Let $f: A → B$ be a homomorphism of a finite abelian groups. For a pairing $A × A → \Bbb{Q}/\Bbb{Z}$, we denote the orthogonal complement of $\mathrm{Ker} f ⊂ A$ by $(\mathrm{Ker} f)^⊥$. When the pairing is non-degenerate, is it true that $\#(\mathrm{Ker} f)^⊥=\#\mathrm{Image}(f^*)$ ? Here, $f^*...

I am having trouble understanding this claim about the existence of the sets $U_j$ in the following theorem from Spivak's differential geometry textbook. I tried proving like this: Firstly, we may take a covering which has no subcover and refine it to produce open sets $U_j$ diffeomorphic to $\...

Let $f: A → B$ be a homomorphism of a finite abelian groups. For a pairing $A × A → \Bbb{Q}/\Bbb{Z}$, we denote the orthogonal complement of $\mathrm{Ker} f ⊂ A$ by $(\mathrm{Ker} f)^⊥$. When the pairing is non-degenerate, is it true that $\#(\mathrm{Ker} f)^⊥=\#\mathrm{Image}(f^*)$ ? Here, $f^*...