In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.


In the introduction the authors mention that there should be a description of the underlying topological space $|\text{Spec}(A)|$ of an affine scheme ($A$ algebra object in $C$; always commutative) in terms of ideals, at least if $C$ satisfies some reasonable properties (which one?). But they do not elaborate this. The definition of $|X|$ in the paper can be found in the end of section 2.4 and is rather sketchy and indirect: The category of (functorial) Zariski opens in $X$ is equivalent to the category of open subsets of a topological space $|X|$ by some abstract result.


**Question 1.** Is there any more concrete description of $|X|$, or at least of $|\text{Spec}(A)|$?


If $C$ was also assumed to be abelian, I would define $|\text{Spec}(A)|$ just as follows: First, an *ideal* of $A$ is the kernel of a morphism of algebras $A \to B$. Equivalently, it is a subobject $I \subseteq A$ such that $I \otimes A \to A \otimes A \to A$ factors through $I$. It is clear how to define *ideal sum* and (finite) *ideal intersection*. If $I,J$ are ideals of $A$, then define the ideal product $I\*J$ to be the kernel of $I \to I \otimes A/J$. A prime ideal is a proper ideal $\mathfrak{p}$ of $A$ such that for all ideals $I,J$ of $A$, we have $I \* J \subseteq \mathfrak{p} \Rightarrow I \subseteq \mathfrak{p} \vee J \subseteq \mathfrak{p}$. For an ideal $I$, let $V(I)$ be the set of prime ideals $\mathfrak{p}$ satisfying $I \subseteq \mathfrak{p}$. Then as usual we get a topological space $|\text{Spec}(A)|$, the *spectrum* of $A$.


**Question 2.** Has this definition already been studied somewhere? 


One way to "test" the above definition of the spectrum is to test if $\text{Spec}(\mathcal{O}\_X)$ turns out to be $X$; here $X$ is a (nice) scheme and $\mathcal{O}\_X$ is our algebra object in $C=\text{Qcoh}(X)$. Now it is not hard to check that we have an injective map $X \to \text{Spec}(\mathcal{O}\_X)$ sending $x$ to the vanishing ideal of $\overline{\{x\}}$ and that for noetherian schemes $X$ (actually I only need that $\text{rad}(\mathcal{O}\_X)^n=0$ for some $n$), this is an isomorphism. Again I don't know if this is well-known at all. I also wonder what happens if $X$ is more general, say quasi-compact and quasi-separated.


But back to the general setting relative to $C$:


**Question 3.** If we use the above definition of the spectrum of $A$, how can we define the structure sheaf?



Spectrum of an algebra object and Reconstruction of Schemes
Florian Marty studied this question in his thesis. The relevant chapter is available as [arXiv:0712.3676](http://arxiv.org/abs/0712.3676) (otherwise, the thesis is available [here](http://thesesups.ups-tlse.fr/540/)). He describes the space $|\mathrm{Spec}(A)|$ as the set of prime ideals endowed with the Zariski topology. He also proves that a basis of this topology is given by the subspaces $|\mathrm{Spec}(A[f^{-1}]|$, from which one deduces easily the description of the structural sheaf of $\mathrm{Spec}(A)$.


