funlean.
Step 58 / 64

Master’s level

Deformation rings

Organize all permitted lifts through the maps from one algebraic object.

Start with a concrete example

Knowing the remainder is 2 leaves candidates such as 2,5,8. With richer mathematical data there may be many allowed ways to lift. A deformation ring organizes those allowed possibilities and their maps; it does not choose an arbitrary answer.

A small check: try the idea

Think with the concrete example above. You can try again.

Does a deformation ring mean we may pick any lift without checking conditions?

Read the answer and reasoning

No; it organizes lifts satisfying specified conditions. Just as a remainder leaves many candidates, lifting data can have possibilities. The deformation problem specifies which possibilities and maps are allowed.

Build the idea in three steps

  1. Fix residual representation data and specify which lifts are allowed.
  2. A ring is a system with addition and multiplication; a deformation ring packages these lifts through maps to other suitable rings.
  3. Universality says every allowed candidate is supplied by a unique compatible map. The long conditions in Lean specify what “allowed” means.

A small example

The universal order form supplies a map for every permitted lift. It does not accept an arbitrary sandwich as a lift.

Lean definition · original excerpt

FLT · aa2d8b3 · Def_GaloisRep_DeformationRingData.lean

  universal : ∀ (A : Type) [CommRing A] [IsLocalRing A] [IsNoetherianRing A]      [IsAdicComplete (IsLocalRing.maximalIdeal A) A] [Algebra 𝒪 A] [IsLocalHom (algebraMap 𝒪 A)],      Function.Surjective (IsLocalRing.residue A ∘ algebraMap 𝒪 A) →      ∀ ρA : GaloisRepAdic A, 𝒟 ρA →        ρA.residual.IsEquiv          (ρbar.baseChangeAlong (IsLocalRing.ResidueField.map (algebraMap 𝒪 A))) →        ∃! φ : R →ₐ[𝒪] A, ∃ hφ : IsLocalHom (φ : R →+* A),

How to read this Lean

This is the universal field of DeformationRingData. ∃! means there exists exactly one map φ; →ₐ[𝒪] says the map respects the stated algebra structure.

Original source ↗ · L33–39Open the complete saved source

Where this idea is used

12. The workshops agree13. Pip’s impossible quilt
I’m lost · Help

Mika’s help desk

Deformation rings

One thing at a time

Read the three explanation steps first. The Lean source is there for a closer look afterwards; you do not have to read it first.

In simpler words

Organize all permitted lifts through the maps from one algebraic object.

Earlier ideas are listed under “Start with these ideas” on this page.

A small clue

The universal order form supplies a map for every permitted lift. It does not accept an arbitrary sandwich as a lift.

Close this panel to return to the exact place you were reading. These are prepared hints for this page.

Start with the first small story →