TITLE: global section of affine $C^\infty$-scheme
QUESTION [5 upvotes]: I'm reading Algebraic Geometry over $C^\infty$-rings.
It is written that "If $\mathfrak{C}$ is not finitely generated then $\Phi_{\mathfrak{C}}:\mathfrak{C}\rightarrow \Gamma(\text{Spec}\mathfrak{C})$ need not surjective, so $\Gamma(\text{Spec} \mathfrak{C})$ can be larger than $\mathfrak{C}$"(p.26).
However, I think as follows.
Let $U_1=\{ x\in \text{Spec}\mathfrak{C}|x(1)\neq0\}$, then $U_1 = \text{Spec}\mathfrak{C} $ because for every element $x:\mathfrak{C}\rightarrow \mathbb{R}$ in $\text{Spec}\mathfrak{C}, x(1)=1\neq 0$. Hence $\Gamma(\text{Spec}\mathfrak{C})=\mathcal{O}_{\text{Spec}\mathfrak{C}}(\text{Spec}\mathfrak{C})=\mathcal{O}_{\text{Spec}\mathfrak{C}}(U_1)=\mathfrak{C}[1^{-1}]=\mathfrak{C}$.
Where am I wrong?

REPLY [4 votes]: My apologies, Omar is correct. There was an mistake in Definition 4.12 of version 5 of my paper Algebraic Geometry over $C^\infty$-rings (posted 24 September 2015). I have replaced this by version 6 (posted 11 November 2015), which I hope is correct.