
and the inclusion maps are continuous with dense images. Here $...

3
votes

0
answers

84
views

### Tensor product of functors, central Hopf monad and star-autonomy

Setting.
Let $\mathcal{C}$ be a category and $(\mathcal{V},\otimes,I, \multimap)$ a (symmetric) closed monoidal category. Let $F:\mathcal{C}\rightarrow \mathcal{V}$ be a functor and $X\in \mathcal{V}$ ...

4
votes

0
answers

169
views

### Is the test function topology a Mackey topology?

I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...

1
vote

0
answers

71
views

### Knot invariants in WZW CFT via Holographic Principle

In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...

3
votes

1
answer

379
views

### Global duality theorem for 2-part

$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field.
Let $E/K$ be an elliptic curve over $K$.
Suppose finiteness of $\Sha(E/K)$.
According to Global duality ...

0
votes

1
answer

26
views

### Calculating vertex potentials from optimal matchings

Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...

2
votes

0
answers

53
views

### Tangential normal invariant isomorphism

Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...

2
votes

0
answers

144
views

### Why do von Neumann algebras possess identity?

My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...

2
votes

0
answers

132
views

### Explicit S-duality map

$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...

5
votes

0
answers

132
views

### Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)

Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...

7
votes

2
answers

248
views

### Double dual of free $\mathbb{Z}_{(p)}$-modules

For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D_{(p)}A=\text{Hom}(A,\mathbb{Z}_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $...

9
votes

1
answer

527
views

### Isbell Duality and Dualizing Scheme Objects

I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...

12
votes

2
answers

1k
views

### Does Poincaré duality preserve algebraic cycles?

Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...

1
vote

1
answer

46
views

### From relative convexity to modulus of continuity estimates for the dual gradient mapping

Let $F: \mathbf{R}^d \to \mathbf{R}$ be a convex function, let $m > 0$, and define $Q_m: \mathbf{R}^d \to \mathbf{R}$ to be the mapping $x \mapsto \frac{m}{2} \| x \|_2^2$. One says that $F$ is $m$-...

3
votes

0
answers

128
views

### Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$

(This question has also been asked on Math StackExchange.)
Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about ...

5
votes

1
answer

167
views

### Complemented subspaces of a dual Banach space

Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$.
My question reads as follows:
Does there exist $\kappa$ for which ...

1
vote

1
answer

166
views

### Homology of complements and homotopy equivalence

Albrecht Dold gave a short proof of the Jordan-Alexander complement theorem in the following form.
Given two closed sets $A$, $B \subset {\bf R}^n$ that are homeomorphic, the singular homology groups $...

1
vote

0
answers

141
views

### $BMO$ is not reflexive

It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is ...

1
vote

3
answers

457
views

### Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space?

I am aware that the dual of $C^\infty(\mathbb{R}^n)$ is the space of distributions (not necessarily tempered) with compact support.
However, if we fix a compact set $K \subset \mathbb{R}^n$, is the ...

4
votes

0
answers

424
views

### Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces

Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...

1
vote

1
answer

82
views

### Legendre transformation of vector valued function

Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?

6
votes

0
answers

155
views

### Explicit computations of Serre duality for elliptic curves

I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$
H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R.
$$
Clearly we have that $H^0(E, \Omega_E^1)=R \...

4
votes

0
answers

100
views

### Duality for finite quotient groups of finitely generated free abelian groups

$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...

2
votes

0
answers

85
views

### Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance:
https://math.stackexchange.com/a/14469/454016
Poincaré duality is explained through a duality of ...

10
votes

0
answers

503
views

### Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...

3
votes

1
answer

210
views

### Riesz representation theorem for duals of spaces of continuously differentiable functions

Let $k$ be a positive integer. I am looking for a possibly exhaustive reference discussing representation of dual spaces of $C^k_b(\mathbb{R}^d)$, $C^k_0(\mathbb{R}^d)$, or at least $C^k(K)$ for ...

1
vote

0
answers

62
views

### Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?

Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...

5
votes

0
answers

147
views

### Faltings' Cartier duality for A-modules in terms of Hopf algebras

$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its ...

10
votes

1
answer

317
views

### Finite domination and Poincaré duality spaces

Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...

4
votes

1
answer

202
views

### $\ast$-autonomous categories with non-invertible dualizing object?

1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...

1
vote

0
answers

48
views

### Application of the $\operatorname{BMO}$, $H^1$ duality

Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...

3
votes

1
answer

237
views

### When Alexander dual of a simplicial complex is a matroid?

Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$.
The Alexander dual $D(C)$ ...

4
votes

0
answers

177
views

### Spanier-Whitehead dual of space of natural transformations

Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...

2
votes

1
answer

96
views

### Is the polar dual of a semi-algebraic convex body also semi-algebraic?

Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as
$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$
with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a ...

7
votes

1
answer

954
views

### Compactness of the unit ball of a Banach space for topologies finer than the weak* topology

Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...

8
votes

1
answer

880
views

### On the Euler characteristic of a Poincaré duality space

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the ...

2
votes

1
answer

587
views

### Interpreting mincost flow dual variables

Consider the task of finding flow of size $b$ with minimum possible cost.
It may be formulated as linear programming in a following way:
$$\boxed{\begin{gather}
\min\limits_{f_{ij} \in \mathbb R} &...

0
votes

1
answer

167
views

### $f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere

Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?