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40
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. ...
12
votes
1answer
949 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 ...
4
votes
0answers
150 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 ...
2
votes
2answers
247 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 ...
1
vote
1answer
187 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 ...