\chapter{Nomenclature of pairings and forms}
Nondegenerate
Perfect pairing
Unimodular
Distinguish two things:
- a bilinear form on two different spaces, aka, a pairing
- a bilinear form on a single space
(related to quadratic form)
\section{Pairings}
A pairing is \Def{nondegenerate} if
for all $v \in V$, there is a $w \in W$ such that
$\left \neq 0$, and conversely:
for all $w \in W$, there is a $v \in V$ such that
$\left \neq 0$.
In other words, the maps $V \to W^*$ and $W \to V^*$ are injective.
A pairing is \Def{perfect} (a \Def{perfect pairing}) if
the maps $V \to W^*$ and $W \to V^*$ are isomorphisms.
A perfect pairing is modeled after the pairing between a vector space and its dual:
$V^* \times V \to K$.
For finite dimensional vector spaces, nondegenerate means perfect,
but for modules (and maybe infinite dimensional vector spaces?),
a pairing can be nondegenerate but not perfect.
For instance, the pairing $\bZ \times \bZ \to \bZ$
by $\left=2xy$ is nondegenerate but not perfect.
This example is clear (because we've written down coordinates),
but in general we apply this to abstract modules.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Examples}
A key example is Poincar\'e duality, which is a perfect pairing
on the homology of an oriented manifold
$$H_k(M;\bZ) \times H_{n-k}(M;\bZ) \to \bZ$$
(there are some issues with torsion).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Forms}
unimodular only makes sense on a fixed space;
it's the equivalent of perfect pairing