A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element , which by Grothendieck’s scheme-philosophy should determine a functor
from finite Abelian groups to sets, typically giving pretty small sets . Using the Fun mantra that should be an algebra over any [...]
Last time we have seen how an affine -algebra R gives us a maxi-functor (because the associated sets are typically huge)
Substantially smaller sets are produced from finitely generated -algebras S (therefore called mini-functors)
Both these functors are ‘represented’ by existing geometrical objects, for a maxi-functor by the complex affine variety (the set [...]
A couple of weeks ago, Alain Connes and Katia Consani arXived their paper “On the notion of geometry over “. Their subtle definition is phrased entirely in Grothendieck’s scheme-theoretic language of representable functors and may be somewhat hard to get through if you only had a few years of mathematics.
I’ll try to give the [...]