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ag.algebraic geometry – Are all formal schemes *actually* Ind-schemes? Answer

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ag.algebraic geometry – Are all formal schemes *actually* Ind-schemes?

$newcommandLRS{mathsf{LRS}}newcommandFormalSch{mathsf{FormalSch}}DeclareMathOperatorSpf{Spf}newcommandIndSch{mathsf{IndSch}}newcommandALRS{mathsf{ALRS}}newcommandFSch{mathsf{FSch}}$I’m attempting to grasp whether or not there is a absolutely loyal functor $LRS supset FormalSch to IndSch$ and in what sense. Here’s my progress to this point:

Let $mathsf{A}$ breathe the class of adic rings. The objects are topological rings whose topology is generated by a descending filtration of beliefs whose intersection is ${0}$. Morphisms are steady homomorphism of rings.

There’s a functor $Spf: mathsf{A} to IndSch$ which takes an adic ring to the formal spectrum which is of course a filtered colimit of (affine) schemes). The goal of the functor may breathe that of adic domestically ringed areas (topological areas with sheaves of adic rings and morphisms between for which the comorphism of sheaves is steady). Denote this class $ALRS$.

In $ALRS$ now we have an adjunction with the “continuous” international part functor $Gamma_{textual content{cont}} dashv Spf $. Continuous right here simply means it remembers the topology (i.e. the filtration).

Now the definition of formal schemes feels sure:

Definition: A formal strategy is an adic domestically ringed area domestically isomorphic to
a proper spectrum of an adic ring. Denote the subcategory of formal schemes by $FSchsubset ALRS$.

This raises an issue although. There’s no patent route to rotate a “formal scheme” on this sense into an ind-schemes (that are mighty extra handy for inescapable functions). We may attempt to outline the ind-scheme because the formal colimit over the Čech nerve of a selected protecting by formal spectra (that are themselves filtered colimits of affine schemes). However, that is in all probability a really evil thought since it’ll more than likely reckon on the altenative of protecting.

Question: Can we assemble a functor $FSch to IndSch$ with some good properties? (Hopefully absolutely loyal but when not possibly at the very least complete.) If not is there a greater definition of a proper strategy which lets you toy in each worlds (ind schemes and domestically ringed areas)?

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