I am trying to read the book "Compact Complex Surfaces" by Barth and friends. Maybe the starting point is a little bit high for me so even when I am reading the preliminaries I find it hard to follow.
The question I would like to ask:
Given a lattice $L$ (f.g. free $\mathbb{Z}$-module) with integral bilinear form $<\cdot,\cdot >$, if $L$ is unimodular (i.e. for any basis $\{e_i\}$ of $L$, the determinant of the matrix $(<e_i, e_j>)$ is $\pm 1$), then any element in its dual $L^\vee$ can be written in the form $<\cdot, x>$ for $x \in L$.
The book just moved on without explaining why this is true... I wonder if this is really obvious. The only thing this reminds me of is the Riesz Representation Theorem, but since $\mathbb{Z}$ is not a field, I cannot prove it in the same way. Otherwise I don't really know what the significance of unimodularity in this. Could anyone explain this to me?
I have never seen lattices like this before (but the discrete groups). I would greatly appreciate it if someone could suggest some books/notes to me.
You have a homomorphism $L\to L^\check{}$ sending $x\mapsto \langle \cdot,x\rangle$ and you want to show that it is an isomorphism. What is the matrix for this linear map with respect to the basis $\{e_i\}$ of $L$ and basis $\{{e}^i\}$ of $L^\check{}$? It suffices to show that the determinant of this matrix is $\pm1$. Here $e^i\colon L \to \mathbb{Z}$ denotes the map sending $e_i\mapsto1$ and $e_j\mapsto 0$ for $j\neq i$.