Questions tagged [duality]

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

Filter by
Sorted by
Tagged with
95 votes
36 answers
16k 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....
49 votes
4 answers
4k views

Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety ...
Yonatan Harpaz's user avatar
16 votes
1 answer
2k 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 ...
Pete L. Clark's user avatar
11 votes
1 answer
509 views

Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714). Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
Gabriel Nivasch's user avatar
3 votes
0 answers
73 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 $...
Isaac's user avatar
  • 2,727
3 votes
2 answers
762 views

Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
Question Mark's user avatar
35 votes
12 answers
3k views

No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
19 votes
4 answers
1k views

Applications of linear programming duality in combinatorics

So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
amakelov's user avatar
  • 987
12 votes
1 answer
553 views

Uniqueness of dualizing objects

One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
Mike Shulman's user avatar
10 votes
1 answer
976 views

Characterization of schemes whose dualizing complex is perfect

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, ...
Yuhao Huang's user avatar
  • 4,982
10 votes
3 answers
2k views

Dual cell structures on manifolds

Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $...
Matthew Kahle's user avatar
9 votes
2 answers
313 views

Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
M. Winter's user avatar
  • 12.5k
8 votes
2 answers
406 views

Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al

$\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\tensor}{\otimes}$ $\DeclareMathOperator{\Sp}{Sp}$ This question is about comparing the approaches for a formal Wirthmüller ...
Bastiaan Cnossen's user avatar
7 votes
1 answer
895 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
  • 2,174
7 votes
2 answers
240 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
7 votes
2 answers
763 views

Criterion for being reflexive via Ext

In this question it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a ...
Hans's user avatar
  • 2,863
6 votes
0 answers
128 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
geodude's user avatar
  • 2,129
6 votes
3 answers
1k views

opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$. Is ${op}$ the instance in Cat of a more ...
Bob's user avatar
  • 476
5 votes
0 answers
324 views

Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\...
AlexE's user avatar
  • 2,926
4 votes
1 answer
198 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
4 votes
1 answer
412 views

Which topological spaces admit embeddings into Euclidean spaces

I'm interested in the dual question to: continuous images of open intervals, about surjections onto open intervals. Namely, if $X$ is a topological space, when can we guarantee that there exists a ...
ABIM's user avatar
  • 5,019
2 votes
1 answer
722 views

Why is the Tate local duality pairing compatible with the Cartier duality pairing?

This question is a follow up to Why is the norm map dual to restriction under Tate local duality? Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
Question Mark's user avatar
2 votes
3 answers
3k views

dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
CodeGolf's user avatar
  • 1,837