Solid modules #
This file contains the definition of a solid R
-module: CondensedMod.isSolid R
. Solid modules
groups were introduced in [scholze2019condensed], Definition 5.1.
Main definition #
CondensedMod.IsSolid R
: the predicate on condensed abelian groups describing the property of being solid.
TODO (hard): prove that ((profiniteSolid ℤ).obj S).IsSolid
for S : Profinite
.
TODO (slightly easier): prove that ((profiniteSolid 𝔽ₚ).obj S).IsSolid
for S : Profinite
.
@[reducible, inline]
The free condensed abelian group on a finite set.
Equations
- Condensed.finFree R = FintypeCat.toProfinite.comp (profiniteToCondensed.comp (Condensed.free R))
Instances For
@[reducible, inline]
The free condensed abelian group on a profinite space.
Equations
Instances For
The functor sending a profinite space S
to the condensed abelian group R[S]^\solid
.
Equations
Instances For
The natural transformation R[S] ⟶ R[S]^\solid
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The predicate on condensed abelian groups describing the property of being solid.
- isIso_solidification_map : ∀ (X : Profinite), CategoryTheory.IsIso ((CategoryTheory.yoneda.obj A).map ((Condensed.profiniteSolidification R).app X).op)
Instances
theorem
CondensedMod.IsSolid.isIso_solidification_map
{R : Type (u + 1)}
[Ring R]
{A : CondensedMod R}
[self : CondensedMod.IsSolid R A]
(X : Profinite)
:
CategoryTheory.IsIso ((CategoryTheory.yoneda.obj A).map ((Condensed.profiniteSolidification R).app X).op)