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

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12
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0answers
723 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. ...
10
votes
0answers
366 views

Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
9
votes
0answers
273 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 ...
7
votes
0answers
132 views

Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
6
votes
0answers
466 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 ...
5
votes
0answers
167 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 ...
4
votes
0answers
61 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 ...
4
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0answers
211 views

Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
4
votes
0answers
288 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 ...
4
votes
0answers
643 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 ...
3
votes
0answers
133 views

Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...
3
votes
0answers
55 views

Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...
3
votes
0answers
52 views

Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space. A poset $(P,\leq)$ is called ...
3
votes
0answers
130 views

Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in ...
2
votes
0answers
62 views

Conditions under which the dual function is self-concordant

Consider the following optimization problem \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,m\\ \nonumber \quad&x\in ...
2
votes
0answers
113 views

Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$

Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
2
votes
0answers
85 views

Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
2
votes
0answers
147 views

Reference Request: Algebraic Serre's Duality Theorem for Curves

Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne ...
2
votes
0answers
76 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
2
votes
0answers
152 views

compact embedding in duals of weighted Sobolev spaces

On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding $$ L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset ...
2
votes
0answers
284 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 ...
1
vote
0answers
66 views

How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$. Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in ...
1
vote
0answers
24 views

Duality of plurisubharmonic functions

Let $F$ be a cone of upper bounded upper semicontinuous functions on a compact set set $X$ containing all the constants. Let $z\in X $ and define a class of positive measure by $$M_z^F=\{ \mu : ...
1
vote
0answers
79 views

Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...
1
vote
0answers
134 views

Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?

Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare. Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where ...
1
vote
0answers
62 views

hyperfunctions and analytic duals

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
1
vote
0answers
317 views

Matlis' dual of injective modules

Let $(R, \mathfrak{m})$ be a commutative Noetherian complete local rings ($R$ can be regular, if you need). Let $E(R/\mathfrak{p})$ be injective hull of $R/\mathfrak{p}$, if $\mathfrak{p}= ...
1
vote
0answers
67 views

Looking for a uniform explanation of algebras with canonical generators.

Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...
0
votes
0answers
107 views

Duality results for quadratic equality constrained optimization problem

Consider the optimization problem $$ \begin{align} \min_{x\in\mathbb{R}^n}&\quad f(x) = x^TA_0x+b_0^Tx+c_0\tag{P1}\\ \nonumber \text{subject to } \quad & g_i(x) = x^TA_ix+b_i^Tx+c_i = ...