A Fourier-Hopf inversion and spectral sequences 1. \underline{Fourier-Hopf inversion}

Maps on the right : A ⟶^{f} B , a ↦ af = [a]f.

• R : commutative ring  DATA
• H : Hopf algebra free over R,

with involutive antipode ; so

H ⟶^{ε} R  counit
H ⟶^{Δ} H ⊗_{R} H  comultiplication
x ↦ ∑_{i} xu_{i} ⊗ xv_{i}
H ⟶^{S} H  antipode, S^{2} = id
• K ⊆ H : normal Hopf subalgebra free over R;

i.e. KΔ ⊆ K ⊗_{R} K, KS ⊆ K, and

\begin{matrix}∑_{i} xu_{i} · k · xv_{i}S & ∈ & K \\ ∑_{i} xu_{i}S · k · xv_{i} & ∈ & K \end{matrix} \} for k ∈ K, x ∈ H
• M : left H-module

Write K^{+} := Kern(K ⟶^{ε} R).

Then K^{+}H = HK^{+} ⊆ H is a Hopf ideal, so H̄ := H/HK^{+} is a Hopf algebra.

Write x̄ := x + HK^{+} for x ∈ H, etc.

Examples

• G group, N ⊴ G, H = RG, K = RN, H̄ = R(G/N) (⇝ Lyndon-Hochschild-Serre).

• 𝔤 Lie algebra free/R, 𝔫 ⊴ 𝔤 such that 𝔫 and 𝔤/𝔫 free/R, H = U(𝔤), K = U(𝔫), H̄ = U(𝔤/𝔫) (⇝ Hochschild-Serre).

Two H̄-modules

(a) Hom_{K}(H, M) is a left H̄-module via

[h](x̄ · f) := _{∑i} xu_{i} · [xv_{i}S · h]f for f ∈ Hom_{K}(H, M), h, x ∈ H.

(b) Hom_{R}(H̄, M) is a left H̄-module via

[h̄](x̄ · g) := [h̄ · x̄]g

for g ∈ Hom_{R}(H̄, M), h, x ∈ H.

With help of G. Carnovale :

\underline{Lemma} (Fourier-Hopf inversion =: FH ) There exist mutually inverse isos of H̄-modules :

[figure: f; (fΦ : h̄ ↦ ∑_{i} hu_{i} · [hv_{i}S]f); Hom_{K}(H, M); Φ; ∼; Hom_{R}(H̄, M); Ψ; (gΨ : h ↦ ∑_{i} hv_{i} · [\overline{hu_{i}S}]g); g]

2. \underline{Spectral sequence comparisons}

Assume : H, K, H̄ free over R, and H free over K

Write

(H -Mod)^{∘} × H -Mod \underrightarrow{U} H̄ -Mod
(X , X′)  ↦ U(X, X′) := \underbrace{Hom_{K}(X, X′)}_{cf. 1.(a)}
(H̄ -Mod)^{∘} × H̄ -Mod \underrightarrow{V} R -Mod
(Y , Y′)  ↦ V (Y, Y′) := HomH̄(Y, Y′)

Three spectral sequences

(a) The composition

\xymatrix{H -Mod \ar@{|->}[r]^{U(R,−)}_{K-fixed points} & H̄ -Mod \ar@{|->}[r]^{V (R,−)}_{H̄-fixed points} & R -Mod}

gives rise to the Grothendieck sp. seq.

E_{U(R,−),V (R,−)}(M) .

(b) The composition

\xymatrix{(H -Mod)^{∘} \ar@{|->}[r]^{U(−,M)} & H̄ -Mod \ar@{|->}[r]^{V (R,−)} & R -Mod}

gives rise to the Grothendieck sp. seq.

E_{U(−,M),V (R,−)}(R)

But …

… \underline{existence} of this sp. seq. needs : P proj./H ⇒^{!} U(P, M) acyclic w.r.t. V (R, −) = HomH̄(R, −). Suffices P := H, so U(P, M) = U(H, M). Now

U(H, M) =^{def} Hom_{K}(H, M) ≃^{FH} Hom_{R}(H̄, M) , and

Exti ; H̄(R, Hom_{R}(H̄, M)) ≃ Ext_{R}^{i}(\underbrace{H̄ ⊗H̄ R}_{= R} , M) ≃ 0

for i ⩾ 1. So \underline{existence} is ok.

(c) B : proj. res. of R over H. B̃ : proj. res. of R over H̄. Replacing R by B resp. by B̃, we obtain a double complex V (B̃, U(B, M)) ≃ Hom_{H}(B̃ ⊗_{R} B, M) , whence a Hochschild-Serre type sp. seq.

E( Hom_{H}(B̃ ⊗_{R} B, M)) .

\underline{Theorem} We have

E_{U(R,−),V (R,−)}(M)_{≀}  (a), abstract, using inj. E_{U(−,M),V (R,−)}(R)_{≀}  (b), intermediate, FH ! E( Hom_{H}(B̃ ⊗_{R} B, M)) (c), concrete, using proj.

Note : both (a) and (c) a priori have

E_{2}^{p,q} = Extp ; H̄( Ext_{K}^{q}(R, M))

and converge to

Ext_{H}^{p+q}(R, M) .

The problem in comparing them is the differentials.

\underline{Related work}

• Beyl : case of groups

• Haas : naturality of spectral sequences

• Barnes : comparison theorem in different setup