I'm reading [Algebraic Geometry over $C^\infty$-rings](http://arxiv.org/abs/1001.0023).
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?



global section of affine $C^\infty$-scheme
I think the error is in Joyce's definition 4.12 of $\mathop{\mathrm{Spec}}\mathfrak{C}$! Consider the case when $\mathfrak{C}$ has no real points (example below): then for all $c \in \mathfrak{C}$, $U\_c = \emptyset$ (as they are subsets of $X\_\mathfrak{C} = \emptyset$), but definition 4.12 attempts to set $O\_{X\_\mathfrak{C}}(U\_c) = \mathfrak{C}[c^{-1}]$ which is doubly wrong: (1) it depends on $c$, (2) should be $0$ if the construction were really a sheaf.


Notice that Joyce prefaces the definition with "As in Moerdijk, van Quê and Reyes [49, §1 & §3]" but I looked up that paper and did not find exactly this definition of $\mathop{\mathrm{Spec}}\mathfrak{C}$:


* In §1 (as Joyce also explains), the authors don't deal with this version of the spectrum, but rather one based on all $C^\infty$-radical prime ideals and I think that in that setting, just as in algebraic geometry, the definition $O(U\_c) = \mathfrak{C}[c^{-1}]$ is not problematic.
* In §3 they do define another version of the spectrum due to Eduardo Dubuc and based on real points, but not through explicit formulas, instead giving a categorical argument for the existence of an adjoint to the functor of global sections defined on locally $C^\infty$-ringed spaces whose stalks have residue field $\mathbb{R}$.


So the problematic formula $O\_{X\_\mathfrak{C}}(U\_c) = \mathfrak{C}[c^{-1}]$ does not actually appear in Moerdijk, van Quê and Reyes, nor, in Dubuc's *$C^\infty$ Schemes*, where the real-points spectrum was originally defined. 


The construction in definition 4.12 can't be too far from correct (and is quite possibly completely correct in the special case of fair $C^\infty$-rings), but I don't know exactly how to fix it.


**Example of a $C^\infty$-ring with no real points.** Take $\mathfrak{C} = C^\infty(\mathbb{R})/I$ where $I$ is the ideal of $C^\infty$ functions $\mathbb{R} \to \mathbb{R}$ with compact support. Suppose there was a morphism of $C^\infty$-rings $f : \mathfrak{C} \to \mathbb{R}$, and consider the composite $f \circ p$ where $p : C^\infty(\mathbb{R}) \to \mathfrak{C}$ is the canonical projection. That is a real point of $C^\infty(\mathbb{R})$ and is thus given by evaluation at some real number $x$, but since functions of compact support can take arbitrary values at $x$, evaluation at $x$ cannot descend to the quotient $\mathfrak{C}$, a contradiction.


Since $X\_{\mathfrak{C}}$ is empty, the only possible value for $\Gamma(\mathop{\mathrm{Spec}}\mathfrak{C})$ is the zero ring, which is what proposition 4.15 says should happen: this example is not fair and its fairification $R^{fa}\_{fg}(\mathfrak{C})$ is indeed the zero ring.


