We will show that every open affine subset $U$ of $X$ has a canonical moment map. We note that the abelian group $M_{U}:={\mathscr{O}}(U)^{\times}/K^{\times}$ is free of finite rank (see [Sa66], Lemme 1). Here, we use that $K$ is algebraically closed (or at least that $X$ is geometrically reduced). We choose representatives $\varphi_{1},\dots,\varphi_{r}$ in ${\mathscr{O}}(U)^{\times}$ of a basis. This leads to a moment map $\varphi_{U}:U\rightarrow T_{U}={\rm Spec}(K[M_{U}])$. By construction, $\varphi_{U}$ refines every other moment map on $U$. Note that this moment map $\varphi_{U}$ is canonical up to (multiplicative) translation by an element of $T_{U}(K)$.

Let $f:X^{\prime}\rightarrow X$ be a morphism of algebraic varieties over $K$ and let $U^{\prime}$ is an open subset of $X^{\prime}$ with $f(U^{\prime})\subset U$. Then $f^{\sharp}$ induces a homomorphism $M_{U}\rightarrow M_{U^{\prime}}$ of lattices. We get a canonical affine homomorphism $\psi_{U,U^{\prime}}:T_{U^{\prime}}\rightarrow T_{U}$ of the canonical tori with $\psi_{U,U^{\prime}}\circ\varphi_{U^{\prime}}=\varphi_{U}\circ f$. This will be applied very often in the case where $U^{\prime}$ is an open subset of $U$ in $X^{\prime}=X$ and $f={\rm id}$. Then we get a canonical affine homomorphism $\psi_{U,U^{\prime}}:T_{U^{\prime}}\rightarrow T_{U}$.