ADAMS AND WIRTHMÜLLER ISOMORPHISMS FOR COMPACT p-ADIC ANALYTIC GROUPS

MARK BEHRENS

Warning: this document is preliminary, completely formal, and may contain errors. In particular, Halvard Fausk alerted me to the fact that my treatment of the norm map (the map ω) is completely misguided, and I think outright wrong. So at this stage treat the existence of the map ω in Lemma 2.3 as an axiom.

1. The derived Pontryagin category

All groups we shall consider are p-torsion free compact p-adic analytic groups. In particular, they are virtually pro-p. Let P_{p}(G) be the Pontryagin category. All of our group-rings are complete group rings, all of our tensor products are complete tensor products and all of our hom sets are continuous hom sets. As such, P_{p}(G) is a closed symmetric monoidal category, with tensor given by −⊗_{ℤ_{p}} − (with diagonal action) and cotensor given by Hom_{ℤ_{p}}(−, −) (with conjugation action). If G is the trivial group, we shall write P_{p} = P_{p}(e).

Suppose that f : G_{1} → G_{2} is a homomorphism. We have the following induced functors:

f_{!} :P_{p}(G_{1}) → P_{p}(G_{2})
M ↦ ℤ_{p}[G_{2}] ⊗_{ℤ_{p}[G_{1}]} M

f_{∗} :P_{p}(G_{1}) → P_{p}(G_{2})
M ↦ Hom_{ℤ_{p}[G_{1}]}(ℤ_{p}[G_{2}], M)

f^{∗} :P_{p}(G_{2}) → P_{p}(G_{1})
A ↦ A

The functor f^{∗} is monoidal and exact, with left adjoint f_{!} and right adjoint f_{∗}. Two cases of special interest follow. The first is the map

i : H → G

given as the inclusion of a closed subgroup. Then we have

i_{!}M = Ind_{H}^{G} M
i_{∗}M = Coind_{H}^{G} M
i^{∗}A = Res_{H}^{G} A

The second is the map

r : G → e.

We have

r_{!}M = M_{G}
r_{∗}M = M^{G}
r_{∗}A = A (with trivial G-action)

Let DP_{p}(G) be the unbounded derived category of P_{p}(G). Then we have derived functors

Lf_{!} :P_{p}(G_{1}) → P_{p}(G_{2})
Rf_{∗} :P_{p}(G_{1}) → P_{p}(G_{2})

as well as the left derived tensor −⊗^{L}− and the right derived cotensor R Hom(−, −). Under these operations, DP_{p}(G) is a triangulated symmetric monoidal category. The functor f^{∗} is exact. Letting f^{!} be defined to be f^{∗}, we find ourselves in what Fausk, Hu, and May [1] refer to as the Wirthmüller context for abstract duality theorems.

From now on we shall always be working in the derived category DP_{p}(G), and so we will drop the L’s and R’s from our notation for the sake of readability. We shall always be implicitly taking the derived versions of our functors.

Given an object M in the derived Pontryagin category, we shall write π_{∗}(M) to denote the homology groups of the underlying chain complex. The (continuous) group homology and cohomology is given by

π_{n}(r_{!}M) = H_{n}(G; M)
π_{−n}(r_{∗}M) = H^{n}(G; M).

2. Poincaré duality and the Adams isomorphism

Let G be a compact p-torsion free p-adic analytic group of dimension d.

Definition 2.1. Let ρ : G → GL(V) be a ℚ_{p}-representation of G of dimension d. Then the determinant of this representation

det(ρ) : G → ℚ_{p}^{×}

must factor through ℤ_{p}^{×} since G is compact. We define an invertible object Λ_{c}^{top}V ∈ DP_{p}(G) to be Σ^{−d}ℤ_{p}[det(ρ)].

Remark 2.2. Note that for p-adic representations V and W, we have

Λ_{c}^{top}(V ⊕ W) ≅ Λ_{c}^{top}(V) ⊗ Λ_{c}^{top}W.

Lemma 2.3. There exists a dualizing object C_{r} ∈ DP_{p}(G) and a map

ω : r_{∗}ℤ_{p} → r_{!}C_{r}

which is an equivalence. There is an equivalence

C_{r} ≃ Λ_{c}^{top}L(G).

Proof. We shall temporarily suspend our convention of working in the derived category and work in the point-set category of unbounded chain complexes in P_{p}(G). Let P → ℤ_{p} be a free resolution of ℤ_{p} by ℤ_{p}[G]-modules. Then define C_{r} to be the complex Hom_{ℤ_{p}[G]}(P, ℤ_{p}[G]), with G-action given by (g · f)(x) = f(x) · g^{−1} for g ∈ G, f ∈ C_{r}, and x ∈ P. Then we have the following key computation [3, Cor. 5.2.5], [2]

H^{n}(G; ℤ_{p}[G]) = π_{−n}(Hom_{ℤ_{p}[G]}(P, ℤ_{p}[G])) ≅ \begin{cases}ℤ_{p}[det(L(G))] & n = d \\ 0 & n ≠ d \end{cases}

where the isomorphism is an isomorphism of G-modules. Therefore, there is an equivalence

C_{r} → Λ_{c}^{top}L(G).

Now, letting ϵ : ℤ_{p}[G] → ℤ_{p}, we get an induced map

ϵ_{∗} : C_{r} = Hom_{ℤ_{p}[G]}(P, ℤ_{p}[G]) → Hom_{ℤ[G]}(P, ℤ_{p}) = r^{∗}r_{∗}ℤ_{p}.

The adjoint of this map is a map

ϵ̃_{∗} : r_{!}C_{r} → r_{∗}ℤ_{p}

which is an equivalence by [3, Theorem 4.4.3]. The inverse of ϵ̃_{∗} in the derived category is our map ω. □

Theorem 2.4 (Adams isomorphism). The ω of the previous lemma then extends to an equivalence

ω : r_{∗}M → r_{!}(M ⊗ C_{r})

for any M ∈ DP_{p}(G).

Proof. The general approach of [1] produces the map ω. To show ω is an equivalence, we must verify that it induces an isomorphism on homology. The induced map

H^{n}(G; M) = π_{−n}(r_{∗}M) ⟶^{ω_{∗}} π_{−n}(r_{!}(M⊗Σ^{−d}ℤ_{p}[det(L(G))])) = H_{d−n}(G; M⊗ℤ_{p}[det(L(G))])

is shown to be an isomorphism in [3, Prop. 4.4.3]. □

Remark 2.5. We say that this is an “Adams isomorphism” because it gives a twisted equivalence between homotopy orbits and homotopy fixed points.

3. The Wirthmüller isomorphism

Consider the following commuting diagram of homomorphisms.

\xymatrix{H \ar[rr]^{s} \ar[dr]_{i} &  & e \\  & G \ar[ur]_{r}}

This induces the following collection of functors.

[figure: DP_{p}(H); s_{!}; s_{∗}; i_{!}; i_{∗}; DP_{p}; s^{∗}; r^{∗}; DP_{p}(G); i^{∗}; r_{!}; r_{∗}]

It is clear that s^{∗} = i^{∗}r^{∗}. The equivalence s_{!} ≃ r_{!}i_{!} is Shapiro’s lemma.

In the previous section, we saw that there existed dualizing objects C_{s} ∈ DP_{p}(H) and C_{r} ∈ DP_{p}(G) and equivalences

ω_{s} :s_{!}C_{s} ⟶^{≃} s_{∗}ℤ_{p}
ω_{r} :r_{!}C_{r} ⟶^{≃} r_{∗}ℤ_{p}.

These dualizing objects are given by

C_{s} = Λ_{c}^{top}L(G)
C_{r} = Λ_{c}^{top}L(H)

In particular, they are invertible objects. Define C_{i} ∈ DP_{p}(H) by

C_{i} = i^{∗}C_{r} ⊗ C_{s}^{−1}.

Theorem 3.1 (The Wirthmüller isomorphism). For M ∈ DP_{p}(H), there is an equivalence

r_{∗}i_{!}M ≃ s_{∗}(C_{i} ⊗ M).

Proof. We compute:

r_{∗}i_{!}M ≃ r_{!}(C_{r} ⊗ i_{!}M)  (duality for G)
≃ r_{!}i_{!}(i^{∗}C_{r} ⊗ M)  (projection formula)
≃ s_{!}(i^{∗}C_{r} ⊗ M)  (Shapiro’s lemma)
≃ s_{!}(C_{s} ⊗ C_{i} ⊗ M)  (definition of C_{i})
≃ s_{∗}(C_{i} ⊗ M)  (duality for H).

□

This gains the more familiar form of a Wirthmüller isomorphism with the following explicit description of the object C_{i}.

Proposition 3.2. Let L(G/H) be the representation of H given by the tangent space T_{eH}(G/H). This is a free ℤ_{p}-module of rank equal to the codimension of H in G. Then there is an equivalence

C_{i} ≃ Λ_{c}^{top}L(G/H).

Proof. The the maps in the sequence

H ⟶^{i} G → pG/H

respect the conjugation action of H on each of the terms. Of course, the conjugation action of H on G/H is the action by left multiplication. Since H = p^{−1}(eH), we have a short exact sequence of the induced representations H-representations on tangent spaces

0 → L(H) ⟶^{D_{e}i} i^{∗}L(G) ⟶^{D_{e}p} L(G/H) → 0

The existence of an Ad-invariant metric (?) on L(G) allows us to conclude that this sequence splits, and thus we have an isomorphism

L(G/H) ⊕ L(H) ≅ i^{∗}L(G).

We conclude (Remark 2.2) that

C_{i} ⊗ C_{s} ≃ Λ_{c}^{top}L(G/H) ⊗ Λ_{c}^{top}L(H) ≅ i^{∗}Λ_{c}^{top}L(G) ≃ i^{∗}C_{r}

ADAMS AND WIRTHMÜLLER ISOMORPHISMS FOR COMPACT p-ADIC ANALYTIC GROUPS 5

from which the result follows. □

Applying homology to the equivalence of Theorem 3.1 together with Proposition 3.2, we get a group cohomological statement. Corollary 3.3. For H a codimension d closed subgroup of G, M ∈ P_{p}(H), there is an isomorphism

H^{n}(G; Ind_{H}^{G}(M)) ≅ H^{n−d}(H; M[det(L(G/H))]).

References

[1] H. Fausk, P. Hu, J.P. May.

[2] M. Lazard.

[3] P. Symonds, T. Weigel.