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Zoran,
concerning your paper with Durov and the sheaf category defined on p. 22, I am wondering:
it would almost seem as if something essentially equivalent is obtained if we would very slightly change the definition of the site (Rings with a chosen nilpotent ideal) and think of it as the tangent category of the category of rings, i.e. of Mod, thought of as being the category of square-0-extensions of rings.
So I am suggesting that we look at sheaves on (the opposite of) Mod
Do you see what I mean?
Intuitively this is kind of obvious, however I never produced a formal statement. It would be also very good to compare this to the method of Pareigis and Morris, doi, who also study formal schemes over an arbitrary base, not necessarily pseudocompact or noetherian local ring. A neat thing is that they can extend the Yoneda to kind of topological Yoneda, it is like considering some ind-objects as topological rings and then reflecting this in some sort of enriched Yoneda way. The ringed object O over the site becomes topological ringed object.
Durov is considering just presheaves, so the category is bigger than the true category of formal schemes. Pareigis tries to find the appropriate subcategory of formal schemes. If you can not access the find I can send it to you (though I did long time ago).
Thanks, Zoran.
Right this moment I am on the train and cannot access the article. If you could send it, that would be nice.
I sent it to you. Maybe we should put it as a file on the nlab. I wrote a stub Morris-Pareigis formal scheme with redirect Morris-Pareigis formal group.
In any approach the formal schemes make a subcategory of the category of ind-schemes. Now the question is how to neatly characterize the diagrams corresponding to the objects in the subcategory. Thickening devices are a tool to do that.
Edit: yet another question is how to see the thickenings from the point of view of the category of qoch sheaves and the structure of its subcategories. In Abelian case this leads to the study of certain filtrations of subcategories for which the only known reference is Lunts-Rosenberg Max Planck preprint
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