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Questions tagged [duality]

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

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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 ...
KAK's user avatar
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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{...
Isaac's user avatar
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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 ...
Isaac's user avatar
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4 votes
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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}(\...
G. Blaickner's user avatar
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1 answer
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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: \...
kindasorta's user avatar
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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': ...
Yilmaz Caddesi's user avatar
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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. ...
Duality's user avatar
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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 ...
user avatar
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}^{\...
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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 \...
JBuck's user avatar
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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&...
Sam Hopkins's user avatar
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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 $[-,-]$ ...
Max Demirdilek's user avatar
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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 $...
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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 $X\in \mathcal{V}$ ...
Max Demirdilek's user avatar
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^{\...
Jon's user avatar
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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 ...
Student's user avatar
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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 ...
Duality's user avatar
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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 ...
Manfred Weis's user avatar
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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 ...
Sagnik Biswas ma20d013's user avatar
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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 ...
MrPajeet's user avatar
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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^...
Sagnik Biswas ma20d013's user avatar
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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 ...
Gro-Tsen's user avatar
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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 $...
Neil Strickland's user avatar
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}\...
LiminalSpace's user avatar
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 ...
user45397's user avatar
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1 vote
1 answer
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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 $m$-...
πr8's user avatar
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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 ...
David Gao's user avatar
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1 answer
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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 ...
Damian Sobota's user avatar
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 $...
coudy's user avatar
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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 ...
Bazin's user avatar
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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 ...
Isaac's user avatar
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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 ...
Andromeda's user avatar
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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 $f: \mathbb{R}^n \to \mathbb{R}^n$?
Dmitry Vilensky's user avatar
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 \...
marco's user avatar
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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 ...
Mikhail Borovoi's user avatar
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 ...
Alexander Praehauser's user avatar
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) ...
Ilk's user avatar
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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 ...
ajr's user avatar
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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 $\...
Anil Bagchi.'s user avatar
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 ...
Homotopy theorist 's user avatar
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 ...
John Klein's user avatar
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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 ...
Max Demirdilek's user avatar
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\...
Ferry Tau's user avatar
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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 $D(C)$ ...
Connor's user avatar
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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). ...
stableunknown's user avatar
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 ...
M. Winter's user avatar
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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 $\...
Goulifet's user avatar
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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 ...
John Klein's user avatar
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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} &...
Oleksandr  Kulkov's user avatar
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 $\Omega$?
Billal Elhamza's user avatar

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