# Algebraic Torus: Etymology

I was wondering if anyone might be able to tell me why algebraic tori are called specifically algebraic tori? I find it difficult to see exactly how "an algebraic group that can be described as a direct product of finitely many multiplicative groups" is in any sense similar to the more standard notion of a doughnut.

Any and all help is appreciated.

• The algebraic $n$-torus $(\mathbb{C}^*)^n$ is homotopy-equivalent to the topological $n$-torus $(\mathbb{S}^1)^n$. From here it's easy to generalize to arbitrary rings $R$ – leibnewtz Jun 5 '18 at 18:10

## 1 Answer

Consider $S^1×S^1.$ This is a torus, and the product of two multiplicative groups (the complex numbers of modulus $1.$)

• If I'm reading the Wikipedia article correctly, one requires the group to be a produce of entire multiplicative groups. $S^1$ is not the multiplicative group of any field, just a subgroup of such. – Wojowu Jun 5 '18 at 18:08
• Actually, I see that the Wikipedia article on "Algebraic torus" says that the name comes from analogy with the theory of tori in Lie group theory. I don't know anything about that. I wouldn't be so strict with etymology though. – saulspatz Jun 5 '18 at 18:13