The (p, [ϖ])-adic topology depends only on the radical of (ϖ)

Let R be a commutative ring and ϖ, ϖ' : R. If ϖ and ϖ' generate ideals with the same radical, that is, each divides a power of the other, then the Teichmüller representatives [ϖ] and [ϖ'] divide powers of each other in 𝕎 R. The ideals (p, [ϖ]) and (p, [ϖ']) of 𝕎 R then have the same radical, so they define the same adic topology. For the ring of integers 𝒪_F of a nonarchimedean field F, any two pseudouniformisers satisfy this hypothesis, so the topology of A_inf = W(𝒪_F) does not depend on the choice of pseudouniformiser.

Main results

• WittVector.teichmuller_mem_radical_span_singleton : [ϖ] lies in the radical of ([ϖ']) when ϖ lies in the radical of (ϖ').
• WittVector.radical_span_p_teichmuller_eq : (p, [ϖ]) and (p, [ϖ']) have the same radical when (ϖ) and (ϖ') do.
• TauCeti.WittVector.isAdic_span_p_teichmuller_iff : a topology on 𝕎 R is (p, [ϖ])-adic exactly when it is (p, [ϖ'])-adic.

References

• K. S. Kedlaya, Sheaves, stacks, and shtukas, lecture notes, Arizona Winter School 2017, §3.1.

theorem WittVector.teichmuller_mem_radical_span_singleton {p : ℕ} [Fact (Nat.Prime p)] {R : Type u_1} [CommRing R] {ϖ ϖ' : R} (h : ϖ ∈ (Ideal.span {ϖ'}).radical) : (teichmuller p) ϖ ∈ (Ideal.span {(teichmuller p) ϖ'}).radical

If ϖ lies in the radical of (ϖ'), then [ϖ] lies in the radical of ([ϖ']): ϖ' ∣ ϖ ^ n gives [ϖ'] ∣ [ϖ] ^ n.

theorem WittVector.radical_span_p_teichmuller_eq {p : ℕ} [Fact (Nat.Prime p)] {R : Type u_1} [CommRing R] {ϖ ϖ' : R} (h : (Ideal.span {ϖ}).radical = (Ideal.span {ϖ'}).radical) : (Ideal.span {↑p, (teichmuller p) ϖ}).radical = (Ideal.span {↑p, (teichmuller p) ϖ'}).radical

The radical of (p, [ϖ]) depends only on the radical of (ϖ). If ϖ and ϖ' generate ideals of R with the same radical, then (p, [ϖ]) and (p, [ϖ']) have the same radical in 𝕎 R.

theorem TauCeti.WittVector.isAdic_span_p_teichmuller_iff {p : ℕ} [Fact (Nat.Prime p)] {R : Type u_1} [CommRing R] [TopologicalSpace (WittVector p R)] {ϖ ϖ' : R} (h : (Ideal.span {ϖ}).radical = (Ideal.span {ϖ'}).radical) : IsAdic (Ideal.span {↑p, (WittVector.teichmuller p) ϖ}) ↔ IsAdic (Ideal.span {↑p, (WittVector.teichmuller p) ϖ'})

The (p, [ϖ])-adic topology depends only on the radical of (ϖ). If ϖ and ϖ' generate ideals of R with the same radical, then a topology on 𝕎 R is the (p, [ϖ])-adic one exactly when it is the (p, [ϖ'])-adic one.