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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:Aformal strategyis 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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