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I split off an entry dg-geometry from the entry on Hochschild geometry, since it really deserves a stand-alone discussion.
Eventually somebody should add the references by Kapranov et al on dg-schemes etc. And much more.
I think that dg-schemes of Kapranov are pretty much different from the geometry based on dg-categories. Dg-schemes of Kapranov are dg-ringed topological spaces – the structure sheaf is a dg-algebra, and the underlying topological space is a usual one. It is much more special and concrete framework than what Toen et al. or Kontsevich et al. do with dg-categories and A-infinity categories.
there is a notion of scheme in any “HAG-context” in the language o Toen-Vezzosi and with respect to every “geometry” in Lurie’s terminologyy. Kapranov’s dg-schemes should be schemes in dg-geometry, in this sense.
Urs, the mathematics terminology is not made optimal according to your nice idealistic wishes. Kapranov-Ciocan-Fountaine dg-schemes are an intermediate notion at the beginning of the subject of derived geometry, the concept which has nice examples which is consistent, but rather special and abandoned soon after the publication. It does not cover the concept of scheme in some modern geometry based on dg-categories; it is an alternative representation for a class of very special cases. I am not even sure if they form a full subcategory.
Do you want to put Kapranov’s definition of dg-scheme onto the Lab? Then we can see.
All these notions will try to formalize the notion “locally equivalent to spec of a dg-algebra”.
Well, most of people nowdays do spec of dg-category viewed as enhanced derived category of qcoh sheaves; the point of view of Kapranov et al. is I think to extend the flexibility notion of structure sheaf of rings. Of course, there is some embedding of the picture by taking a generator which is often a dg-algebra. The question of reconstruction from enhanced derived category is subtle.
I will put dg-schemes of that kind in the nlab at least once I am back to normal work (I just came back from Hungary; wednesday is the deadline for material spending for this year. I have a visitor from Thursday to Sunday. Small trip out of Zagreb Sun/Monday etc…) if not before.
I am not against putting Kapranov’s refernce (on the contrary). I am just against a light conclusion that it is the same level of generality and even cautious again about full and faithful embedding.
okay, thanks, no rush. I have no time for it either. But if you have a rough idea, maybe you could just write a single sentence at “dg-geometry” that mentions Kapranov. I feel he deserves to be menioned as one of the first (if I understand correctly) who thought in this direction at all, even if maybe his definitions don’t quite survive.
I did mention those papers in the first versions of derived algebraic geometry. It may be more fruitful to have general picture of any kind of derived geometry under derived geometry than to discuss models of derived schemes in the sense of algebraic geometry at both places in detail.
Where the term dg-geometry is actually used ?
I made up that particular term. I needed a more specific term than “derived algebraic geometry”. In Toen-Vezzosi they have the concept of an “HAG-context”. One of those that they discuss is $cdgAlg_k^- \hookrightarrow cdgAlg_k$. I needed a name for that.
A little later, when they work entirely over unbounded dg-algebra, they call this the “complicial HAG context”. But I don’t feel we can use that term here, as it collides with other things.
I think some term is necessary: there are other contexts. For instance if over a field not of char 0, there is no model in terms of cdg-algebras at all.
(Okay, maybe we should rename “dg-geometry” to “cdg-geometry”.)
I like complicial though. dg-geometry may be OK, but we should think more through terminology. It is nice seeing how you are getting more familiar with algebraic geometry terminology and literature in general. Soon you will teach algebraic geometers in that subject :)
I am currently expanding the entry dg-geometry by listing more of the results by Toën-Vezzosi.
Turns out that several of the statements that I was working on and discussing here recently are all proven there (unsurprisingly, I have to admit…)
I made a few changes to dg-scheme.
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