I've started to read about toric varieties and I have a couple of questions about the definition. There is an example that says the following:
"Given a lattice $N$, an isomorphism $N \simeq \mathbb{Z}^n$ induces an isomorphism $$N \otimes_{\mathbb{Z}} \mathbb{C}^{*} \simeq (\mathbb{C}^{*})^n$$. The text defines this tensor product as the Torus of $N$.
My doubts are regarding the tensor product, I don't really have an intuitive understanding of what this tensor product means. Could anyone explain this or suggest any reading that could clarify my doubts?
Thanks!
At the moment I am reading "Toric Varieties", by Cox, Little and Schenk. Here are my thoughts about " what this tensor product means " in this scenario:
The lattice N is the set of one-parameter subgroups of the torus. In other words, N consists of curves on the torus that are also subgroups. The tensor product identifies points in the torus with specializations of these curves.
For more details I encourage you to read the first pages of the book.