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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?

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The description of the space underlying $Spec(A)$ is studied by Florian Marty in his thesis. The corresponding chapter is available as arXiv:0712.3676 Marty answers Question 1 in the way you expect. He also proves that the $Spec(A[f^{-1}]$'s form a basis of the Zariski topology, from which it is easy to answer Question 3. –  Denis-Charles Cisinski Jul 18 '11 at 19:45
    
Could you add this as an answer? Thanks! –  Martin Brandenburg Jul 18 '11 at 20:02
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Florian Marty studied this question in his thesis. The relevant chapter is available as arXiv:0712.3676 (otherwise, the thesis is available here). 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 $\mathrm{Spec}(A)$.

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This thesis is very interesting. Unfortunately, the theorems only work for relative contexts, where one assumes that $\hom(1,-)$ maps regular epimorphisms to surjections. This excludes many "global" examples such as $C=\mathsf{Sh}(X)$ for some space $X$ or $C=\mathsf{Qcoh}(X)$ for some scheme $X$. –  Martin Brandenburg Oct 4 '13 at 15:39
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