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?

  • $\begingroup$ 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. $\endgroup$ Jul 18 '11 at 19:45
  • $\begingroup$ Could you add this as an answer? Thanks! $\endgroup$ Jul 18 '11 at 20:02

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 $\mathrm{Spec}(A)$.

  • $\begingroup$ 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 $X$. $\endgroup$ Oct 4 '13 at 15:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.