Schlessinger's criterion

Stream: maths

[Paul Lezeau]

I've started looking into formalizing Schlessinger's criterion for pro-representable functors. The two main references I know of are Schlessinger's original paper and Stacks. Would anyone have any recommendations on which one to follow? I am more familiar with the first one, but the second one does things in more generality (but also would require more work setting up things).

[Filippo A. E. Nuccio]

I think this would be great, and probably following Stacks would be better. Considering that a good deal of the Category theory part of the libraries is already inspired by Stacks, this might be less painful than imagined.

[Kevin Buzzard]

@Mark Dickinson do you have an opinion?

[Kevin Buzzard]

The motivation is R=T stuff

[Kevin Buzzard]

How much work would the R=T theorem in your thesis be?

[Kevin Buzzard]

Oh I guess constructing the map R->T is a long way away. Maybe just assume that the map exists?

[Mark Dickinson]

Kevin Buzzard said:

Mark Dickinson do you have an opinion?

It's been a long time since my head was in this space, and I don't think I'm qualified to have an opinion at this point (though I'd be extremely interested to watch the developments). I'm not familiar with Stacks, but I do remember that when I originally read Schlessinger's paper it felt as though some bits were more complex / less natural than they could have been. (I can't remember specifics, though; just the general impression.)

[Mark Dickinson]

Apparently I felt strongly enough about this to write up a note on a criterion for existence of a universal deformation ring that didn't use Schlessinger's criteria (in PCMS 9), but in that note, past me couldn't be bothered to explain to future me exactly why avoiding Schlessinger's criterion was a good thing.

[Paul Lezeau]

Filippo A. E. Nuccio said:

I think this would be great, and probably following Stacks would be better. Considering that a good deal of the Category theory part of the libraries is already inspired by Stacks, this might be less painful than imagined.

Ok thanks for the advice! I'll start looking into that.