Classification of functors between categories of G-sets

S. Bouc

Abstract. Let G and H be finite groups. I give a one to one correspondence between the isomorphism classes of functors from G-sets to H-sets which preserve disjoint unions ans cartesian squares, and isomorphism classes of Hsets-G. The composition of functors leads to a new product on sets with a double action, which is a subset of the usual one. The associated double Burnside rings and categories are better behaved than the usual ones with respect to semi-simplicity. I give some applications to Hochschild (co)-homology and to Thévenaz-Webb’s version of Alperin’s conjecture.

1. Introduction

Let G be a finite group, and R be a commutative ring. There are several equivalent definitions of Mackey functors for G with values in the category R-mod of left R-modules (cf. [5]), and one of them is in terms of the category G-set of finite sets with a left G-action: a Mackey functor M for G with values in R-mod is a bifunctor from G-set to R-mod, i.e. a couple M = (M_{∗}, M^{∗}) of functors with M_{∗} covariant and M^{∗} contravariant, which coïncide on objets (i.e. M_{∗}(X) = M^{∗}(X) for any G-set X). This bifunctor is supposed to have the following two properties:

1. It transforms disjoint unions into direct sums: if X ∐ Y is the disjoint union of X and Y, and i_{X} and i_{Y} are the canonical injections from X and Y into X ∐ Y, then the morphisms M_{∗}(i_{X}) ⊕ M_{∗}(i_{Y}) and M^{∗}(i_{X}) ⊕ M^{∗}(i_{Y}) are mutual inverse isomorphisms from M(X) ⊕ M(Y) onto M(X ∐ Y).

2. It commutes on cartesian squares: if

X \underrightarrow{a} Y
b ↓  ↓ c
Z \overrightarrow{d} T

is a cartesian (or pullback) square, then

M_{∗}(b)M^{∗}(a) = M^{∗}(d)M_{∗}(c)

A morphism of Mackey functors is a natural transformation of bifunctors. The Mackey functors for G in R-mod form an abelian category Mack_{R}(G).

This definition suggests the following question: if G and H are two groups, are there functors F from G-set to H-set such that if M is a Mackey functor for H, then the composite functor M ∘F is a Mackey functor for G? There are well known examples of that situation: for instance, if H is a subgroup of G, and M a Mackey functor for H, then the induced functor Ind_{H}^{G}M is defined over the G-set X as

1991 Mathematics Subject Classification. Primary 20J15; Secundary 19A22.

(Ind_{H}^{G}M)(X) = M(Res_{H}^{G}X)

and so it is composed of M and the restriction functor from G-set to H-set.

In order to generalize that situation, it seems natural to look for functors F from G-set to H-set with the following two properties:

1. The functor F transforms disjoint unions into disjoint unions: with the same notation as above, the map F (i_{X}) ∐ F (i_{Y}) is a bijection of F (X) ∐ F (Y) into F (X ∐ Y).

2. The functor F transforms a cartesian square into a cartesian square. The object of the present work is to give a classification of such functors F.

2. Sets with a double action

2.1. Definitions. Let G and H be two groups. A set with a double action, or G-set-H, is a set U with a left G-action and a right H-action which commute, i.e. are such that

(g.u).h = g.(u.h) ∀(g, u, h) ∈ G × U × H

It turns out that the (isomorphism classes of) functors F from G-set to H-set which preserve disjoint unions and pullback squares are in one to one correspondence whith the (isomorphism classes of) finite H-sets-G. In order to give a precise statement, I need to define the following product on sets with a double action: if G, H and K are three groups, if U is a G-set-H, and V a H-set-K, then let

U ∘ V = {(u, v) ∈ U × V | ∀h ∈ H, u.h = u ⇒ ∃k ∈ K, h.v = v.k}

It is the set of couples (u, v) such that the left orbit of v under the right stabilizer of u is contained in the right orbit of v.

The group H acts on the right on U ∘ V by

(u, v).h = (u.h, h^{−1}.v)

and I define U ∘_{H} V as the quotient set

U ∘_{H} V = (U ∘ V)/H

The set U ∘_{H} V is a G-set-K with the following double action

g.(u, v).k = (g.u, v.k)

It is a subset of the usual reduced product U ×_{H} V, but it is a strict subset in general.

2.2. Classification of functors. Let G and H be two groups, and U be a finite H-set-G. If X is a G-set, or a G-set-{1}, then U ∘_{G} X is a H-set-{1}, i.e. an H-set. This provides a functor U ∘_{H} − from G-set to H-set. The precise statement mentioned above is the following:

Theorem 2.1. Let G and H be two groups. If F is a functor from G-set to H-set which preserves disjoint unions and cartesian squares, then there exists an H-set-G U, unique up to isomorphism of H-sets-G, such that F is isomorphic to the functor U ∘_{G} −. Conversally, if U is a H-set-G, then the functor U ∘_{G} − preserves disjoint unions and cartesian squares.

FUNCTORS BETWEEN CATEGORIES OF G-SETS 3

The following corollary is probably well-known:

Corollary 2.2. Let G and H be finite groups. Then the categories G-set and H-set are equivalent if and only if G and H are isomorphic.

2.3. Mackey formula.

Notation 2.3. If G and H are groups, and L is a subgroup of the product G × H, then let p_{1}(L) (resp. p_{2}(L)) be the projection of L in G (resp. in H), and let k_{1}(L) (resp. k_{2}(L)) be the (projection of the) intersection of L with G × {1} (resp. with {1} × H), i.e.

k_{1}(L) = {g ∈ G | (g, 1) ∈ L}

Let (G × H)/L be the set of left classes of L in G × H, viewed as a G-set-H with the following double action

g.(x, y)L.h = (gx, h^{−1}y)L

If K is another group, and M is a subgroup of H × K, let L ∗ M the subgroup of G × K defined by

L ∗ M = {(g, k) ∈ G × K | ∃h ∈ H, (g, h) ∈ L, (h, k) ∈ M}

With those notations:

Proposition 2.4. Let G, H and K be finite groups. If L is a subgroup of G × H and M is a subgroup of H × K, then

(G × H)/L ∘_{H} (H × K)/M = ∑_{{x∈p_{2}(L)\H/p_{1}(M) \atop k_{2}(L)^{x}⊆p_{1}(M)}}(G × K)/(L ∗^{(x,1)}M)

Remark 2.5. There is an analogue formula for the usual product ×_{H}, obtained by forgetting the second summation condition k_{2}(L)^{x} ⊆ p_{1}(M).

Proposition 2.6. Let G and H be finite groups, and L be a subgroup of H×G. If X is a G-set, then X^{k_{2}(L)} has a natural structure of p_{1}(L)-set, and

((H × G)/L) ∘_{G} X ≃ Ind_{p_{1}(L)}^{H}X^{k_{2}(L)}

Example 2.7. 1) Let H be a subgroup of G. If U is the group G, viewed as an H-set-G, then the functor U ∘_{G} − is isomorphic to the restriction functor from G-set to H-set. It leads to the induction functor between the corresponding categories of Mackey functors. If V is the group G, viewed as a G-set-H, then the functor V ∘_{H} − is the induction functor from G-set to H-set, which gives the restriction functor from Mack_{R}(G) to Mack_{R}(H).

2) Let N be a normal subgroup of G, and H = G/N. If U is the group H, viewed as an H-set-G, then the functor U ∘_{G} − is isomorphic to the fixed points functors X ↦ X^{N} from G-set to H-set. The corresponding functor between categories of Mackey functors is the inflation functor from Mack_{R}(H) to Mack_{R}(G). If V is the group H, viewed as an G-set-H, then the functor V ∘_{H} − is isomorphic to the inflation functor from H-set to G-set, and is associated to the co-inflation functor from Mack_{R}(G) to Mack_{R}(H).

3) If θ is an isomorphism of G on H, then H is an H-set-G with the right action of G given by θ. It corresponds of course to the transport by θ of G-set to H-set, and of Mack_{R}(H) to Mack_{R}(G).

4) It’s easy to see that any functor from G-set to H-set preserving disjoint unions and cartesian squares is a disjoint union of functors which are composed functors of the previous five types.

3. The associated double Burnside ring

The product on sets with a double action defined in the previous section is associative and left and right distributive with respect to disjoint unions. The products ∘_{G} gives a ring structure on the Grothendieck group Γ(G) of G-sets-G, which is a kind of double Burnside ring. The unit element of that ring is the group G, viewed as a G-set-G by left and right multiplication.

3.1. A subring of Γ(G). Let X and M be subgroups of G such that X ⊆ N_{G}(M). Let

Δ_{X,M} = {(a, b) ∈ G × G | a ∈ X, ab^{−1} ∈ M}

It is a subgroup of G × G if X ⊆ N_{G}(M). Let

t_{X,M} = (G × G)/Δ_{X,M}

Lemma 3.1. Let (X, M) and (Y, N) be couples of subgroups of G such that X ⊆ N_{G}(M) and Y ⊆ N_{G}(N). Then for any x ∈ G, the group X ∩^{x}Y normalizes M.^{x}N, which is a subgroup of G if M^{x} ⊆ Y. Moreover

t_{X,M} ∘_{G} t_{Y,N} = ∑_{{x∈X\G/Y \atop M^{x}⊆Y}} t_{X∩^{x}Y,M.^{x}N}

This lemma shows that the elements t_{X,N} generate a subring of Γ(G).

3.2. Projectors in the Burnside ring. The usual Burnside ring b(G), which is the Grothendieck group of G-set, has a natural structure of Γ(G)-module, obtained by extending the map X ↦ U ∘_{G} X by linearity. This gives a natural morphism Φ of rings from Γ(G) into End_{Z}(b(G)).

In [1], I introduced orthogonal projectors of the Burnside ring, associated to some families \underline{F} of subgroups of G having the following properties:

1. The family \underline{F} is stable under G-conjugacy.

2. The family \underline{F} is stable under product: if H and K are in \underline{F}, and if H ⊆ N_{G}(K), then H.K ∈ \underline{F}.

3. The trivial subgroup is in \underline{F}. The projector associated to the element P of \underline{F} could be defined by

π_{P}^{G}(X) = − ∑_{{inf s=P \atop s∈Sd(\underline{F})/N_{G}(P)}}(−1)^{|s|}Ind_{N_{G}(s)}^{G}X^{sup s}

where Sd(\underline{F}) is the set of strictly increasing sequences of elements of \underline{F}.

A natural question is then to ask if those endomorphisms of b(G) can be lifted through Φ to an orthogonal family of idempotents of Γ(G). Of course, the answer is yes:

Proposition 3.2. Let P an element of \underline{F}. Define E_{P}^{G} ∈ Γ(G) by

E_{P}^{G} = − ∑_{{inf s=P \atop s∈Sd(\underline{F})/N_{G}(P)}}(−1)^{|s|}t_{N_{G}(s),sup s}

Then Φ(E_{P}^{G}) = π_{P}^{G}, and the elements E_{P}^{G}, for P ∈ \underline{F} /G, form an orthogonal family of idempotents of Γ(G), and their sum is the identity.

3.3. A morphism from b(G) to Γ(G). The natural morphism from b(G) to End_{Z}(b(G)) induced by multiplication can also be lifted to a ring morphism from b(G) to Γ(G):

Lemma 3.3. Let X be a G-set, and X̃ be the set G × X, viewed as a G-set-G by

g.(a, x).g′ = (gag′, g.x)

Then the map X ↦ X̃ extends to a ring homomorphism from b(G) to Γ(G), and for any G-set Y

X̃ ∘_{G} Y = X̃ ×_{G} Y ≃ X × Y

The main consequence of this lemma is that any decomposition of unity into orthogonal idempotents in b(G) can be transported to Γ(G). In particular, if K is a field of characteristic 0 or coprime to the order of G, then the Burnside ring b_{K}(G) = K ⊗_{Z} b(G) is semi-simple, and there are explicit formulae, due to Gluck ([3]), giving its primitive idempotents: they are indexed by the conjugacy classes of subgroups of G, and the idempotent e_{H}^{G} associated to the class of the subgroup H is given by

e_{H}^{G} = \frac{1}{|N_{G}(H)|} ∑_{K⊆H} |K|χ̃]K, H[.(G/K)

where χ̃]K, H[ is the reduced Euler-Poincaré characteristic of the “open interval” ]K, H[ in the poset of subgroups of G.

The family ẽ_{H}^{G} gives then another decomposition of unity in orthogonal idempotents in the ring Γ_{K}(G) = K ⊗ Γ(G).

3.4. The idempotents F_{K,H}^{G}. Taking for \underline{F} the family of all subgroups of G, I have now two different decomposition of unity in orthogonal idempotents in Γ_{K}(G) (namely the E_{H}^{G} and the ẽ_{H}^{G}). It can be shown that the products ẽ_{G}^{G} ∘_{G} E_{H}^{G} are zero unless H is a normal subgroup of G, and a few transformations lead to the following proposition:

Proposition 3.4. Let K be a field of characteristic 0 or coprime to |G|. Let H\underline{◁}K be subgroups of G. Define

F_{K,H}^{G} = \frac{1}{|N_{G}(K, H)|} ∑_{{X⊆K \atop H⊆N\underline{◁}K}} |X|χ̃]X, K[χ̃]H, N[^{K}.t_{X,N}

where χ̃]H, N[^{K} is the reduced Euler-Poincaré characteristic of the “open interval” ]H, N[^{K} in the poset of normal subgroups of K.

Then the F_{K,H}^{G}, as (K, H) runs through a set of representatives of conjugacy classes of couples of subgroups of G such that H\underline{◁}K, are a set of mutually orthogonal idempotents of Γ_{K}(G), and their sum is the identity.

3.5. Central idempotents of Γ_{K}(G). The sum of idempotents F_{K,H}^{G} associated to couples (K, H) for which K/H is isomorphic to a given group is a central idempotent of Γ_{K}(G):

Proposition 3.5. Let H\underline{◁}K and H′\underline{◁}K′ be subgroups of G. If
F_{K′,H′}^{G} ∘_{G} Γ_{K}(G) ∘_{G} F_{K,H}^{G} ≠ 0

then the quotients K/H and K′/H′ are isomorphic.

Corollary 3.6. Let S be a finite group. Define

B_{S} = ∑_{(K,H) mod. G ; {H\underline{◁}K \atop K/H≃S}} F_{K,H}^{G}

Then the B_{S}, as S runs through a set of representatives of isomorphism classes of sections of G, are mutually orthogonal idempotent of the center of Γ_{K}(G), and their sum is the identity.

3.6. Identification of Γ_{K}(G). The previous proposition provides a decomposition of Γ_{K}(G) into a direct product of algebras B_{S} ∘_{G} Γ_{K}(G). Each of those pieces can be completely decribed up to isomorphism. Let me first fix some notations:

Notation 3.7. Let S be a finite group, and R_{S} be a system of representatives of conjugacy classes of couples (K, H) of subgroups of G such that H\underline{◁}K and K/H ≃ S. For (K, H) ∈ R_{S}, choose an isomorphism ι_{K,H} from K/H to S. It induces a morphism a_{K,H} of the group N_{G}(K, H)/K to the group Out(S) of outer automorphisms of S.

With those notations:

Theorem 3.8. Let K be a field of characteristic 0 or coprime to |G|. Then

1. The algebra Γ_{K}(G) is isomorphic to the direct product of the algebras B_{S} ∘_{G} Γ_{K}(G), as S runs through a set of representatives of isomorphism classes of sections of G.

2. For any section S of G, the algebra B_{S} ∘_{G} Γ_{K}(G) is isomorphic to the Hecke algebra

End_{Out(S)}( ⊕_{(K,H)∈R_{S}} Ind_{a_{K,H}(N_{G}(K,H)/K)}^{Out(S)}K)

Corollary 3.9. If the characteristic of K is coprime to |G| and |Out(S)| for all sections S of G, then the algebra Γ_{K}(G) is semi-simple.

4. Associated categories

The product ∘_{G} is associative and posesses a unit. It is natural to try to associate to it a category which objects are the finite groups, the morphisms the sets with a double action, the composition of morphisms being given by the product ∘_{G}. Its actually possible to refine this definition.

4.1. P-free-Q sets. Let P be a non-empty family of finite groups which is stable under taking subgroups, taking quotients, and taking extensions. This is equivalent to say that P is the family of finite groups with composition factors in a given family of simple groups.

If P and Q are two such families, if G and H are finite groups, and U is a G-set-H, I will say that U is P-free-Q if for any u ∈ U, the left stabilizer of u in G is in P and its right stabilizer is in Q. In the case where P and Q are reduced to the trivial group, a P-free-Q set is just a set which is free on the left and on the right.

Lemma 4.1. Let G, H and K be groups. Let A be a G-set-H and B be a H-set-K. If A and B are P-free-Q, then so is A ∘_{H} B.

4.2. The categories C(P, Q) and F_{R}(P, Q). Let C(P, Q) be the following category: the objects are the finite groups. A morphism in C(P, Q) from G to H is an element of the Grothendieck group of the category of H-sets-G which are P-free-Q: such elements are difference of two H-sets-G which are P-free-Q. The product of the morphism A − B from G to H and the morphism C − D from H to K is defined as

(C − D) ∘ (A − B) = ((C ∘_{H} A) ∐(D ∘_{H} B)) − ((C ∘_{H} B) ∐(D ∘_{H} A))

More generally, if R is a commutative ring, let C_{R}(P, Q) be the category R⊗C(P, Q), defined the same way with

Hom_{C_{R}(P,Q)}(G, H) = R ⊗ Hom_{C(P,Q)}(G, H)

Finally, let F_{R}(P, Q) be the category of R-linear functors from C_{R}(P, Q) to R-mod.

The category F_{R}(P, Q) is an abelian category, and I can talk about simple, projective, injective objects.

Proposition 4.2. The simple objects of F_{R}(P, Q) are in one to one correspondence with the isomorphism classes of couples (H, V), where H is a finite group, and V an ROut(H)-simple-module. If P ⊆ Q, then for any group G the value of the simple functor S_{H,V} on G is given by

S_{H,V} (G) ≃ ⊕_{(G_{1},N_{1})}T r_{1}^{N_{G}(G_{1},N_{1})/G_{1}}(V)

where (G_{1}, N_{1}) runs through a set of representatives of conjugacy classes of couples of subgroups of G such that N_{1}\underline{◁}G_{1}, N_{1} ∈ P, and G_{1}/N_{1} ≃ H.

4.3. Semi-simplicity. If G is a finite group, there is a largest normal subgroup of G which is in P, which I shall denote by O_{P}(G).

If K is a subgroup of G, and P a normal subgroup of K which is in P, let

Φ_{K,P}^{G} = ∑_{H\underline{◁}K ; {O_{P} (H)=P \atop H mod. N_{G}(K,P)}} F_{K,H}^{G}

Then it can be shown that the elements Φ_{K,P}^{G} are in fact in End_{C_{K}(P,P)}(G), which I shall denote by End_{K,P}(G). They give a decomposition of unity in mutually orthogonal idempotents of that algebra, and lead to a complete description of End_{K,P}(G), similar to theorem 3.8, proving that those algebras are also semi-simple.

They also provide a good notion of residue for an object M of F_{K}(P, P): the residue \overline{M}(G) of M at G is defined by

\overline{M}(G) = Im M(Φ_{1}^{G})

It can also be identified with the quotient of M(G) by the submodule ∑ M(f)(M(H)), for |H| < |G| and f ∈ Hom_{C_{K}(P,P)}(H, G).

Those residues provide the following decomposition:

Proposition 4.3. Let M be an object of F_{K}(P, P), and G a finite group. Then

M(G) ≃ ⊕_{(K,P)}\overline{M}(K/P)^{N_{G}(K,P)/K}

where (K, P) runs through a set of representatives of conjugacy classes of couples of subgroups of G such that P \underline{◁}K and P ∈ P.

A little more work gives then the following:

Theorem 4.4. If K is a field of characteristic 0, then the category F_{K}(P, P) is semi-simple: any object M splits as

M ≃ ⊕_{Γ}S_{Γ,\overline{M}(Γ)}

where Γ runs in a set of representatives of isomorphism classes of finite groups. A further decomposition of \overline{M}(Γ) into a sum of KOut(Γ)-simple modules gives a decomposition of M into a sum of simple functors.

5. Applications

5.1. Mackey functors. The main motivation of that work was the construction of functors between categories of Mackey functors, and the sets with a double action provide indeed such constructions: let G and H be finite groups, and U be a H-set-G. If M is a Mackey functor for the group H, and X is a G-set, define

(M ∘ U)(X) = M(U ∘_{G} X)

For a map of G-sets f : X → Y define

(M ∘ U)_{∗}(f) = M_{∗}(U ∘_{G} f) (M ∘ U)^{∗}(f) = M^{∗}(U ∘_{G} f)

Proposition 5.1. If M is a Mackey functor for H, and U is a H-set-G, then M ∘ U is a Mackey functor for G. The correspondence M ↦ M ∘ U is a functor from Mack_{R}(H) to Mack_{R}(G).

5.2. Hochschild (co)homology. Let G be a finite group, and [G] be a set of representatives of the conjugacy classes of G. Define

c_{G} = ∐_{g∈[G]} t_{C_{G}(g),<g>}

It is a G-set-G, and can be identified with its image in Γ(G). If M is a Mackey functor for G, let HM = M ∘ c_{G} be the Mackey functor for G obtained by composition with c_{G}. Then:

Proposition 5.2. Let M is a Mackey functor for G over the ring R, and HM = M ∘ c_{G}. Then for any subgroup K of G

HM(K) = ⊕_{k∈[K]}M(C_{K}(k))

In particular, if M is the i-th functor of cohomology H^{i}(−, R) (resp. the i-th functor of homology H_{i}(−, R)), then HM(K) is isomorphic to the i-th group of Hochschild cohomology HH^{i}(K, R) (resp. the i-th group of Hochschild homology HH_{i}(K, R)).

5.3. Mackey functors and Alperin’s conjecture. J.Thévenaz and P.Webb have proposed the following equivalent version of Alperin’s conjecture (cf. [4])

Conjecture 5.3. For any finite group G, and any prime number p, there exists Mackey functors M_{1} and M_{2}, over a field R of characteristic 0 or coprime to |G|, such that

1. For any subgroup H of G, the restrictions Res_{H}^{G}M_{1} and Res_{H}^{G}M_{2} are projective with respect to p-local subgroups of H.

2. For any subgroup H of G

dim_{R} M_{1}(H) − dim_{R} M_{2}(G) = np(H)

where np(H) is the number of simple non-projective kH-modules over an algebraically closed field k of characteristic p.

The formalism of sets with a double action provides an explicit form of that conjecture: let s_{G} be the idempotent E_{1}^{G}, when the family \underline{F} = \underline{s}_{p}(G) is the family of p-subgroups of G, i.e.

s_{G} = − ∑_{{inf s=\{1\} \atop s∈Sd(\underline{s}_{p}(G))/G}}(−1)^{|s|}t_{N_{G}(s),sup s}

Let

c′_{G} = ∐_{g∈[G_{p′}]} t_{C_{G}(g),<g>}

where the sum runs over a set or representatives of the conjugacy classes of p-regular elements of G.

Let Alp_{G} = c_{G} ∘_{G} s_{G}. Its possible to show that Alp_{G} is also equal to c′_{G} ∘_{G} s_{G}. Let R be any commutatice ring, and F P_{R} be the fixed points Mackey functor for the trivial RG-module R. Then:

Proposition 5.4. The virtual Mackey functor M = F P_{R} ∘ (c′_{G} − Alp_{G}) is the difference of Mackey functors M_{1} and M_{2} such that

1. For any subgroup H of G, the functors Res_{H}^{G}M_{1} and Res_{H}^{G}M_{2} are projective with respect to p-local subgroups of H.

2. For any subgroup H of G, the modules M_{1}(H) and M_{2}(H) are free over R and

rank_{R}M_{1}(H)−rank_{R}M_{2}(H) = l(H)−|H\Alp_{H}/H| = − ∑_{s∈Sd(s_{p}(H))/H}(−1)^{|s|}l(N_{H}(s))

where s_{p}(H) is the poset of non-trivial p-subgroups of H. In particular, Alperin’s conjecture is equivalent to

|G\Alp_{G}/G| = f_{0}(G)

for any finite group G.

5.4. Steinberg residues. Let M be an object of F_{R}(P, P), where P contains finite p-groups. Define the Steinberg residue of M at the group G by

SM(G) = Im M(s_{G})

Steinberg residues provide decomposition of objects of F_{R}(P, P), generalizing those obtained in [1]:

Proposition 5.5. If P contains p-groups, and M is an object of F_{R}(P, P), then for any group G

M(G) ≃ ⊕_{P ∈\underline{s}_{p}(G)/G}SM(N_{G}(P)/P )

5.5. Adjunction and generalized Steinberg modules. Let G and H be finite groups, and U be a H-set-G. The functor M ↦ M ∘ U from Mack_{R}(H) to Mack_{R}(G) admits a left adjoint N ↦ L_{U}(N) and a right adjoint N ↦ R_{U}(N), constructed in [2]. The functor L_{U} is left adjoint to an exact functor, and so it maps projectives to projectives. It is also easy to see that if P = \underline{s}_{p}(G), and if U is P-free-P, then L_{U}(N) is projective with respect to p-subgroups if N is.

Now if R is a field of characteristic p, then evaluation at the trivial subgroup gives a one to one correspondence between the (isomorphism classes of) projective Mackey functors which are projective with respect to p-subgroups and the (isomorphism classes of) trivial source modules. This allows a translation of the functor L_{U} in the category of trivial source modules: if L is such a module, this construction gives a module U ∘ L, which is a sum of trivial source modules.

It is easy to see that the virtual module s_{G} ∘ L obtained that way is what I called the generalized Steinberg module St(G, L) in [1].

References

1. Serge Bouc, Projecteurs dans l’anneau de Burnside, projecteurs dans l’anneau de Green, et modules de Steinberg généralisés, J. of Algebra 139 (1991), no. 2, 395–445.

2. , Foncteurs de Green et G-ensembles, Preprint, 1996.

3. D. Gluck, Idempotent formula for the Burnside ring with applications to the p-subgroup simplicial complex, Illinois J. Math. 25 (1981), 63–67.

4. Jacques Thévenaz and Peter Webb, A Mackey functor version of a conjecture of Alperin, Astérisque 181-182 (1990), 263–272.

5. , The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865– 1961.

S. Bouc, Equipe des groupes finis, UFR de Mathématiques, Université Paris 7-Denis Diderot, 2 place Jussieu, 75231. Paris Cedex 05, France

    E-mail address: bouc@mathp7.jussieu.fr