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Local ring object? Local rings are not too far from fields, unlike arbitrary rings.
I don’t understand the question.
I believe that Mateo is referring to Example 3.2 at field. One has to be a bit careful about this example (which was added by Ingo Blechschmidt in Revision 36). What is true is that the canonical ring object of the big Zariski topos is a ’field object’, where this latter is defined using the internal logic of the Zariski topos in exactly the same way as in Definition 1.1 of the nLab article. A proof is given Proposition 2.2 in this paper of Anders Kock.
In particular, this is not a field in the usual sense, Mateo, so from that point of view your question does not parse. However, I believe (though Ingo, if he sees this thread, would be much more knowledgeable than me), that the philosophy, when working with algebraic geometry via the internal logic of the big Zariski topos over a field $k$, is that one can view this field object as precisely analogous to $k$. So in a sense, the answer to your question is: yes (to ’obtain’ $k$, just take the big Zariski topos over it).
By “field in the usual sense” I guess is meant a classical field in the category $Set$.
The notion of field is tricky: moving from classical to constructive settings, it bifurcates into a multitude of concepts, and one has to decide which notion is right for a given context.
Yes, by ’field in the usual sense’, I was referring to Definition 1.1 at field as interpreted in the classical category of sets, as I assumed that this is what Mateo had in mind in his question.
As I wrote, this is exactly the definition that Anders Kock is using in his article, when formulated in the internal logic of the big Zariski topos. And, whilst I haven’t checked it very carefully, I believe that Kock’s proof is actually constructive, so that even we often think of something other than Definition 1.1 as the definition of a field in a constructive setting, it seems to be able to be used here.
Oh, I see, I guess Mateo was asking whether every field occurs as the canonical ring object in some Zariski topos?
Perhaps Example 3.2 on field should be clarified to emphasize that this is an internal field object, since the rest of the page is about “ordinary” fields in $Set$.
Take the scheme in #1 to be Spec(k), no?
David Roberts: no. Example 3.2 at field is specific to the big Zariski topos. Edit: if you mean the base scheme, then, yes, this is true in a sense, namely the sense in which I discussed in my first comment.
Yes, the big Zariski topos over $Spec(k)$ is what I meant. In this sense, every field is the canonical internal local ring object of a big topos, but then surely so is every local ring and indeed more generally. I don’t see how this singles out fields in any sense.
David Roberts: I’m not quite sure what you’re getting at. The fact under discussion is that the canonical local ring object of the big Zariski topos is moreover in fact a field object. That is to say, one obtains only local ring objects that are in fact field objects in this way. When the base scheme is an affine scheme $Spec R$, I believe that one should think of this ’internal field’ as analogous to the field of fractions of $R$. For an arbitrary base scheme $S$, I believe one should think of it as the sheaf version of this, i.e. as locally, where $S$ is isomorphic to an affine scheme $Spec R$, being the field of fractions of $R$.
I find this point of view interesting. As Kock hints at in his article, it suggests the possibility of viewing commutative ring theory / algebraic geometry, as linear algebra / local linear algebra.
I have now slightly edited Example 3.2 at field, by the way, to try to make it clearer.
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