What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}\_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes?


Is there, perhaps, even an intrinstic characterization of those locally ringed toposes which are equivalent to $(\mathbf{Sh}(X),\mathcal{O}\_X)$ for some scheme $X$?



Properties of the petit Zariski topos
Unfortunately I don't know an interesting intrinsically formulated sufficient criterion for a locally ringed topos to be the little Zariski topos of a scheme. This is an extremely interesting question!


There are necessary conditions, for instance (formulated in the internal language of the topos):



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> For any element $f : \mathcal{O}\_X$: If $f$ is not invertible, then $f$ is nilpotent.
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While this condition does exclude some locally ringed toposes, it doesn't exclude the locally ringed topos given by a smooth manifold (if a smooth function has the property that the only open subset on which it is (multiplicately) invertible is the empty set, then it is zero).


On a reduced scheme, where it holds that $\mathcal{O}\_X$ is internally a reduced ring, a consequence of this condition is that $\mathcal{O}\_X$ is "$\neg\neg$-separated": For any $f : \mathcal{O}\_X$, if $\neg\neg(f = 0)$, then in fact $f = 0$.


**Update**: There is a further, but more convoluted, property which is enjoyed by the little Zariski topos of any scheme but not in general by the topos given by a smooth manifold:



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> For any element $f : \mathcal{O}\_X$ the localized module $\mathcal{O}\_X[f^{-1}]$ is a sheaf with respect to the internal modality $\Box$, where $\Box\varphi :\equiv (\text{$f$ invertible} \Rightarrow \varphi)$.
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This condition can be rephrased so as to not refer to internal modalities:



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> For any element $f : \mathcal{O}\_X$ it holds that:
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> 1. For any $s : \mathcal{O}\_X$ such that $\text{$f$ invertible} \Rightarrow s = 0$, there is a natural number $n$ such that $f^n s = 0$.
> 2. For any subset $K \subseteq \mathcal{O}\_X$ such that $\text{$f$ invertible} \Rightarrow \text{$K$ is a singleton}$, there is a natural number $n$ and an element $s : \mathcal{O}\_X$ such that $\text{$f$ invertible} \Rightarrow f^{-n} s \in K$.
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The first part is satisfied by (the topos of sheaves over a) manifold, but the second part is not. A counterexample is $X = \mathbb{R}^1$, $f(x) = x$, $K = \text{"$e^{1/x}$ on $x \neq 0$"}$. In fact a manifold satisfies this condition if and only if it is empty.


The *big* Zariski topos of a scheme $X$, that is approximately the topos of sheaves on the Zariski site $\mathrm{Sch}/X$ with structure sheaf $\mathbb{A}^1 : T \mapsto \mathcal{O}\_T(T)$, has the following special properties, which are more helpful in distinguishing it from other locally ringed toposes:



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> 1. For any element $f : \mathbb{A}^1$: If $f$ is not zero, then $f$ is invertible.
> 2. For any finitely presented $\mathbb{A}^1$-algebra $A$, the canonical map $A \to \mathrm{Hom}(\mathrm{Hom}\_{\mathbb{A}^1\mathrm{-Alg}}(A, \mathbb{A}^1), \mathbb{A}^1)$ is an isomorphism of $\mathbb{A}^1$-algebras.
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In the special case that $A = \mathbb{A}^1[T]$, the second condition yields



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> Any map $\mathbb{A}^1 \to \mathbb{A}^1$ is given by a unique polynomial in $\mathbb{A}^1[T]$.
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which somehow expresses the vague intution that "in algebraic geometry, every morphism is a polynomial".


