Let $N$ be a lattice (free $\mathbb Z$ - module of rank $n$) with dual $M=\operatorname{Hom}_{\mathbb Z - \text{mod}}(N,\mathbb Z)$ and let $\sigma$ be a cone in $N\otimes\mathbb R\cong\mathbb R^n$. Let $U_\sigma=\operatorname{Hom}_{sg}(\sigma^\vee\cap M,\mathbb C)$ be the affine toric variety corresponding to $\sigma$ (where Hom$_{sg}$ means semi group homomorphisms). I am trying to understand the proof of the following lemma -
Lemma : Let $\sigma$ be an $n$ dimensional cone in $N$ then $U_\sigma$ is contractible.
Proof : Let $v\in\sigma\cap N$ be in the relative interior of $\sigma$. Let $H:U_\sigma\times[0,1]\to U_\sigma$ be defined as follows - for any $x\in U_\sigma$ and $0\le t\le1$, let $H(x,t)(u)=t^{\langle u,v\rangle}x(u)$ for all $u\in\sigma^\vee\cap M$. Note that $H(x,t)$ is a homomorphism of semi groups $\sigma^\vee\cap M\to \mathbb C$ and hence may be regarded as an element of $U_\sigma$. Since $\langle u,v\rangle\ge0$ for $u\in\sigma^\vee\cap M$, $H$ is continuous. Clearly, $H(\_,1)=id_{U_\sigma}$ and$H(\_,0)$ is the constant map $x\mapsto x_\sigma$, where $x_\sigma(u)$ is $1$ if $u\in\sigma^\perp$ and $0$ otherwise.
My question is, which part of this proof uses that $\sigma$ is $n$ dimensional?
Thank you.