A general theory of canonical induction formulae^{∗}

Robert Boltje Institut für Mathematik Universität Augsburg 86135 Augsburg Germany email: boltje@math.uni-augsburg.de

September 25, 1996

(revised: November 7, 1997)

Abstract

So far there exist three independent constructions of two different canonical versions of Brauer’s induction theorem for complex characters due to V. Snaith, P. Symonds, and the author (see [23],[27],[2]). ‘Canonical’ in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows to construct canonical induction formulae for arbitrary Mackey functor. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as for example the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes.

Introduction

It is always very pleasant if a question about some higher dimensional object can be reduced to one-dimensional objects, as it is the case with the splitting principle for vector bundles (cf. [18, 17.5]) or with Brauer’s induction theorem for character rings of a finite group (cf. [8]). In both cases one knows that for each higher dimensional object there exists a family of one-dimensional objects by which the object itself or at least its interesting invariants are determined. These existence theorems have already fundamental consequences. In the case of Brauer’s theorem (and this was Brauer’s incentive) one deduces easily from the classically known facts for one-dimensional representations that the Artin L-function of an arbitrary complex Galois representation of a number field is meromorphically extendible to the whole complex plane and satisfies a functional equation. For this kind of qualitative questions the existence statement in Brauer’s theorem is sufficient. If one wants to study more detailed properties, as for example Artin’s conjecture, stating that the Artin L-function of a non-trivial irreducible representation is holomorphic, a more explicit version of this theorem would be desirable.

∗AMS Subject Classification 19A22, 20C11, 20C15, 20C20

Another important area in number theory is the conjectured Langlands correspondence between Galois representations of a local field K and representations of GL_{n}(K) for varying n ∈ ℕ. In this case again, one has a correspondence with all the required properties for the subsets of one-dimensional Galois representations and representations of GL_{1}(K) via local class field theory. By Brauer’s theorem one can associate to each Galois representation a family of one-dimensional Galois representations on finite extension fields; this determines via class field theory a family of representations of the unit groups of these extension fields, and one would like to have an induction process which associates to this family a (virtual) representation of GL_{n}(K) for some n ∈ ℕ. Let us assume for the moment that there is such a notion of induction. Then still one has the problem that Brauer’s theorem is only an existence theorem, and one has to show that a construction as indicated above would not depend on the special choice of a family of one-dimensional representations.

Surprisingly there are canonical choices for Brauer’s induction theorem, as Snaith proved in [23] by topological methods. Independently, using an algebraic approach the author introduced another canonical choice (cf. [2]). Both choices share the property of functoriality with respect to group homomorphisms. However, while the topological formula is not additive but gives a topological interpretation to the multiplicities of the occurring one-dimensional representations in terms of Euler characteristics (showing a priori that they are integers), the algebraic formula is additive but produces a priori only multiplicities in ℚ. In [2] we gave a proof for the integrality of these coefficients which made heavy use of the fact that we are dealing with complex characters, in contrast to the construction of the coefficients which was quite formal and invited to generalization. There is also a geometric interpretation of the algebraic formula due to Symonds, cf. [27], but like the topological approach this method seems not to be apt to generalize to repesentations over other rings than ℂ or ℝ.

There exists already a variety of applications of these canonical Brauer induction formulae, cf. [5], [24], [25], [26], and [27], and one would certainly like to have similar constructions for various other representation rings besides the character ring. The aim of this paper is to introduce the basic constructions, notions and results about a very general approach, producing canonical choices of induction formulae for many sorts of representation rings. We consider this article as a reference for future examples and applications. Therefore some parts are kept in a more general framework than necessary for the five examples of canonical induction formulae we already introduce here (see Examples 1.8 and 6.13), namely for the classical case of the character ring, the ring of Brauer characters, the Grothendieck group of projective modules for a finite group over a complete discrete valuation ring O, and the representation rings of trivial source modules and linear source modules over O.

The language of Mackey functors on a finite group G over a commutative base ring k is perfectly suited to formulate our results and constructions, and also motivates the idea of considering functorial choices of induction formulae. The notion of a Mackey functor axiomatizes structures allowing induction, restriction, and conjugation maps, as for example character rings or cohomology groups of G-modules.

We introduce Mackey functors and related structures (restriction functors and conjugation functors) in Section 1, where we also provide the necessary facts about them. Restriction functors and conjugation functors arise from Mackey functors by forgetting the induction maps and then also the restriction maps. In Section 2 we define basic functors between these three categories, namely the consturctions −_{+} and −^{+} which arise from adjoints of the forgetful functors. The functor −_{+} generalizes the construction of the Burnside ring from the constant restriction functor ℤ, and there is also a generalization of the mark homomorphism on the Burnside ring. In Section 3 we motivate and define the notion of a canonical induction formula as a morphism of restriction functors a: M → A_{+}, which splits the induction morphism b: A_{+} → M, for a Mackey functor M and a restriction subfunctor A of M, where A_{+}(G) is a quotient of ⨁_{H≤G} A(H) and b: A_{+}(G) → M(G) is induced by the induction maps. If M is the character ring Mackey functor and A ⊆ M is spanned by the linear characters, we obtain as an example the canonical induction formula in [2].

Section 4 gives a glimpse of future applications by extending morphisms on A to morphisms on M using a canonical induction formula. We will apply these results in subsequent papers to constructions of Adams operations and Chern classes on various representation rings different from the character ring.

In Section 5 we obtain for suitable base rings a parametrization of all morphisms of restriction functors a: M → A_{+} by the more convenient set of morphisms p: M → A of conjugation functors. Section 6 is devoted to the situation where the order of G is invertible in the base ring. Under this hypothesis we obtain complete answers in terms of p to the questions, if a is a splitting for b, a morphism of Mackey functors, or a ring homomorphism. In Section 7 we study the effect of a change of base rings with respect to the functors −_{+} and −^{+}, in order to be able to pass from results over ℚ, which we proved in Section 7, to result over ℤ, the case we are most interested in.

We place ourselves in a more specific situation in Section 8, where we work with the base ring ℤ and impose in Hypothesis 8.1 conditions on M and A which are satisfied in all the examples we are interested in. Crucial among them for what follows is the notion of a stable basis B of A, which, as for example the set of one-dimensional characters, is a ℤ-basis which is stable under conjugation maps and whose positive span is stable under restriction maps. For M, A, p: M → A, and B as in Hypothesis 8.1 we may tensor all objects with ℚ over ℤ in order to obtain an associated map a: ℚ ⊗ M → ℚ ⊗ A_{+} by our previous results. The most interesting question then is, whether a is integral, i.e. a maps M to A_{+}. In Section 9 we transform an explicit alternating sum formula for a in terms of p, in which the denominator |H| occurs for each subgroup H ≤ G, by refining the index set from the simplicial complex of chains of subgroups of G to the simplicial chain complex of a poset associated to the stable basis B. A further subtle refinement and rearrangement in this alternating sum makes apparent that under some condition (∗_{π}) on M, A, p, and B (π being a set of primes) we obtain an alternating sum formula whose denominators divide |G|_{π′}. This is the statement of Theorem 9.3 which provides a general tool for integrality proofs. We show that (∗_{π}) is satisfied for the set π of all primes in the cases of the character ring and the Brauer character ring, and for the set π of primes distinct from the prime characteristic l of the residue field of O in the case of the Grothendieck group of projective OG-modules. In the first two cases this completes the integrality proof, and in the third case we give an argument which shows that the remaining l-power denominators also vanish. An integrality proof using Theorem 9.3 for the introduced induction formulae in the case of trivial source modules and linear source can be found in [4].

In Section 10 we adopt a global point of view by considering M and A to be defined on all finite groups instead of only the subgroups of a given finite group, and we show how our previous results can be extended to this point of view. Moreover, we assume that there are restriction maps for all group homomorphisms, not only the subgroup inclusions, and show that the canonical induction formulae also commute with these restriction maps.

Finally, in Section 11 we give a method of computing a canonical induction formula a: M → A_{+} in the standard situation of Hypothesis 8.1 by giving a matrix equation Γ · α(m) = β(m) for the coefficients α(m) of a_{G}(m) ∈ A_{+}(G), m ∈ M(G), with respect to a special basis in A_{+}(G), where Γ is an upper triangular quadratic matrix with coefficients in ℤ which is independent of m.

Notation

Let G be a finite group, g ∈ G, and K, H ≤ G be subgroups of G. We set gH = gHg^{−1} and H^{g} = g^{−1}Hg, write H < G if H is a proper subgroup of G, and K =_{G} H if K and H are conjugate under G. For a set π of primes we denote by g_{π} the element of < g > whose order is precisely the π-part of the order of g. The exponent of G is denoted by exp(G) and the set of complex irreducible characters of G by Irr(G). For a G-set S we denote by S^{G} the set of G-fixed points. If k is a commutative ring, kG denotes the group ring and I(kG) the augmentation ideal of kG, i.e. the k-submodule of kG generated by the elements g − 1, g ∈ G. All rings and algebras are unitary, and homomorphisms of rings and algebras always preserve unities. For a ring R its group of units is denoted by R^{×}, the category of R-modules by R−Mod, and the full subcategory of finitely generated R-modules by R−mod. For the rank of a free R-module M ∈ R−mod we write rk_{R}M. Unadorned tensor products are taken over ℤ.

1 Mackey functors and related structures

Throughout this section G denotes a finite group and k a commutative ring.

1.1 Definition (a) A k-conjugation-functor (resp. k-algebra conjugation functor) on G is a pair (X, c) consisting of a family of k-modules (resp. k-algebras) X(H), H ≤ G, and a family of k-module (resp. k-algebra) homomorphisms c_{g,H}: X(H) → X(^{g}H), the conjugation maps, for H ≤ G and g ∈ G, satisfying the axioms

(C1) c_{h,H} = id_{X(H)} (Triviality),

(C2) c_{g′g,H} = c_{g′,^{g}H} ∘ c_{g,H} (Transitivity),

for all h ∈ H ≤ G and g, g′ ∈ G.

(b) A k-restriction functor (resp. k-algebra restriction functor) on G is a triple (A, c, res) consisting of a k-conjugation (resp. k-algebra conjugation) functor (A, c) on G together with a family of k-module (resp. k-algebra) homomorphisms res_{K}^{H}: A(H) → A(K), the restriction maps, for K ≤ H ≤ G, satisfying the axioms

(R1) res_{H}^{H} = id_{A(H)} (Triviality),

(R2) res_{L}^{K} ∘ res_{K}^{H} = res_{L}^{H} (Transitivity),

(R3) c_{g,K} ∘ res_{K}^{H} = res _{^{g}K}^{^{g}H} ∘ c_{g,H} (G-equivariance), for all L ≤ K ≤ H ≤ G and g ∈ G.

(c) A k-Mackey functor on G is a quadruple (M, c, res, ind) consisting of a k-restriction functor (M, c, res) and a family of k-module homomorphisms ind_{K}^{H}: M(K) → M(H), the induction maps, for K ≤ H ≤ G, satisfying the axioms

(M1) ind_{H}^{H} = id_{M(H)} (Triviality),

(M2) ind_{K}^{H} ∘ ind_{L}^{K} = ind_{L}^{H} (Transitivity),

(M3) c_{g,H} ∘ ind_{K}^{H} = ind _{^{g}K}^{^{g}H} ∘ c_{g,K} (G-equivariance),

(M4) res_{U}^{H} ∘ ind_{K}^{H} = ∑_{h∈U\H/K} ind_{U∩^{h}K}^{U} ∘ res_{U∩^{h}K}^{^{h}K} ∘ c_{h,K} (Mackey-formula),

for all L ≤ K ≤ H ≤ G, U ≤ H, and g ∈ G, where in (M4), h runs through a set of representatives in H for the double cosets U\H/K.

A k-Green functor on G is a k-Mackey functor (M, c, res, ind) on G such that each M(H), H ≤ G, is a k-algebra, the conjugation and restriction maps are k-algebra homomorphisms and the axioms

(M5) x · ind_{K}^{H}(y) = ind_{K}^{H}(res_{K}^{H}(x) · y),

ind_{K}^{H}(y) · x = ind_{K}^{H}(y · res_{K}^{H}(x)), (Frobenius axioms)

are satisfied for all K ≤ H ≤ G, x ∈ M(H), and y ∈ M(K).

(d) A morphism f: X → Y of k-conjugation (resp. k-restriction, resp. k-Mackey functors) X and Y on G (by abuse of notation we often write X instead of (X, c), etc.) is a family of k-module homomorphisms f_{H}: X(H) → Y (H), H ≤ G, commuting with conjugation maps (resp. conjugation and restriction maps, resp. conjugation, restriction, and induction maps). For a morphism of kalgebra conjugation functors (resp. k-algebra restriction functors, resp. k-Green functors) on G we require additionally that f_{H} is a k-algebra homomorphism for all H ≤ G.

1.2 Remark (a) From Definition 1.1 we obtain categories k−Con(G), k−Con_{alg}(G), k−Res(G), k−Res_{alg}(G), k−Mack(G), and k−Mack_{alg}(G). In each of these six categories we have an obvious notion of subfunctors. Injectivity (resp. surjectivity) of a morphism f: X → Y means injectivity (resp. surjectivity) of all f_{H}: X(H) → Y (H), H ≤ G. The categories k−Con(G), k−Res(G), and k−Mack(G) are abelian. For any map of commutative rings k → k′ there are scalar extension functors

k′ ⊗_{k} −: k−Con(G) → k′−Con(G), …,
k−Mack_{alg}(G) → k′−Mack_{alg}(G).

(b) Let X be an object in one of the six categories in part (a). Then the axioms (C1) and (C2) imply that the conjugation maps are isomorphisms and that they provide each X(H), H ≤ G, with a module structure over the group ring kN_{G}(H)/H. In the sequel we will often write ^{g}x instead of c_{g,H}(x) for g ∈ G, H ≤ G, and x ∈ X(H). If R_{G} denotes a set of representatives for the conjugacy classes of subgroups of G, then clearly k−Con(G) ≅ ∏_{H∈R_{G}} kN_{G}(H)/H − Mod.

1.3 Example (a) The Burnside rings Ω(H) and the cohomology groups H^{n}(H, V), H ≤ G, for fixed n ∈ ℕ_{0} and V ∈ ℤG−Mod, are examples of ℤ-Mackey functors on G (even of a ℤ-Green functor in the Burnside ring case). We refer to [12, §80] for the notation and basic results concerning the Burnside ring.

(b) There is a constant k-algebra restriction functor \underline{k} with \underline{k}(H) = k for all H ≤ G, and with all conjugation and restriction maps being the identity. Obviously \underline{k} is an initial object in the category k−Res_{alg}(G). In Section 2 we will see that the Burnside ring Green functor k ⊗ Ω arises from \underline{k} by a functor −_{+}.

1.4 Definition (cf. [15], [31], [28])

(a) Let X, Y, Z ∈ k−Mack(G). A pairing X ⊗_{k} Y → Z is a family of k-module homomorphisms

X(H) ⊗_{k} Y (H) → Z(H), x ⊗_{k} y ↦ x · y,

satisfying the axioms (P1) ^{g}(x · y) = ^{g}x · ^{g}y (G-equivariance), (P2) res_{K}^{H}(x · y) = res_{K}^{H}(x) · res_{K}^{H}(y) (compatibility with restrictions), (P3) ind_{K}^{H}(a) · b = ind_{K}^{H}(a · res_{K}^{H}(b)),

a′ · ind_{K}^{H}(b′) = ind_{K}^{H}(res_{K}^{H}(a′) · b′), (Frobenius axioms),

for K ≤ H ≤ G, g ∈ G, a′, x ∈ X(H), b, y ∈ Y (H), a ∈ X(K), b′ ∈ Y (K).

(b) Let A be a k-Green functor on G. A k-Mackey functor M on G together with a pairing A⊗_{k} M → M is called an A-module, if A(H)⊗_{k} M(H) → M(H) provides each M(H), H ≤ G, with an A(H)-module structure.

If A(H) is commutative for all H ≤ G, then an A-algebra is a k-Green functor B on G together with a morphism i: A → B of k-Green functors such that each B(H), H ≤ G, is an A(H)-algebra via i_{H}.

For a finite left G-set S we denote the class of S in the Burnside ring by [S] ∈ Ω(G). The elements [G/H] ∈ Ω(G), where H runs through a set of representatives for the conjugacy classes of subgroups of G, form a ℤ-basis of Ω(G) and also a k-basis of k ⊗ Ω(G). The following proposition (whose proof is left to the reader) points out the distinguished role the Burnside ring functor k⊗ Ω ∈ k−Mack_{alg}(G) plays for the categories k−Mack(G) and k−Mack_{alg}(G), namely the same role that ℤ plays for the categories of abelian groups (i.e. ℤmodules) and the categories of rings (i.e. ℤ-algebras).

1.5 Proposition ([15, Prop. 4.2], [31, Ex. 2.11], [28, Prop. 6.1])

(i) Every k-Mackey functor M on G has a unique structure of a k⊗Ω-module, namely

(k ⊗ Ω(H)) ⊗_{k} M(H) → M(H), [H/K] ⊗_{k} m ↦ ind_{K}^{H}(res_{K}^{H}(m)),

for K ≤ H ≤ G and m ∈ M(H).

(ii) For every k-Green functor A on G there is a unique morphism i: k⊗Ω → A of k-Green functors on G, namely

i_{H}: k ⊗ Ω(H) → A(H), [H/K] ↦ ind_{K}^{H}(1_{A(K)}),

for K ≤ H ≤ G, i.e. k ⊗ Ω is an initial object in k−Mack_{alg}(G). In particular, A has a unique k ⊗Ω-algebra structure. Moreover, i_{H} induces the unique k ⊗Ωmodule structure of A.

1.6 For a k-Mackey functor M on G and H ≤ G we define the k-submodule

I(M)(H) := ∑_{K<H} ind_{K}^{H}(M(K)) = ∑_{K<H} im(ind_{K}^{H}: M(K) → M(H))

of M(H). Axiom (M3) implies that I(M) is a k-conjugation subfunctor of M on G. Since morphisms of k-Mackey functors commute with induction maps, these submodules are preserved under such morphisms, and we obtain a functor

I: k−Mack(G) → k−Con(G).

Note that if M ∈ k−Mack_{alg}(G), then I(M)(H) is an ideal of M(H) for H ≤ G by the Frobenius axiom (M5).

Following Thévenaz (cf. [28]), we call a subgroup H of G primordial for M, if I(M)(H) ≠ M(H), i.e. if there is an element in M(H) which can not be obtained as a sum of properly induced elements. We denote the set of primordial subgroup for M by P(M). Note that for H ≤ G one has M(H) = ∑_{K≤H,K∈P(M)} ind_{K}^{H}(M(K)).

1.7 Dually to 1.6 we define for A ∈ k−Res(G) and H ≤ G the k-submodule

K(A)(H) := ⋂_{K<H} ker(res_{K}^{H}: A(H) → A(K))

of A(H). These submodules form a k-conjugation subfunctor of A and they are preserved under morphisms of restriction functors on G. Hence, we obtain a functor

K: k−Res(G) → k−Con(G).

Note that if A ∈ k−Res_{alg}(G), then K(A)(H) is an ideal in A(H), since the restriction maps are k-algebra homomorphisms.

A subgroup H ≤ G is called coprimordial for A, if K(A)(H) ≠ 0, i.e. if the elements in A(H) are not uniquely determined by proper restriction maps. We denote the set of coprimordial subgroups for A by C(A). Note that for H ≤ G two elements x, y ∈ A(H) are equal if and only if res_{K}^{H}(x) = res_{K}^{H}(y) for all K ≤ H with K ∈ C(A). More about coprimordial subgroups and the connection to primordial subgroups can be found in [1, III.1.11–1.18].

1.8 Example For a ring A and a group H we set Ĥ(A) := Hom(H, A^{×}), and regard it as a multiplicative abelian group in the case that A is commutative. In this paper we will mainly be interested in the following examples.

(a) The character rings R(H), H ≤ G, i.e. the free abelian groups on the sets Irr(H) of complex irreducible characters of H, form a ℤ-Green functor on G with the usual conjugation, restriction and induction maps. For H ≤ G let R^{ab}(H) ⊆ R(H) denote the ℤ-span of the subset Ĥ := Ĥ(ℂ) ⊂ Irr(H) of linear characters. Then R^{ab} with the inherited conjugation and restriction maps is a ℤ-algebra restriction functor on G. Note that, since an induced linear character may have non-linear constituents, R^{ab} ⊆ R is not a Mackey subfunctor. The set C(R) of coprimordial subgroups for R consists of the cyclic subgroups and the set P(R) of primordial subgroups consists of the elementary subgroups of G.

(b) Let F be an algebraically closed field of prime characteristic l > 0 . The Grothendieck rings R_{F} (H) of F H − mod, H ≤ G, with respect to short exact sequences, which we identify with the free abelian group on the set of isomorphism classes [V] of irreducible F H-modules V, form a ℤ-Green functor on G, and the subrings R_{F}^{ab}(H), H ≤ G, generated by the isomorphism classes [V] of F H-modules V with dim_{F} V = 1 form a ℤ-algebra restriction subfunctor of R_{F}. For a homomorphism φ: H → F^{×} we denote by F_{φ} the F -vector space F endowed with the H-action h · α = φ(h)α for h ∈ H, α ∈ F. For φ = 1 we write F instead of F_{1}. Then the map φ ↦ F_{φ} induces a bijection between Ĥ(F) and the set of isomorphism classes of one-dimensional F H-modules, and we will often identify φ with [F_{φ}], thus considering Ĥ(F) as ℤ-basis of R_{F}^{ab}(H). From the theory of Brauer characters it follows that the set C(R_{F}) of coprimordial subgroups is the set of cyclic l′-subgroups.

(c) Let O be a complete discrete valuation ring of characterstic zero containing a primitive exp(G)-th root of unity and assume that the residue field O/rad(O) is of positive characteristic l and algebrically closed. A linear source OG-module M is an OG-module, free of finite rank as O-module, such that the source of any of its indecomposable direct summands is a module of O-rank one. Equivalently, M is isomorphic to a direct summand of a monomial OGmodule, i.e. a direct sum of modules of the form ind_{H}^{G}(O_{φ}) for some H ≤ G and φ ∈ Ĥ(O). Still equivalent is the condition that res_{P}^{G}(M) is a monomial OP -module for all l-subgroups (or equivalently for a Sylow l-subgroup) P of G. These equivalences can be proved like the analogous statements on trivial source modules in [10]. Together with the usual conjugation, restriction, and induction maps, the Grothendieck rings L_{O}(H) with respect to direct sums of the categories OH−lin of linear source OH-modules, H ≤ G, form a ℤ-Green functor on G. Since direct summands of linear source modules are again linear source modules, L_{O}(G) is free on the isomorphism classes of indecomposable linear source OG-modules.

Note that, as in (b), the map φ ↦ O_{φ}, for φ ∈ Ĝ(O), induces a bijection between Ĝ(O) and the set of isomorphism classes of linear source OG-modules of O-rank one. For H ≤ G we denote the span of [O_{φ}] ∈ L_{O}(H), φ ∈ Ĥ(O), by L_{O}^{ab}(H), and obtain a ℤ-algebra restriction subfunctor L_{O}^{ab} ⊆ L_{O}. For more on linear source modules, i.e. a proof of the semisimplicity of ℂ ⊗ L_{O}(G) and the determination of its species, i.e. algebra homomorphisms ℂ ⊗ L_{O}(G) → ℂ, see [4].

(d) Let O be as in (c). There is a full subcategory OG−triv ⊆ OG−lin of trivial source OG-modules, i.e. linear source OG-modules, all of whose indecomposable summands have trivial source. For H ≤ G the Grothendieck ring T_{O}(H) of OH−triv is a subring of L_{O}(H), and they form a ℤ-Green subfunctor T_{O} of L_{O}. It is easy to see that an OH-module of the form O_{φ}, φ ∈ Ĥ(O), is a trivial source module, if and only if H/ker(φ) is an l′-group. We define T_{O}^{ab}(H) ⊆ L_{O}^{ab}(H) as the span of the elements [O_{φ}], φ ∈ Ĥ(O)_{l′}. Clearly, T_{O}^{ab} is a ℤ-algebra restriction subfunctor of T_{O}.

Note that in the literature trivial source OG-modules are often called lpermutation modules, since they are exactly those OG-modules M whose restriction res_{P}^{G}(M) is a permutation module for all l-subgroups (or equivalently for a Sylow l-subgroup) P of G. Also equivalent is the condition that M is isomorphic to a direct summand of a permutation module. Since a direct summand of a trivial source module is again a trivial source module, T_{O}(G) is a free abelian group on the set of isomorphism classes of indecomposable trivial source OG-modules.

We remark that for a field F of characteristic l as in (b), the similarly defined Green functor T_{F} is isomorphic to T_{O}, and T_{F}^{ab} is isomorphic to T_{O}^{ab}, cf. [10]. In fact, reduction modulo rad(O) induces an isomorphism T_{O} ≅ T_{O/rad(O)}.

Since T_{O}(H) and L_{O}(H) are ℤ-free for all H ≤ G, we have equalities C(T_{O}) = C(ℚ ⊗ T_{O}) and C(L_{O}) = C(ℚ ⊗ L_{O}). Moreover, in Proposition 6.2 we will show that C(ℚ ⊗ T_{O}) = C(ℚ ⊗ L_{O}), since T_{O} ⊆ L_{O} is an inclusion of Green functors. From Conlon’s induction theorem (cf. [12, Cor. 81.32]) it follows that C(ℚ ⊗ T_{O}) is the set of l-hypo-elementary subgroups, i.e. subgroups H, whose biggest normal l-subgroup has a cyclic l′-group as factor group, i.e. H/O_{l}(H) is cyclic.

(e) Let O be as in (c) and let OG−proj be the category of finitely generated projective OG-modules. Since each projective OG-module is a direct summand of a free module, OG−proj is a full subcategory of OG−triv. The Grothendieck groups P_{O}(H), of OH−proj, H ≤ G, with respect to direct sums, form a Mackey subfunctor P_{O} ⊆ T_{O} ⊆ L_{O} and are free abelian groups on the isomorphism classes of indecomposable projective OH-modules. A rank-one module O_{φ}, φ ∈ Ĥ(O), H ≤ G, is projective, if and only if H is an l′-group. Hence, we set P_{O}^{ab}(H) = T_{O}^{ab}(H) = L_{O}^{ab}(H) for l′-subgroups H ≤ G, and P_{O}^{ab}(H) = 0 otherwise, and obtain a ℤ-restriction subfunctor P_{O}^{ab} ⊆ P_{O}.

Note again that if F is a field of characteristic l > 0 as in (b), then for the similarly defined Mackey functor P_{F} and restriction functor P_{F}^{ab} we have P_{F} ≅ P_{O} and P_{F}^{ab} ≅ P_{O}^{ab}.

Moreover, by [22, Théorèmes 34, 36], the Mackey functor P_{F} is isomorphic to the subfunctor of the character ring functor R consisting of those virtual characters vanishing on l-singular elements, i.e. elements of order divisible by l. This shows that C(P_{F}) = C(P_{O}) is the set of cyclic l′-subgroups of G.

2 The two plus-constructions and the mark morphism

Let G and k be given as in Section 1. We are going to define two functors

−^{+} : k−Con(G) → k−Mack(G)

and

−_{+} : k−Res(G) → k−Mack(G)

together with a natural transformation

ρ^{A}: A_{+} → A^{+}

for every A ∈ k−Res(G). These constructions generalize well-known features of the Burnside ring. It will be obvious from the constructions that

\underline{k}_{+} ≅ k ⊗ Ω(G) and \underline{k}^{+}(G) ≅ ( ∏_{H≤G} k)^{G},

where G acts by permuting the components in the last product according to the conjugation action on the subgroups H ≤ G, and that under these identifications the map ρ\underline{ℤ} ; G is just the classical mark homomorphism

ρ_{G}: Ω(G) → ( ∏_{H≤G} ℤ)^{G}, [S] ↦ (|S^{H}|)_{H≤G}.

The functor −^{+} is part of Thévenaz’ definition of the twin functor of a Mackey functor (cf. [28, Sect. 4]). The functor −_{+} was considered in special cases by Deligne in [13] and also by Dress in [14].

2.1 For X ∈ k−Con(G) we define (X^{+}, c^{+}, res^{+}, ind^{+}) ∈ k−Mack(G) by

X^{+}(H) := ( ∏_{K≤H} X(K))^{H}

for H ≤ G, where h ∈ H acts on ∏_{K≤H} X(K) by the conjugation maps (c_{h,K})_{K≤H}. Each g ∈ G induces for H ≤ G a map

c_{g,H}^{+} := ∏_{K≤H} c_{g,K}: ( ∏_{K≤H} X(K))^{H} → ( ∏_{K≤H} X(^{g}K))^{^{g}H}.

For K ≤ H ≤ G we define

res^{+}_{K}^{H}: ( ∏_{L≤H} X(L))^{H} → ( ∏_{L≤K} X(L))^{K}

as the obvious projection map, and

ind^{+}_{K}^{H} := ∑_{h∈H/K} c_{h,K}^{+} : ( ∏_{L≤K} X(L))^{K} → ( ∏_{L≤H} X(L))^{H}

as the relative norm map, where we view the image of c_{h,K}^{+} as contained in ∏_{L≤H} X(L) filling up the additional components with zeros.

For a morphism f: X → Y in k−Con(G) we define

f_{H}^{+} := ∏_{K≤H} f_{K}: ( ∏_{K≤H} X(K))^{H} → ( ∏_{K≤H} Y (K))^{H}

for H ≤ G. It is a straight forward verification to show that −^{+} is a functor from k−Con(G) to k−Mack(G) and that the same definitions yield a functor −^{+} : k−Con_{alg}(G) → k−Mack_{alg}(G).

2.2 For (A, c, res)∈k−Res(G) we define (A_{+}, c_{+}, res_{+}, ind_{+})∈k−Mack(G) as follows. For H ≤ G let

A_{+}(H) := ( ⨁_{K≤H} A(K))_{H},

where we view ⨁_{K≤H} A(K) as a kH-module via the sum of the conjugation maps c_{h,K}: A(K) → A(^{h}K), h ∈ H, and where for a kH-module M the kmodule M_{H} of coinvariants is defined as M/N, N being the smallest KHsubmodule such that H acts trivially on M/N, i.e. N = I(kH)M, where I(kH) is the augmentation ideal of kH. We will abbreviate the image of a ∈ A(K), K ≤ H, in A_{+}(H) by [K, a]_{H}. Thus, if R_{H} is a set of representatives for the conjugacy classes of subgroups of H, we can write each element x ∈ A_{+}(H) as

x = ∑_{K∈R_{H}}[K, a_{K}]_{H}

for certain elements a_{K} ∈ A(K), K ∈ R_{H}, with

∑_{K∈R_{H}}[K, a_{K}]_{H} = ∑_{K∈R_{H}}[K, a′_{K}]_{H}

for elements a′_{K} ∈ A(K), K ∈ R_{H}, if and only if there exist n_{K} ∈ N_{H}(K) for each K ∈ R_{H} with a_{K} = ^{n_{K}}(a′_{K}).

For K ≤ H ≤ G and g ∈ G we define maps

c_{+}_{g,H}: A_{+}(H) → A_{+}(^{g}H), [U, a]_{H} ↦ [^{g}U, ^{g}a]_{^{g}H},
res_{+}_{K}^{H}: A_{+}(H) → A_{+}(K), [U, a]_{H} ↦ ∑_{h∈K\H/U}[K ∩ ^{h}U, res_{K∩^{h}U}^{^{h}U}(^{h}a)]_{K},
ind_{+}_{K}^{H}: A_{+}(K) → A_{+}(H), [V, b]_{K} ↦ [V, b]_{H},

where U ≤ H, V ≤ K, a ∈ A(U), and b ∈ A(V).

For a morphism f: A → B in k−Res(G) and H ≤ G we set

f_{+}_{H}: A_{+}(H) → B_{+}(H), [K, a]_{H} ↦ [K, f_{K}(a)]_{H},

where K ≤ H and a ∈ A(K). Then again, straight forward calculations show that −_{+} is a functor from k−Res(G) to k−Mack(G), and that the same definitions yield a functor −_{+} : k−Res_{alg}(G) → k−Mack_{alg}(G), where the k-algebra structure on A_{+}(H) for A ∈ k−Res_{alg}(G) and H ≤ G is defined by

[U, a]_{H} · [V, b]_{H} := ∑_{h∈U\H/V} [U ∩ ^{h}V, res_{U∩^{h}V}^{U} (a) · res_{U∩^{h}V}^{^{h}V} (^{h}b)]_{H},

where U, V ≤ H, a ∈ A(U), and b ∈ A(V). If each A(H), H ≤ G, is commutative, so is each A_{+}(H).

For A ∈ k−Res(G) (resp. A ∈ k−Res_{alg}(G)) and H ≤ G we define the map

ι_{H}^{A}: A(H) → A_{+}(H), a ↦ [H, a]_{H}

which is injective by Axiom (C1). This defines a morphism ι^{A}: A → A_{+} of k-restriction functors (resp. k-algebra restriction functors) on G, which is functorial in A.

Note that if A ∈ k−Res_{alg}(G) is commutative then A_{+}(H) is an A(H)- algebra via ι^{A}. If A is not commutative, this map is still a unitary ring homomorphism providing A_{+}(H) with an A(H)-module structure: a · [K, b]_{H} = [K, res_{K}^{H}(a)b]_{H}, for a ∈ A(H), b ∈ A(K), K ≤ H ≤ G.

2.3 For A ∈ k−Res(G) and H ≤ G we define

π_{H}^{A}: A_{+}(H) → A(H), [K, a]_{H} ↦ \begin{cases}a, & if K = H, \\ 0, & if K < H. \end{cases}

By Axiom (C1), π_{H}^{A} is well-defined, and it is clear that π^{A}: A_{+} → A is a morphism of k-conjugation functors on G. Since π_{H}^{A}: A_{+}(H) → A(H) is surjective with kernel ∑_{K<H} ind_{+}K^{H}(A_{+}(K)) for H ≤ G, we call π^{A} the Brauer morphism in analogy of the Brauer map in modular representation theory, cf. [12, §58A], and also [28, p. 29]. Moreover, for H ≤ G and x ∈ A_{+}(H) we call π_{H}^{A}(x) ∈ A(H) the residue of x, a notion introduced and studied by Puig in [20] and by Thévenaz in [28].

Next, for H ≤ G, we define

ρ_{H}^{A} := (π_{K}^{A} ∘ res_{+}_{K}^{H})_{K≤H}: A_{+}(H) → A^{+}(H) = ( ∏_{K≤H} A(K))^{H}.

Again by routine calculations one can show that ρ^{A}: A_{+} → A^{+} is a morphism in k−Mack(G) which is natural in A ∈ k−Res(G). If A ∈ k−Res_{alg}(G), then π^{A}: A_{+} → A is in k−Con_{alg}(G), and therefore, ρ^{A}: A_{+} → A^{+} is a morphism in k−Mack_{alg}(G).

As the following proposition shows, for each H ≤ G, the map

σ_{H}^{A}: A^{+}(H) → A_{+}(H),  (2.3.a)
(a_{K})_{K≤H} ↦ ∑_{L≤K≤H} |L|μ(L, K)[L, res_{L}^{K}(a_{K})]_{H},

is almost inverse to ρ_{H}^{A}, where μ(L, K) denotes the Möbius function of the poset of subgroups of H evaluated at (L, K), cf. [21] for generalities on Möbius functions.

2.4 Proposition For A ∈ k−Res(G) and H ≤ G one has

σ_{H}^{A} ∘ ρ_{H}^{A} = |H| · id_{A_{+}(H)} and ρ_{H}^{A} ∘ σ_{H}^{A} = |H| · id_{A^{+}(H)}.

In particular, the kernels and cokernels of ρ_{H}^{A} and σ_{H}^{A} are annihilated by |H|. If A_{+}(H) has trivial |H|-torsion, then ρ_{H}^{A} is injective, and if |H| is invertible in k, then ρ_{H}^{A} is an isomorphism.

Proof For U ≤ H and a ∈ A(U) we have

ρ_{H}^{A}([U, a]_{H}) = ((π_{K}^{A} ∘ res_{+}_{K}^{H})([U, a]_{H}))_{K≤H}

= ( ∑_{h∈K\H/U} π_{K}^{A}([K ∩ ^{h}U, res_{K∩^{h}U}^{^{h}U}(^{h}a)]_{K}))_{K≤H}

= ( ∑_{{h∈H/U \atop K≤^{h}U}} res_{K}^{^{h}U}(^{h}a))_{K≤H},

since for K ≤ H the relation K ≤ ^{h}U implies KhU = hU. Applying σ_{H}^{A} to this family we obtain

(σ_{H}^{A} ∘ ρ_{H}^{A})([U, a]_{H}) = ∑_{L≤K≤H} |L|μ(L, K) ∑_{{h∈H/U \atop K≤^{h}U}} [L, res_{L}^{^{h}U}(^{h}a)]_{H}

= \frac{1}{|U|} ∑_{h∈H} ∑_{L≤K≤^{h}U} μ(L, K)|L|[L, res_{L}^{^{h}U}(^{h}a)]_{H}.

Considering for h ∈ H the map

f: {L ≤ ^{h}U} → A_{+}(H), L ↦ |L|[L, res_{L}^{^{h}U}(^{h}a)]_{H},

the inner sum collapses by Möbius inversion (cf. [21, 3., Prop. 2]) to the element |^{h}U|[^{h}U, ^{h}a]_{H} = |U|[U, a]_{H} which yields

(σ_{H}^{A} ∘ ρ_{H}^{A})([U, a]_{H}) = \frac{1}{|U|} ∑_{h∈H} |U|[U, a]_{H} = |H|[U, a]_{H}.

Conversely, let (a_{K})_{K≤H} ∈ A^{+}(H). Then the U-component, U ≤ H, of the element (ρ_{H}^{A} ∘ σ_{H}^{A})((a_{K})_{K≤H}) ∈ A^{+}(H) is given by

(π_{U}^{A} ∘ res_{+}_{U}^{H} ∘ σ_{H}^{A})((a_{K})_{K≤H})

= ∑_{L≤K≤H} |L|μ(L, K)π_{U}^{A}(res_{+}_{U}^{H}([L, res_{L}^{K}(a_{K})]_{H}))

= ∑_{L≤K≤H} |L|μ(L, K) ∑_{{h∈H/L \atop U≤^{h}L}} res_{U}^{^{h}L}(^{h}(res_{L}^{K}(a_{K})))

= ∑_{L≤K≤H} μ(L, K) ∑_{{h∈H \atop U≤^{h}L}} res_{U}^{^{h}K}(^{h}(a_{K}))

= ∑_{h∈H} ∑_{U^{h}≤L≤K≤H} μ(L, K)res_{U}^{^{h}K}(^{h}(a_{K})).

Considering for each h ∈ H the map

f: {U^{h} ≤ K ≤ H} → A(U), K ↦ res_{U}^{^{h}K}(^{h}a_{K}),

the inner sum collapses by Möbius inversion (cf. [21, 3., Prop. 2]) to ^{h}a_{(U^{h})} = a_{U}, and we have

(π_{U}^{A} ∘ res_{+}_{U}^{H} ∘ σ_{H}^{A})((a_{K})_{K≤H}) = ∑_{h∈H} a_{U} = |H|a_{U},

which completes the proof of the proposition.

2.5 Remark Note that the above proposition reproves the explicit formula of Gluck, cf. [16] and [32], for the idempotents e_{H}^{(G)} ∈ ℚ ⊗ Ω(G), H ≤ G, (and also in k ⊗ Ω(G), if |G| is invertible in G), where e_{H}^{(G)} = ρ_{G}^{−1}((a_{K})_{K≤G}) with a_{K} = 1 if K =_{G} H, and a_{K} = 0 otherwise:

e_{H}^{(G)} = \frac{1}{|N_{G}(H)|} ∑_{L≤H} |L|μ(L, H)[G/L].  (2.5.a)

3 The definition of a canonical induction formula

Throughout this section let k be a commutative ring, G a finite group, M a k-Mackey functor on G, and A ⊆ M a k-restriction subfunctor of M, i.e. A(H) ⊆ M(H), H ≤ G, are k-submodules and stable under the conjugation and restriction maps of M.

3.1 For H ≤ G we define

b_{H}^{M,A}: A_{+}(H) → M(H), [K, a]_{H} ↦ ind_{K}^{H}(a),

where K ≤ H and a ∈ A(K). It is easy to see that b^{M,A}: A_{+} → M is a morphism of k-Mackey functors on G, which we will call the induction morphism of M from A. If M is a k-Green functor and A ⊆ M a k-algebra restriction subfunctor, then b^{M,A} is a morphism of k-Green functors on G.

3.2 Remark and Example Let k = ℤ, M = R the character ring Green functor, and A = R^{ab} as in Example 1.8 (a). By Brauer’s induction theorem (cf. [22, Théorème 20]), the induction morphism b^{R,R^{ab}} : R_{+}^{ab} → R is surjective. For fixed H ≤ G and χ ∈ R(H), the ‘different ways’ of writing χ as a ℤlinear combination of induced linear characters correspond bijectively to the set of elements in A_{+}(H) that are mapped to χ under b_{H}^{R,R^{ab}}, provided that one agrees to identify two such linear combinations, if one arises from the other by replacing sumands ind_{K}^{H}(ψ) by conjugate summands ind_{^{h}K}^{H}(^{h}ψ) for K ≤ H,

ψ ∈ K̂, and h ∈ H.

Now we fix H ≤ G. If we want to specify for each χ ∈ R(H) a preferred way of writing χ as a ℤ-linear combination of induced linear characters, this amounts to choosing a map a_{H}: R(H) → R_{+}^{ab}(H) such that b_{H}^{R,R^{ab}} ∘ a_{H} = id_{R(H)}. Of course there are many different possible choices for a_{H} in general.

Next we consider all subgroups H ≤ G simultaneously. Knowing that R_{+}^{ab} and R are Mackey functors on G and that b^{R,R^{ab}} is a morphism of Mackey functors, it is just natural to require that a: R → R_{+}^{ab} be also a morphism of Mackey functors. However, there is no family of maps a_{H}: R(H) → R_{+}^{ab}(H),

H ≤ G, commuting with induction maps and satisfying b_{H}^{R,R^{ab}} ∘ a_{H} = id_{R(H)} for all H ≤ G. In fact, if we assume that (a_{H})_{H≤G} is such a family, then Artin’s induction theorem (cf. [12, 15.4]), namely

|H| · R(H) ⊆ ∑_{{K≤H \atop K cyclic}} ind_{K}^{H}(R(K))

for H ≤ G implies that

|H| · a_{H}(R(H)) ⊆ ∑_{{K≤H \atop K cyclic}} ind_{+}_{K}^{H}(R_{+}^{ab}(K))

for H ≤ G. Recalling the definition of ind_{+}_{K}^{H} for K ≤ H ≤ G from 2.2 we observe from this inclusion that |H| · a_{H}(R(H)) is contained in the ℤ-span of the elements [K, ψ]_{H}, where K ≤ H is a cyclic group and ψ ∈ K̂. Since these elements are part of a ℤ-basis of R_{+}^{ab}(G) (as we will see in Lemma 7.2), also

a_{H}(R(H)) must be contained in this span. But then b_{H}^{R,R^{ab}} ∘ a_{H} = id_{R(H)} implies

R(H) = ∑_{{K≤H \atop K cyclic}} ind_{K}^{H}(R(K))

which is certainly not true for arbitrary finite groups H. The best we can hope for in this example is a morphism a: R → R_{+}^{ab} of ℤ-restriction functors on G with b^{R,R^{ab}} ∘ a = id_{R}. Such a morphism was constructed in [2].

The above considerations motivate the following definition.

3.3 Definition A canonical induction formula for M from A is a morphism a ∈ k−Res(G)(M, A_{+}) with b^{M,A} ∘ a = id_{M}.

3.4 Remark As examples we will introduce in this paper canonical induction formulae for R from R^{ab} (as already done in [2]), for R_{F} from R_{F}^{ab}, for L_{O} from L_{O}^{ab}, for T_{O} from T_{O}^{ab}, and for P_{O} from P_{O}^{ab}, the notation being the one introduced in Example 1.8.

Note that our notation is different from the one in [2]. What we denote here by R_{+}^{ab}(G) was denoted there by R_{+}(G), which in view of 2.2 has a different meaning here.

4 An application: Extending morphisms

In this section we will indicate how to use canonical induction formulae in order to extend certain morphisms. In subsequent papers we will apply this to construct Adams operations and Chern classes on various representation rings.

4.1 Throughout this section we assume that K is a commutative ring and G is a finite group. We fix a k-restriction functor M on G (not necessarily a Mackey functor) and a k-restriction subfunctor A ⊆ M. Furthermore we assume that there is a morphism a ∈ k−Res(G)(M, A_{+}) such that a_{H}(φ) = [H, φ]_{H} for all H ≤ G and φ ∈ A(H), a condition which will be satisfied in all the examples of canonical induction formulae considered in this paper, namely for M and A as listed in Remark 3.4, cf. Proposition 6.12.

4.2 Definition For M, A, and a: M → A_{+} as in 4.1 and for any N ∈ k−Mack(G) we define a k-linear map

Φ_{N}^{M,A}: k−Res(G)(M, N) → k−Res(G)(A, N), F ↦ F |_{A},

by restricting a morphism on M to one on A. More interestingly, in the other direction, we define a k-linear map

Σ_{N}^{M,A,a}: k−Res(G)(A, N) → k−Res(G)(M, N), f ↦ b^{N,N} ∘ f_{+} ∘ a,

and call Σ_{N}^{M,A,a}(f) the canonical extension of f ∈ k−Res(G)(A, N) with respect to a.

This terminology is justified by the following theorem.

4.3 Theorem Let M, A, and a: M → A_{+} be as in 4.1, and let N ∈ k−Mack(G). Then Φ_{N}^{M,A} ∘ Σ_{N}^{M,A,a} = id. In particular, the map Φ_{N}^{M,A} is surjective, i.e. each morphism f: A → N of k-restriction functors on G can be extended to a morphism F: M → N of restriction functors on G.

Proof We have to show that (b_{H}^{N,N} ∘ f_{+H} ∘ a_{H})(φ) = f_{H}(φ) for all f ∈ k−Res(G)(A, N), H ≤ G, and φ ∈ A(H). Since a_{H}(φ) = [H, φ]_{H}, we obtain

(b_{H}^{N,N} ∘ f_{+H} ∘ a_{H})(φ) = b_{H}^{N,N}(f_{+H}([H, φ]_{H})) = b_{H}^{N,N}([H, f_{H}(φ)]_{H})
= ind_{H}^{H}(f_{H}(φ)) = f_{H}(φ),

which completes the proof.

The following theorem describes a situation where Φ_{N}^{M,A} and Σ_{N}^{M,A,a} are inverse isomorphisms, in particular where the canonical extension map Σ_{N}^{M,A,a} does not depend on a.

4.4 Theorem Let M, A, and a: M → A_{+} be as in 4.1, and let N ∈ k−Mack(G) be such that A(H) = M(H) for all H ∈ C(N). Then Φ_{N}^{M,A} and Σ_{N}^{M,A,a} are inverse isomorphisms.

Proof It suffices to show that Σ_{N}^{M,A,a} ∘ Φ_{N}^{M,A} = id, i.e. b_{H}^{N,N} ∘ (F |_{A})_{+H} ∘ a_{H} = F_{H} for all F ∈ k−Res(G)(M, N) and H ≤ G. By the last property of C(N) mentioned in 1.7, it is enough to show that

res_{K}^{H} ∘ b_{H}^{N,N} ∘ f_{+H} ∘ a_{H} = res_{K}^{H} ∘ F_{H}

for all K ≤ H with K ∈ C(N), where f := F |_{A}. Since b, f_{+}, a, and F commute with restrictions, it suffices to show b_{K}^{N,N} ∘f_{+K} ∘a_{K} = F_{K} for all K ∈ C(N). But for K ∈ C(N) we have A(K) = M(K), hence (b_{K}^{N,N} ∘ f_{+K} ∘ a_{K})(φ) = f_{K}(φ) = F_{K}(φ) for all φ ∈ M(K) as in the proof of Theorem 4.3.

5 Adjunctions

Throughout this section let k be a commutative ring and G a finite group. We will try to obtain an overview of the set of morphisms a: M → A_{+} of k-restriction functors on G in the situation of Definition 3.3.

5.1 For A ∈ k−Res(G) and H ≤ G we recall the map

ι_{H}^{A}: A(H) → A_{+}(H), a ↦ [H, a]_{H},

from 2.2, which is induced by the inclusion A(H) → ⨁_{K≤H} A(K). Note that π^{A} ∘ ι^{A} = id_{A}.

For X ∈ k−Con(G) and H ≤ G we set

ω_{H}^{X}: X^{+}(H) → X(H), (x_{K})_{K≤H} ↦ x_{H},

i.e. ω_{H}^{X} is induced by the projection ∏_{K≤H} X(K) → X(H). Then ω^{X}: X^{+} → X is a morphism of k-conjugation functors on G, even of k-algebra conjugation functors, if X ∈ k−Con_{alg}(G). Moreover, ω^{X} is functorial in X.

A statement similar to part (i) of the following proposition can be found in [7, Lemme 2.10].

5.2 Proposition (i) The forgetful functor k−Mack(G) → k−Res(G) is right adjoint to −_{+} : k−Res(G) → k−Mack(G). More precisely, for A ∈ k−Res(G) and M ∈ k−Mack(G) one has k-linear inverse isomorphisms

κ_{A,M}: k−Mack(G)(A_{+}, M) → k−Res(G)(A, M)
(f_{H})_{H≤G} ↦ (f_{H} ∘ ι_{H}^{A})_{H≤G}
([K, a]_{H} ↦ ind_{K}^{H}(g_{K}(a)))_{H≤G} ←(g_{H})_{H≤G}

which are natural in A and M. The same maps yield inverse natural bijection between k−Mack_{alg}(G)(A_{+}, M) and k−Res_{alg}(G)(A, M), if A ∈ k−Res_{alg}(G) and M ∈ k−Mack_{alg}(G).

(ii) The forgetful functor k−Res(G) → k−Con(G) is left adjoint to the composition k−Con(G) → k−Mack(G) → k−Res(G) of −^{+} with the forgetful functor of part (i). More precisely, for B ∈ k−Res(G) and X ∈ k−Con(G) one has k-linear inverse isomorphism

λ_{B,X}: k−Res(G)(B, X^{+}) → k−Con(G)(B, X)
(r_{H})_{H≤G} ↦ (ω_{H}^{X} ∘ r_{H})_{H≤G}
(b ↦ (p_{K}(res_{K}^{H}(b)))_{K≤H} )_{H≤G} ←(p_{H})_{H≤G}

which are natural in B an X. The same maps yield inverse natural bijections between k−Res_{alg}(G)(B, X^{+}) and k−Con_{alg}(G)(B, X), if B ∈ k−Res_{alg}(G) and X ∈ k−Con_{alg}(G).

Proof All assertions are immediate consequences of the very definitions of the categories and functors involved.

5.3 Our incentive from Section 3 is to obtain an overview of the morphisms in k−Res(G)(M, A_{+}) for M ∈ k−Mack(G) and A ⊆ M a k-restriction subfunctor. More generally, for arbitrary M, A ∈ k−Res(G), we can now define a map

Θ_{M,A}: k−Res(G)(M, A_{+}) → k−Res(G)(M, A^{+}) →^{λ_{M,A}} k−Con(G)(M, A),
a ↦ ρ^{A} ∘ a,

where the first map in this definition is composition which ρ^{A}. Since ω^{A} ∘ ρ^{A} = π^{A}, we have Θ_{M,A}(a) = π^{A}∘a for a ∈ k−Res(G)(M, A_{+}), and we call p := Θ(a) the residue of a. For H ≤ G we can recover a_{H} from p up to |H|-torsion by

|H|a_{H}(m) = ∑_{L≤K≤H} |L|μ(L, K)[L, res_{L}^{K}(p_{K}(res_{K}^{H}(m)))]_{H},  (5.3.a)

for m ∈ M(H). In fact, we just apply the inverse of λ_{M,A} to p and compose λ_{M,A}^{−1}(p)_{H} = ρ_{H}^{A} ∘a_{H} with σ_{H}^{A}, using σ_{H}^{A} ∘ρ_{H}^{A} = |H|id_{A_{+}(H)} from Proposition 2.4.

5.4 Corollary Let M, A ∈ k−Res(G), a ∈ k−Res(G)(M, A_{+}), and let p := Θ_{M,A}(a) = π^{A} ∘ a ∈ k−Con(G)(M, A) be the residue of a.

(i) For each H ≤ G the diagram

\begin{aligned}M(H) & ⟶^{a_{H}} & A_{+}(H) \\ ^{(p_{K}∘res_{K}^{H})_{K≤H}} ↘ & ↓^{ρ_{H}^{A}=(π_{K}^{A}∘res_{+}_{K}^{H})_{K≤H}} \\ & A^{+}(H) \end{aligned}  (5.4.a)

is commutative.

(ii) If A_{+}(H) has trivial |H|-torsion for all H ≤ G, then Θ_{M,A} is injective. (iii) If |G| is invertible in k, then Θ_{M,A} is an isomorphism. (iv) Assume that M, A ∈ k−Res_{alg}(G) and that ρ^{A} is injective. Then a ∈

k−Res_{alg}(G)(M, A_{+}), if and only if p ∈ k−Con_{alg}(G)(M, A).

Proof (i) For K ≤ H ≤ G we have π_{K}^{A} ∘ res_{+}_{K}^{H} ∘ a_{H} = π_{K}^{A} ∘ a_{K} ∘ res_{K}^{H} = p_{K} ∘ res_{K}^{H}.

(ii) This is immediate from Equation (5.3.a).

(iii) If |G| is invertible in k, then ρ^{A} is an isomorphism by Proposition 2.4, and hence Θ_{M,A} is an isomorphism by definition.

(iv) If a ∈ k−Res_{alg}(G)(M, A_{+}), then p = π^{A} ∘ a ∈ k−Con_{alg}(G)(M, A), since π^{A} ∈ k−Con_{alg}(G)(A_{+}, A), cf. 2.3. Now let p ∈ k−Con_{alg}(G)(M, A). Since ρ^{A} is an injective morphism of k-algebra restriction functors, it suffices to show that ρ^{A} ∘ a ∈ k−Res_{alg}(G)(M, A^{+}). But this follows immediately from the commutativity of Diagram (5.4.a), since res_{K}^{H}, K ≤ H ≤ G, are kalgebra homomorphisms and since the k-algebra structure of A^{+}(H) is defined componentwise.

5.5 Definition Let M, A ∈ k−Res(G) and assume that A_{+}(H) has trivial |H|-torsion for each H ≤ G. Then

Θ_{M,A}: k−Res(G)(M, A_{+}) → k−Con(G)(M, A), a ↦ π^{A} ∘ a,

is injective by Corollary 5.4 (ii) and for each p ∈ im(Θ_{M,A}) we define

a^{M,A,p} ∈ k−Res(G)(M, A_{+})

as the unique preimage of p under Θ_{M,A}.

6 Invertible group order

Throughout this section let G be a finite group and k a commutative ring such that |G| is invertible in k.

6.1 Our aim is to find canonical induction formulae a ∈ k−Res(G)(M, A_{+}) for given M ∈ k−Mack(G) and given k-restriction subfunctor A ⊆ M. By Corollary 5.4 (iii) we have an isomorphism

Θ_{M,A}: k−Res(G)(M, A_{+}) → k−Con(G)(M, A), a ↦ π^{A} ∘ a.

For p ∈ k−Con(G)(M, A) and a^{M,A,p} := Θ_{M,A}^{−1}(p), as defined in 5.5, we observe from Equation (5.3.a) the explicit formula

a_{H}^{M,A,p}(m) = \frac{1}{|H|} ∑_{L≤K≤H} |L|μ(L, K)[L, res_{L}^{K}(p_{K}(res_{K}^{H}(m)))]_{H},  (6.1.a)

for all H ≤ G and m ∈ M(H), and from Corollary 5.4 (i) the commutativity of the diagram

\begin{aligned}M(H) & ⟶^{a_{H}^{M,A,p}} & A_{+}(H) \\ ^{(p_{K}∘res_{K}^{H})_{K≤H}} ↘ & ↓^{ρ_{H}^{A}=(π_{K}^{A}∘res_{+}_{K}^{H})_{K≤H}} \\ & A^{+}(H) \end{aligned}  (6.1.b)

for all H ≤ G, where ρ_{H}^{A} is an isomorphism by Proposition 2.4.

In a first step we will determine those p ∈ k−Con(G)(M, A) which correspond under Θ_{M,A} to canonical induction formulae, i.e. which satisfy b^{M,A} ∘ a^{M,A,p} = id_{M}. Before we will do this we have to state a result about a natural decomposition of a Mackey functor in the invertible group order case, parts of which can be found in [15, Thm. 2] and [32, Thm. 4.1’].

For the notation of the following proposition we refer to Remark 2.5, Proposition 1.5 (i), and to 1.6 and 1.7.

6.2 Proposition Let M ∈ k−Mack(G) (resp. M ∈ k−Mack_{alg}(G)). For each H ≤ G we have a decomposition into k-submodules (resp. ideals)

M(H) = e_{H}^{(H)} · M(H) ⊕ (1 − e_{H}^{(H)}) · M(H)  (6.2.a)

and the summands are given by

e_{H}^{(H)} · M(H) = K(M)(H) and (1 − e_{H}^{(H)}) · M(H) = I(M)(H). (6.2.b)

In particular, M decomposes as k-conjugation functor (resp. k-algebra conjugation functor) as M = K(M) ⊕ I(M) (resp. M = K(M) × I(M)), and C(M) = P(M).

Moreover, for an inclusion M ⊆ N of k-Green functors we have C(M) = C(N).

Proof Since |G| is invertible in k we have e_{H}^{(H)} ∈ k ⊗ Ω(H) by Remark 2.5. Since the pairing (k ⊗ Ω) ⊗_{k} M → M of Proposition 1.5 (i) provides M(H) with the structure of a k ⊗ Ω(H)-module, we clearly have a decomposition as in (6.2.a).

For the proof of the remaining equations we first claim that for K < H ≤ G we have res_{K}^{H}(e_{H}^{(H)}) = 0. In fact, Since ρ\underline{k} is an isomorphism by Proposition 2.4, it suffices to show that ρ\underline{k} ; K(res_{K}^{H}(e_{H}^{(H)})) = 0 in \underline{k}^{+}(K). But ρ\underline{k} commutes with restrictions, and the definition of e_{H}^{(H)} in Remark 2.5 together with the definition of restriction on \underline{k}^{+}(H) proves the claim.

Using this we have res_{K}^{H}(e_{H}^{(H)} · M(H)) = res_{K}^{H}(e_{H}^{(H)}) · res_{K}^{H}(M(H)) = 0 for all K < H ≤ G. Conversely, the explicit formula (2.5.a) yields

1 − e_{H}^{(H)} = −\frac{1}{|H|} ∑_{K<H} |K|μ(K, H)[H/K],  (6.2.c)

and the definition of the pairing (k ⊗ Ω) ⊗_{k} M → M shows that K(M)(H) = ⋂_{K<H} ker(res_{K}^{H}: M(H) → M(K)) is annihilated by 1 − e_{H}^{(H)}, and hence contained in e_{H}^{(H)} · M(H). This proves the first part of (6.2.b).

Again the definition of the pairing (k ⊗ Ω) ⊗_{k} M → M and Equation (6.2.c) imply (1 − e_{H}^{(H)}) · M(H) ⊆ I(M)(H). For the opposite inclusion it suffices to show that e_{H}^{(H)} · ind_{K}^{H}(M(K)) = 0 for K < H. But this is immediate from the Frobenius axioms (P3) in Definition 1.4 and the claim above.

If M ⊆ N is an inclusion of k-Green functors, then we have

H ∈ C(M) ⟺ K(M)(H) ≠ 0 ⟺ e_{H}^{(H)} · M(H) ≠ 0 ⟺ e_{H}^{(H)} · 1_{M(H)} ≠ 0,

and similar for N. Since 1_{M(H)} = 1_{N(H)}, this shows C(M) = C(N).

6.3 Corollary Let M ∈ k−Mack(G), H ≤ G, and m ∈ M(H). Then m = 0, if and only if e_{H}^{(H)} · m = 0 and res_{K}^{H}(m) = 0 for all K < H.

Proof The condition res_{K}^{H}(m) = 0 for all K < H is equivalent to m ∈ K(M)(H), hence, by Proposition 6.2, also to m = e_{H}^{(H)} · m, and the result follows.

6.4 Proposition Let M ∈ k−Mack(G), A ⊆ M a k-restriction subfunctor, p ∈ k−Con(G)(M, A), and a^{M,A,p} the corresponding morphism in k−Res(G)(M, A_{+}), cf. Definition 5.5. Then the following are equivalent:

(i) The morphism a^{M,A,p} is a canonical indution formula, i.e. b^{M,A}∘a^{M,A,p} = id_{M}.

(ii) For all H ≤ G and m ∈ M(H) one has e_{H}^{(H)} · (p_{H}(m) − m) = 0, i.e. p_{H}(m) − m ∈ ∑_{K<H} ind_{K}^{H}(M(K)) by Proposition 6.2.

Proof We set a := a^{M,A,p}. From Equation (6.1.a) we obtain

e_{H}^{(H)} · b_{H}^{M,A}(a_{H}(m)) =  (6.4.a)

= \frac{1}{|H|}e_{H}^{(H)} ∑_{L≤K≤H} |L|μ(L, K)(ind_{L}^{H} ∘ res_{L}^{K} ∘ p_{K} ∘ res_{K}^{H})(m)

= e_{H}^{(H)} · p_{H}(m),

for H ≤ G and m ∈ M(H), since e_{H}^{(H)} annihilates all summands with L < H by Proposition 6.2.

If a is a canonical induction formula for M from A, then Equation (6.4.a) shows that e_{H}^{(H)} · m = e_{H}^{(H)} · p_{H}(m).

Conversely, if e_{H}^{(H)} · m = e_{H}^{(H)} · p_{H}(m) for all H ≤ G and m ∈ M(H), then by Corollary 6.3 it suffices to show that

e_{H}^{(H)} · b_{H}^{M,A}(a_{H}(m)) = e_{H}^{(H)} · m and res_{K}^{H}(b_{H}^{M,A}(a_{H}(m))) = res_{K}^{H}(m)

for all K < H ≤ G and m ∈ M(H). Under our assumption, the first equation is just a restatement of Equation (6.4.a), and therefore holds. The second equation follows by induction on |H|. In fact, since b^{M,A} and a commute with restrictions, it suffices to show that (b_{K}^{M,A} ∘ a_{K})(m′) = m′ for all m′ ∈ M(K), K < H. but this follows from the first equation and the induction hypothesis on the second one.

6.5 Corollary Let M and A be as in the above Proposition. If there exists a canonical induction formula for M from A, then e_{H}^{(H)} · M(H) ⊆ A(H) for all H ≤ G.

6.6 Corollary Let M, A, and p be as in Proposition 6.4 and assume that A(H) = M(H) and p_{H} = id_{M(H)} for all H ∈ C(M). Then a^{M,A,p} is a canonical induction formula for M from A.

Proof Condition (ii) in Proposition 6.4 is obviously satisfied for H ∈ C(M), since under our assumption p_{H}(m) − m = 0, and also generally for H ∉ C(M), since e_{H}^{(H)} · M(H) = 0 for H ∉ C(M) by Proposition 6.2.

Assuming the notation of 6.1, we next determine those p∈k−Con(G)(M, A) whose corresponding morphism a^{M,A,p} ∈ k−Res(G)(M, A_{+}) is even a morphism of Mackey functors.

6.7 Proposition Let M ∈ k−Mack(G), A ⊆ M a k-restriction subfunctor, a ∈ k−Res(G)(M, A_{+}), and p := π^{A} ∘ a, i.e. a = a^{M,A,p}. Then the following are equivalent:

(i) a ∈ k−Mack(G)(M, A_{+}).

(ii) p_{H}(ind_{K}^{H}(M(K))) = 0 for all K < H ≤ G.

(iii) p_{H}((1 − e_{H}^{(H)}) · M(H)) = 0 for all H ≤ G.

Proof Statements (ii) and (iii) are equivalent by Proposition 6.2. We show that (i) and (ii) are equivalent, just assuming that ρ^{A} is injective which is of course satisfied if |G| is invertible in k by Proposition 2.4. By definition, a is a morphism of Mackey functors, if and only if

ind_{+}_{K}^{H} ∘ a_{K} = a_{H} ∘ ind_{K}^{H}: M(K) → A_{+}(H)

for all K ≤ H ≤ G. Since ρ_{H}^{A} is injective, this is equivalent to

π_{U}^{A} ∘ res_{+}_{U}^{H} ∘ ind_{+}_{K}^{H} ∘ a_{K} = π_{U}^{A} ∘ res_{+}_{U}^{H} ∘ a_{H} ∘ ind_{K}^{H}: M(K) → A(U)

for all H ≤ G and U, K ≤ H. We transform the left hand side by using the Mackey axiom and commutativity of a with restrictions and conjugations, and we transform the right hand side by using the commutativity of a with restrictions, the Mackey axiom, and the relation p_{U} = π_{U}^{A} ∘ a_{U} to obtain the equivalent condition

∑_{h∈U\H/K} π_{U}^{A}∘ind_{+}_{U∩^{h}K}^{U} ∘ a_{U∩^{h}K} ∘ res_{U∩^{h}K}^{^{h}K} ∘ c_{h,K}

= ∑_{h∈U\H/K} p_{U} ∘ ind_{U∩^{h}K}^{U} ∘ res_{U∩^{h}K}^{^{h}K} ∘ c_{h,K}: M(K) → A(U)

for all H ≤ G and K, U ≤ H. Since π_{U}^{A} ∘ ind_{+}_{U∩^{h}K}^{U} = 0 unless U ≤ ^{h}K, this is equivalent to

∑_{{h∈U\H/K \atop U≤^{h}K}} p_{U} ∘ res_{U}^{^{h}K} ∘ c_{h,K} = ∑_{h∈U\H/K} p_{U} ∘ ind_{U∩^{h}K}^{U} ∘ res_{U∩^{h}K}^{^{h}K} ∘ c_{h,K} (6.7.a)

for all H ≤ G and U, K ≤ H.

Now Equation (6.7.a) implies 0 = p_{H} ∘ ind_{K}^{H} for K < H ≤ G by choosing U = H. Conversely, if p_{H} ∘ ind_{K}^{H} = 0 for all K < H ≤ G, then Equation (6.7.a) holds, since the right hand side reduces to the left hand side.

6.8 Corollary Let M ∈ k−Mack(G), A ⊆ M a k-restriction subfunctor, and p ∈ k−Con(G)(M, A). Then a^{M,A,p} is a canonical induction formula and a morphism of k-Mackey functors on G if and only if

(id_{M(H)} − p_{H})(M(H)) ⊆ (1 − e_{H}^{(H)}) · M(H) ⊆ ker(p_{H})

for all H ≤ G.

Proof This is immediate from Proposition 6.4 and Proposition 6.7.

6.9 Example For each k-Mackey functor M on G there is a canonical induction formula for M of a minimal type (cf. Corollary 6.5), namely from A with A(H) := e_{H}^{(H)} · M(H), H ≤ G, and associated to p_{H}: M(H) → A(H), m ↦ e_{H}^{(H)} · m, H ≤ G. In fact, this choice of p satisfies the condition in Corollary 6.8, hence a^{M,A,p} ∈ k−Mack(G). From Equation (6.1.a) we have the explicit formula

a_{H}^{M,A,p}(m) = \frac{1}{|H|} ∑_{{K≤H \atop K∈C(M)}} |K|[K, e_{K}^{(K)} · res_{K}^{H}(m)]_{H},

for H ≤ G and m ∈ M(H), since res_{L}^{K} is trivial on e_{K}^{(K)} for L < K (as shown in the proof of Proposition 6.2) and e_{K}^{(K)} · M(K) = 0 for K ∉ C(M). Moreover, applying b^{M,A} and expanding e_{K}^{(K)} as in Equation (2.5.a) we obtain

m = \frac{1}{|H|} ∑_{{L≤K≤H \atop K∈C(M)}} |L|μ(L, K)ind_{L}^{H}(res_{L}^{H}(m))  (6.9.a)

for m ∈ M(H), H ≤ G. This generalizes Brauer’s explicit version (cf. [9] or [11, 15.4]) of Artin’s induction theorem for the character ring tensored with ℚ, where C(M) consists of the set of cyclic subgroups of G, and also an explicit version of Conlon’s induction theorem for the Green ring and various subrings tensored with ℚ, where C(M) consists of the set of l-hypo-elementary subgroups of G (cf. [29, Theorem D’] and see also [12, 81.31]).

Recall from 2.2 that for A ∈ k−Res_{alg}(G) the ring A_{+}(H) is a natural A(H)-module for all H ≤ G.

6.10 Proposition Let M ∈ k−Mack_{alg}(G), A ⊆ M a k-algebra restriction subfunctor, and p ∈ k−Con(G)(M, A). Then the maps a_{H}^{M,A,p}: M(H) → A_{+}(H) are k-algebra (resp. A(H)-module) homomorphisms for all H ≤ G, if and only if the maps p_{H}: M(H) → A(H) are k-algebra (resp. A(H)-module) homomorphisms for all H ≤ G.

Proof The statement about the k-algebra structure follows immediately from Corollary 5.4 (iv).

We set a := a^{M,A,p}. The maps a_{H}, H ≤ G, are A(H)-linear, if and only if a_{H}(b · m) = [H, b]_{H} · a_{H}(m) for all b ∈ A(H) and m ∈ M(H). By the injectivity and multiplicativity of ρ^{A} this is equivalent to (ρ_{H}^{A} ∘ a_{H})(b · m) = ρ_{H}^{A}([H, b]_{H}) · ρ_{H}^{A}(a_{H}(m)). Since ρ_{H}^{A} = (π_{K}^{A} ∘ res_{+}_{K}^{H})_{K≤H}, since a commutes with restrictions, since π_{K}^{A} ∘ a_{K} = p_{K}, and since res_{K}^{H}: M(H) → M(K) is multiplicative for K ≤ H ≤ G, this is equivalent to

p_{K}(res_{K}^{H}(b) · res_{K}^{H}(m)) = res_{K}^{H}(b) · p_{K}(res_{K}^{H}(m))

for all K ≤ H ≤ G, m ∈ M(H), and b ∈ A(H). This in turn is equivalent to p_{H} being A(H)-linear for all H ≤ G.

6.11 Proposition Let M, M′ ∈ k−Mack(G), A ⊆ M and A′ ⊆ M′ be k-restriction subfunctors, p: M → A and p′: M′ → A′ be morphisms of kconjugation functors on G, and let f: M → M′ be a morphism of k-restriction functors with f(A) ⊆ A′. Then the diagram

M ⟶^{a^{M,A,p}} A_{+}
f↓  ↓^{f_{+}}
M′ ⟶^{a^{M′,A′,p′}} A′_{+}

is commutative, if and only if the diagram

M ⟶^{p} A
f↓  ↓^{f}
M′ ⟶^{p′} A′

is commutative.

Proof We write a and a′ for a^{M,A,p} and a^{M′,A′,p′} respectively. If the first diagram commutes we compose it with π^{A′}: A′_{+} → A′ and obtain π^{A′} ∘ f_{+} ∘ a = π^{A′} ∘ a′ ∘ f. Since π^{A′} is natural in A′, we have π^{A′} ∘ f_{+} = f ∘ π^{A}, and we also have π^{A} ∘ a = p and π^{A′} ∘ a′ = p′, showing that f ∘ p = p′ ∘ f.

Now we assume that f ∘ p = p′ ∘ f: M → A′. By Proposition 2.4, ρ^{A′} is injective, and for the commutativity of the first diagram it suffices to show that

π_{K}^{A′} ∘ res_{+}_{K}^{H} ∘ f_{+H} ∘ a_{H} = π_{K}^{A′} ∘ res_{+}_{K}^{H} ∘ a′_{H} ∘ f_{H}

for all K ≤ H ≤ G. Now the left hand side is equal to

π_{K}^{A′} ∘ f_{+K} ∘ a_{K} ∘ res_{K}^{H} = f_{K} ∘ π_{K}^{A} ∘ a_{K} ∘ res_{K}^{H} = f_{K} ∘ p_{K} ∘ res_{K}^{H}

and the right hand side equals

π_{K}^{A′} ∘ a′_{K} ∘ f_{K} ∘ res_{K}^{H} = p′_{K} ∘ f_{K} ∘ res_{K}^{H},

and the result follows.

6.12 Proposition Let M ∈ k−Mack(G), A ⊆ M be a k-restriction subfunctor, p ∈ k−Con(G)(M, A), H ≤ G, and m ∈ M(H) such that

res_{K}^{H}(p_{H}(m)) = p_{K}(res_{K}^{H}(m))  (6.12.a)

for all K ≤ H, then a_{H}^{M,A,p}(m) = [H, m]_{H}. In particular this holds for all m ∈ A(H) and all H ≤ G, if p_{H}|_{A(H)} = id_{A(H)} for all H ≤ G.

Proof In view of the explicit formula (6.1.a) we have to show that

∑_{L≤K≤H} μ(L, K)|L|[L, res_{L}^{H}(p_{H}(m))]_{H} = |H|[H, p_{H}(m)]_{H}.

But this follows by Möbius inversion of the function K ↦ |K|[K, res_{K}^{H}(p_{H}(m))]_{H} on the poset of subgroups of H, cf. [21, 3., Prop. 2].

6.13 Example We assume the notation of Example 1.8.

(a) We define p ∈ ℤ−Con(G)(R, R^{ab}) for H ≤ G and χ ∈ Irr(G) by

p_{H}(χ) = \begin{cases}χ, & if χ(1) = 1, \\ 0, & if χ(1) > 1. \end{cases}

Since R(H) = R^{ab}(H) and p_{H} = id_{R(H)} for all cyclic subgroups of G, Corollary 6.5 implies that a = a^{ℚ⊗R,ℚ⊗R^{ab},ℚ⊗p} is a canonical induction formula; but a is not a morphism of Mackey functors, since in general p_{H}(ind_{K}^{H}(χ)) ≠ 0 for K < H ≤ G and χ ∈ R(K), cf. Proposition 6.7 and Remark 3.2. Moreover, a_{H} is R^{ab}(H)-linear, but not a ring homomorphism, cf. Proposition 6.10. By Proposition 6.12, a_{H}(φ) = [H, φ]_{H} for all H ≤ G and φ ∈ R^{ab}(H).

(b) We define p ∈ ℤ−Con(G)(R_{F} , R_{F}^{ab}) for H ≤ G and a simple F H-module V by

p_{H}([V]) = \begin{cases}[V], & if dim_{F} (V) = 1, \\ 0, & if dim_{F} (V) > 1. \end{cases}

Since R_{F} (H) = R_{F}^{ab}(H) for all cyclic l′-subgroups H of G, Corollary 6.5 implies that a = a^{ℚ⊗R_{F} ,ℚ⊗R_{F}^{ab},ℚ⊗p} is a canonical induction formula. As in part (a), a is not a morphism of Mackey functors, and a_{H} is R_{F}^{ab}(H)-linear but not a ring homomorphism for H ≤ G. Note that a_{H}([F_{φ}]) = [H, [F_{φ}]]_{H} for all H ≤ G and all φ ∈ Ĥ(F) by Proposition 6.12.

(c) We define p ∈ ℤ−Con(G)(L_{O}, L_{O}^{ab}) for H ≤ G and an indecomposable linear source F H-module V by

p_{H}([V]) = \begin{cases}[V], & if rk_{O}V = 1, \\ 0, & if rk_{O}V > 1. \end{cases}

Note that p_{H}(T_{O}(H)) ⊆ T_{O}^{ab}(H) and p_{H}(P_{O}(H)) ⊆ P_{O}^{ab}(H). Since P_{O}(H) = P_{O}^{ab}(H) and p_{H} = id_{P_{O}(H)} for all cyclic l′-subgroups of G, Corollary 6.5 implies that a = a^{ℚ⊗P_{O},ℚ⊗P_{O}^{ab},ℚ⊗p} is a canonical induction formula. By the results of this section we see that a is not a morphism of Mackey functors, but that a_{H} is P_{O}^{ab}(H)-linear for all l′-subgroups H ≤ G (note that P_{O}^{ab}(H) = 0, if l divides |H|), and that a_{H}([O_{φ}]) = [H, [O_{φ}]]_{H} for all l′-subgroups H ≤ G and all φ ∈ Ĥ(O).

In order to show that a^{ℚ⊗L_{O},ℚ⊗L_{O}^{ab},ℚ⊗p} and a^{ℚ⊗T_{O},ℚ⊗T_{O}^{ab},ℚ⊗p} are canonical induction formulae we have to go deeper into modular representation theory. This is done in [4] where we also show that a^{ℚ⊗L_{O},ℚ⊗L_{O}^{ab},ℚ⊗p} is integral, i.e. maps L_{O} to L_{O}^{ab}. Note that by Proposition 6.11 these three induction formulae are restrictions of one-another with respect to the inclusions P_{O} ⊆ T_{O} ⊆ L_{O} and P_{O+}^{ab} ⊆ T_{O+}^{ab} ⊆ L_{O+}^{ab}, cf. also Lemma 7.2 and Lemma 7.5.

7 Change of base ring and stable basis

Throughout this section let k be a commutative ring and G a finite group.

In the previous section we could give complete answers to some questions about canonical induction formulae in the situation where |G| is invertible in the base ring k (cf. Propositions 6.4, 6.7, 6.10, and 6.11). However, we are mainly interested in the base ring ℤ and therefore have to study how these results over ℚ can be used for the base ring ℤ. More generally we fix a homomorphism k → k′ between commutative rings for this section and study how the constructions A_{+} and A^{+} for A ∈ k−Res(G) behave under this extension of base rings.

Moreover, in Poposition 2.4 and Corollary 5.4 (ii) we needed the hypothesis that A_{+}(H) has trivial |H|-torsion for all H ≤ G in order to obtain injectivity of ρ^{A} and Θ_{M,A}. First we will establish this situation.

7.1 Definition Let A ∈ k−Res(G). A stable k-basis of A is a family B = (B(H))_{H≤G} of subsets B(H) ⊂ A(H), H ≤ G, such that B(H) is a k-basis of A(H) for each H ≤ G and c_{g,H}(B(H)) = B(^{g}H), for all g ∈ G and H ≤ G.

7.2 Lemma Let A ∈ k−Res(G) and let B = (B(H))_{H≤G} be a stable kbasis of A. For each H ≤ G the set M(H) := {(K, b) | K ≤ H, b ∈ B(K)} is a left H-set via the conjugation maps and isomorphic to the disjoint union ⋃_{K≤H} B(K) as H-set. Moreover, the elements [K, b]_{H} ∈ A_{+}(H), where (K, b) runs through a set R(H) ⊆ M(H) of representatives for the H-orbits of M(H), form a k-basis of A_{+}(H).

Proof The first statement is obvious. For each H ≤ G we have a kHisomorphism ⨁_{K≤H} A(H) ≅ kM(H) and a decomposition

kM(H) = kR(H) ⊕ k{b − ^{h}b | b ∈ R(H), h ∈ H}.

It is clear that the second summand on the right hand side is just I(kH)·kM(H), so that the composition

kR(H) ⊆ kM(H) ≅ ⨁_{K≤H} A(K) → A_{+}(H)

is an isomorphism of k-modules (cf. 2.2) sending (K, b) ∈ R(H) to [K, b]_{H}.

7.3 Remark Note that in the Examples 1.8 (a)–(e) the k-restriction functors R^{ab}, R_{F}^{ab}, L_{O}^{ab}, T_{O}^{ab}, and P_{O}^{ab} have a stable basis, namely, respectively B(H) = Ĥ, Ĥ(F), Ĥ(O), Ĥ(O)_{p′}, and Ĥ(O) if H is an l′-subgroup, ∅ otherwise, for H ≤ G. By Lemma 7.2, the groups R_{+}^{ab}(H), R_{F +}^{ab}(H), L_{O+}^{ab}(H), T_{O+}^{ab}(H), and P_{O+}^{ab}(H) are free abelian with basis [K, φ]_{H}, (K, φ) ∈ H\M(H), for H ≤ G.

7.4 For any X ∈ k−Con(G) (resp. X ∈ k−Con_{alg}(G)) there is a cononical morphism (i.e. natural in X) of k′-Mackey functors (resp. k′-Green functors),

k′ ⊗_{k} X^{+} → (k′ ⊗_{k} X)^{+},

which maps α ⊗_{k} (x_{K})_{K≤H} to (α ⊗_{k} x_{K})_{K≤H} for H ≤ G, α ∈ k′, x_{K} ∈ X(K), K ≤ H. Moreover, there is a canonical morphism of k-Mackey functors (resp. k-Green functors) on G, X^{+} → k′ ⊗_{k} X^{+}, mapping (x_{K})_{K≤H} to 1 ⊗_{k} (x_{K})_{K≤H} for H ≤ G, x_{K} ∈ X(K), K ≤ H.

For any A ∈ k−Res(G) (resp. A ∈ k−Res_{alg}(G)) there is a canonical morphism (i.e. natural in A) of k′-Mackey functors (resp. k′-Green functors),

k′ ⊗_{k} A_{+} → (k′ ⊗_{k} A)_{+},

which maps α ⊗_{k} [K, a]_{H} to [K, α ⊗_{k} a]_{H} for H ≤ G, α ∈ k′, a ∈ A(K), K ≤ H. Moreover, there is a canonical morphism of k-Mackey functors (resp. k-Green functors) on G, A_{+} → k′ ⊗_{k} A_{+}, which maps [K, a]_{H} to [K, 1⊗_{k} a]_{H} for H ≤ G, a ∈ A(K), K ≤ H.

It is easy to see that for A ∈ k−Res(G) the diagram

A_{+} ⟶ k′ ⊗_{k} A_{+} ⟶ (k′ ⊗_{k} A)_{+}
ρ^{A}↓  ↓_{k′⊗_{k}ρ^{A}}  ↓_{ρ^{k′⊗_{k}A}}
A^{+} ⟶ k′ ⊗_{k} A^{+} ⟶ (k′ ⊗_{k} A)^{+}

is commutative.

7.5 Lemma Let X ∈ k−Con(G) and A ∈ k−Res(G).

(i) If k′ is flat over k, then the canonical morphism k′ ⊗_{k} X^{+} → (k′ ⊗_{k} X)^{+} is an isomorphism.

(ii) The canonical morphism k′ ⊗_{k} A_{+} → (k′ ⊗_{k} A)_{+} is always an isomorphism.

(iii) If k′ and X(H), H ≤ G, are flat over k and if k → k′ is injective, then the canonical morphism X^{+} → k′ ⊗_{k} X^{+} is injective.

(iv) If A_{+}(H) is flat over k for all H ≤ G and if k → k′ is injective, then the canonical morphism A_{+} → k′ ⊗_{k} A_{+} is injective.

Proof (i) We show that if M is any kG-module, then the map k′ ⊗_{k} M^{G} → (k′ ⊗_{k} M)^{G}, α ⊗_{k} m ↦ α ⊗_{k} m, where α ∈ k′ and m ∈ M^{G}, is an isomorphism. In fact, this map is the composition of the following sequence of natural isomorphisms

k′ ⊗_{k} M^{G} ≅ Hom_{k′}(k′, k′) ⊗_{k} Hom_{kG}(k, M) ≅ Hom_{k′⊗_{k}kG}(k′ ⊗_{k} k, k′ ⊗_{k} M)
≅ Hom_{k′G}(k′, k′ ⊗_{k} M) ≅ (k′ ⊗_{k} M)^{G},

where we identify k′ with Hom_{k′}(k′, k′) and M^{G} with Hom_{kG}(k, M^{G}), k being the trivial kG-module, and where the second canonical map is an isomorphism, since k′ is flat over k and k is finitely presented as kG-module (cf. [19, Lemma I.4.1 (b)]).

(ii) We show that, for any kG-module M, the map k′ ⊗_{k} M_{G} → (k′ ⊗_{k} M)_{G}, α⊗_{k} (m+I(kG)·M) ↦ (α⊗_{k} m)+I(kG)·(k′ ⊗_{k} M), where α ∈ k′ and m ∈ M, is an isomorphism. In fact, identifying M_{G} with k ⊗_{kG} M, k being the trivial kG-module, via α ⊗_{kG} m ↦ αm + I(kG) · M for α ∈ k′ and m ∈ M, the above map is the composition of the following sequence of isomorphisms

k′ ⊗_{k} M_{G} ≅ k′ ⊗_{k} (k ⊗_{kG} M) ≅ k′ ⊗_{k′G} (k′ ⊗_{k} M) ≅ (k′ ⊗_{k} M)_{G},

where the middle isomorphism is given by α′ ⊗_{k} (α ⊗_{kG} m) ↦ α′ ⊗_{k′G} (α ⊗_{k} m) with inverse α′ ⊗_{k′G} (β′ ⊗_{k} m) ↦ α′β′ ⊗_{k} (1 ⊗_{kG} m) for α ∈ k, α′, β′ ∈ k′, and m ∈ M.

(iii) This follows from the commutativity of the diagram

k ⊗_{k} ( ∏_{K≤H} X(K))^{H} ⟶ k′ ⊗_{k} ( ∏_{K≤H} X(K))^{H}
↓  ↓
k ⊗_{k} ∏_{K≤H} X(K) ⟶ k′ ⊗_{k} ∏_{K≤H} X(K),

from the injectivity of the vertical maps (since k′ is flat over k), and the injectivity of the lower horizontal map (since ∏_{K≤H} X(K) is flat over k).

(iv) This is obvious.

7.6 Corollary Let A ∈ k−Res(G). If A has a stable basis, k′ is flat over k, and k → k′ is injective, then the left horizontal maps in Diagram (7.4.a) are injective, the right horizontal maps are isomorphisms, and the vertical morphisms are injective.

Proof The statement about the vertical morphisms follows from Proposition 2.4 and Lemma 7.2, and the one about the horizontal morphisms follows from Lemma 7.5 and Lemma 7.2.

8 The standard situation

In the sequel we will often assume the following standard situation.

8.1 Hypothesis M is a ℤ-Mackey functor on a finite group G, A is a ℤ-restriction subfunctor of M on G, and p ∈ ℤ−Con(G)(M, A) such that the following conditions are satisfied:

(i) M(H) is a free abelian group for all H ≤ G.

(ii) A has a stable basis B such that for all K ≤ H ≤ G and φ ∈ B(H), the element res_{K}^{H}(φ) ∈ A(K) is a linear combination

res_{K}^{H}(φ) = ∑_{ψ∈B(K)} m_{(K,ψ)}^{(H,φ)} · ψ

of the basis elements ψ ∈ B(K) with non-negative coefficients m_{(K,ψ)}^{(H,φ)} ∈ ℕ_{0}.

8.2 Remark We assume A ⊆ M, B, and p as in Hypothesis 8.1.

(a) Hypothesis 8.1 is clearly designed for the various representation rings of G. Note that the Examples 6.13, namely R^{ab} ⊆ R, R_{F}^{ab} ⊆ R_{F}, L_{O}^{ab} ⊆ L_{O}, T_{O}^{ab} ⊆ T_{O}, and P_{O}^{ab} ⊆ P_{O}, together with the respective morphisms p and the stable bases from Remark 7.3, satisfy this hypothesis.

(b) As a convention, the letters χ, θ, ξ, ζ will always denote elements of M(H), H ≤ G, and φ, ψ, λ, μ will always denote elements of A(H), H ≤ G. This should help the reader to switch from the abstract setting to the standard example R^{ab} ⊆ R.

(c) We call a pair (H, φ) with H ≤ G and φ ∈ B(H) a monomial pair. For H ≤ G the set of monomial pairs (K, ψ) with K ≤ H will be denoted by M(H), cf. the notation in Lemma 7.2. Note that M(H) is an H-poset (i.e. H acts via poset automorphisms) by the following definition

(L, λ) ≤ (K, ψ) : ⟺ L ≤ K and m_{(L,λ)}^{(K,ψ)} > 0,
h(K, ψ) := (^{h}K, c_{h,K}(ψ)),

for (L, λ), (K, ψ) ∈ M(H) and h ∈ H. Note that we need the non-negativity property in Hypothesis 8.1 (ii) to ensure that the relation ≤ is transitive. We write (L, λ) =_{H} (K, ψ) if (L, λ) and (K, ψ) lie in the same H-orbit, and we denote by N_{H}(K, ψ) the stabilizer of (K, ψ) in H. The posets M(H), H ≤ G, are subposets of M(G) and inherit the H-action from M(G).

(d) We recollect some results that hold in the situation of Hypothesis 8.1: For H ≤ G, A_{+}(H) is a free ℤ-module with basis {[K, ψ]_{H}}, where (K, ψ)

runs through a set of representatives of the H-orbits H\M(H), cf. Lemma 7.2. We identify this ℤ-basis of A_{+}(H) with the ℚ-basis {1⊗[K, ψ]_{H}} of ℚ⊗A_{+}(H) and write just [K, ψ]_{H} ∈ ℚ ⊗ A_{+}(H). We will always identify ℚ ⊗ A_{+} with (ℚ ⊗ A)_{+} and ℚ ⊗ A^{+} with (ℚ ⊗ A)^{+} via the natural isomorphisms definied in 7.4, cf. Corollary 7.6. Under these isomorphisms of Mackey functors, the structure morphisms π^{ℚ⊗A} and ℚ ⊗ π^{A} are also identified, as well as the maps ℚ⊗r_{H}^{p}: ℚ⊗M(H) → ℚ⊗A^{+}(H) and r_{H}^{ℚ⊗p}: ℚ⊗M(H) → (ℚ⊗A)^{+}(H), where we set r_{H}^{p} := λ_{M,A}^{−1}(p)_{H} = (p_{K} ∘res_{K}^{H})_{K≤H}, cf. Proposition 5.2 (ii). We will work with the Mackey functors ℚ ⊗ A_{+} and ℚ ⊗ A^{+} and denote the corresponding structural maps again by c_{+}, res_{+}, ind_{+}, c^{+}, res^{+}, ind^{+}, π^{A}, ρ^{A}, r^{p}, b^{M,A}. Note that ρ^{A}: A_{+} → A^{+} is injective and ρ^{A}: ℚ ⊗ A_{+} → ℚ ⊗ A^{+} is an isomorphism by Proposition 2.4. Under the above identifications we obtain a commutative diagram

ℚ−Res(G)(ℚ ⊗ M, ℚ ⊗ A_{+}) ⟶^{Θ_{ℚ⊗M,ℚ⊗A}} ℚ−Con(G)(ℚ ⊗ M, ℚ ⊗ A)
⋃  ⋃
ℤ−Res(G)(M, A_{+})  ⟶^{Θ_{M,A}}  ℤ−Con(G)(M, A),

where the vertical inclusions are described by extensions of scalars, and Θ_{M,A} and Θ_{ℚ⊗M,ℚ⊗A} are the maps of taking residues (cf. 5.3). By Proposition 5.4 (ii) and (iii), Θ_{M,A} is injective and Θ_{ℚ⊗M,ℚ⊗A} is an isomorphism. The set of canonical induction formulae for M from A is a subset of ℤ−Res(G)(M, A_{+}) and can be identified via Θ_{M,A} with a subset of ℤ−Con(G)(M, A). In the next section we will derive sufficient conditions for p ∈ ℤ−Con(G)(M, A) to be the residue of a canonical induction formula for M from A, i.e. that a^{ℚ⊗M,ℚ⊗A,ℚ⊗p}(M) ⊆ A_{+}, in which case we call a^{ℚ⊗M,ℚ⊗A,ℚ⊗p} integral. Note that this is equivalent to the condition p ∈ im(Θ_{M,A}), i.e. p = Θ_{M,A}(a) for a unique a ∈ ℤ−Res(G)(M, A_{+}), namely a = a^{M,A,p} with the notation of Definition 5.5.

(e) Now we assume the integrality condition that p = Θ_{M,A}(a) for some a = a^{M,A,p} ∈ ℤ−Res(G)(M, A_{+}). Then ℚ ⊗ a = a^{ℚ⊗M,ℚ⊗A,ℚ⊗p} under the identifications of part (d). Since we have the commutative diagram

M  ⟶^{a^{M,A,p}}  A_{+}  ⟶^{b^{M,A}}  M
r^{p} ↘  ↙ _{ρ^{A}}
A_{+}
⋂  ⋂  ⋂  ⋂
ℚ ⊗ A_{+}
ℚ⊗r^{p} ↗  ↖ _{ℚ⊗ρ^{A}}
ℚ ⊗ M  ⟶_{ℚ⊗a^{M,A,p}}  ℚ ⊗ A_{+} ⟶_{ℚ⊗b^{M,A}} ℚ ⊗ M

with vertical inclusions, we know that a is an induction formula (resp. a morphism of Mackey functors, resp. a morphism of k-algebra restriction functors, resp. A(H)-linear for all H ≤ G, resp. commuting with a morphism f ∈ ℤ−Res(G)(M, M′ with f(A) ⊆ A′ for some other M′, A′ satisfying the standard hypothesis, cf. Proposition 6.11) if and only if the ℚ-tensored statement holds for ℚ ⊗ a. Hence we can apply all the results developed in Section 6 to ℚ ⊗ a and obtain the desired information about a.

(f) From now on we will write a^{M,A,p} also for a^{ℚ⊗M,ℚ⊗A,ℚ⊗p}. With Proposition 6.4 we already have a nice criterion to decide, whether a^{M,A,p}: ℚ ⊗ M → ℚ ⊗ A_{+} is a canonical induction formula. What we don’t have so far, are conditions on M, A, B, and p implying that a^{M,A,p} is integral. Note that there is the factor 1/|H| in (6.1.a) which makes this task difficult. We will show in the next section that under suitable conditions one can get rid of the denominator |H|. As a first step towards this result we have to refine the summation in Equation (6.1.a) to an alternating sum by expanding the Möbius coefficient and using the simplicial complex of chains of the poset M(G) as index set for the summation.

8.3 We assume M, A, B, and p as in Hypothesis 8.1. For H ≤ G and χ ∈ M(H) we write

p_{H}(χ) = ∑_{φ∈B(H)} m_{φ}(χ) · φ ∈ A(H)

and call m_{φ}(χ) ∈ ℤ the multiplicity of φ in χ. Note that these multiplicities depend on p without being apparent from the notation. Note also that m_{ψ}(res_{K}^{H}(φ)) = m_{(K,ψ)}^{(H,φ)} for K ≤ H, φ ∈ B(H), and ψ ∈ B(K), if p_{K}|_{A(K)} = id_{A(K)}, but that in general this equality does not hold.

We denote the set of strictly ascending chains σ = ((H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n})) of elements in M(H) by Δ(M(H)). We write |σ| = n for the length of the above chain. Note that Δ(M(H)) is an H-set and Δ(M(H)) ⊆ Δ(M(G)) for all H ≤ G. For σ as above we define the multiplicity m_{σ} of σ by

m_{σ} := m_{(H_{0},φ_{0})}^{(H_{1},φ_{1})} ⋯ m_{(H_{n−1},φ_{n−1})}^{(H_{n},φ_{n})} ∈ ℕ_{0},

cf. 8.1 (ii). This multiplicity does not depend on p.

8.4 Lemma Assuming Hypothesis 8.1 and the notation from 8.3, we have

a_{H}^{M,A,p}(χ) = \frac{1}{|H|} ∑_{{σ=((H_{0},φ_{0})<···<(Hn,φn)) \atop ∈Δ(M(H))}}(−1)^{n}|H_{0}|m_{σ}m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H} (8.4.a)

for all H ≤ G and χ ∈ M(H).

Proof We expand the explicit formula (6.1.a) using

μ(L, K) = ∑_{L=H_{0}<···<H_{n}=K}(−1)^{n},

the sum running over all chains connecting L and K, cf. [21, 3, Prop. 6]. This yields

a_{H}^{M,A,p}(χ) = \frac{1}{|H|} ∑_{H_{0}<···<H_{n}≤H}(−1)^{n}|H_{0}|[H_{0}, (res_{H_{0}}^{H_{n}} ∘ p_{H_{n}} ∘ res_{H_{n}}^{H})(χ)]_{H}

= \frac{1}{|H|} ∑_{{H_{0}<···<Hn≤H \atop φn∈B(Hn)}} (−1)^{n}|H_{0}|m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, res_{H_{0}}^{H_{n}}(φ_{n})]_{H}

for H ≤ G and χ ∈ M(H). For a fixed chain H_{0} < ⋯ < H_{n} of subgroups of H and φ_{n} ∈ B(H_{n}) we have

res_{H_{0}}^{H_{n}}(φ_{n}) = (res_{H_{0}}^{H_{1}} ∘ ⋯ ∘ res_{H_{n−1}}^{H_{n}})(φ_{n})

= ∑_{φ_{n−1}∈B(H_{n−1})} m_{(H_{n−1},φ_{n−1})}^{(H_{n},φ_{n})} · (res_{H_{0}}^{H_{1}} ∘ ⋯ ∘ res_{H_{n−2}}^{H_{n−1}})(φ_{n−1})

⋮

= ∑_{φ_{0}∈B(H_{0})} ⋯ ∑_{φ_{n−1}∈B(H_{n−1})} m_{(H_{n−1},φ_{n−1})}^{(H_{n},φ_{n})} ⋯ m_{(H_{0},φ_{0})}^{(H_{1},φ_{1})} · φ_{0}

= ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈Δ(M(H))} m_{σ} · φ_{0},

since m_{(H_{i−1},φ_{i−1})}^{(H_{i},φ_{i})} = 0 unless (H_{i−1}, φ_{i−1}) < (H_{i}, φ_{i}) by definition. Substituting this formula for res_{H_{0}}^{H_{n}}(φ_{n}) into the last alternating sum we obtain the result.

9 An integrality theorem

In this section we are going to transform the alternating sum formula (8.4.a) for a_{H}^{M,A,p} in order to derive an integrality criterion in Theorem 9.3. Throughout this section we assume A ⊆ M, B, and p: M → A as in Hypothesis 8.1 and use the associated multiplicities m_{(K,ψ)}^{(H,φ)}, m_{σ}, and m_{φ}(χ) for (K, ψ), (H, φ) ∈ M(G), χ ∈ M(U), φ ∈ B(U), U ≤ G, σ ∈ Δ(M(G)).

9.1 Let H ≤ G. For

σ = ((H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n})) ∈ Δ(M(H))  (9.1.a)

and a set π of primes we define

N_{H}^{π}(σ) := {s ∈ N_{H}(σ) | s_{π′} ∈ H_{0}},

where N_{H}(σ) denotes the stabilizer of σ in H and s_{π′} the π′-part of s, π′ being the complement of π in the set of all primes. Note that H_{0} ⊴ N_{H}(σ) by axiom (C1) in Definition 1.1 and that for s ∈ N_{H}(σ) the condition s_{π′} ∈ H_{0} is equivalent to sH_{0} being a π-element in N_{H}(σ)/H_{0}. Hence, we have

|N_{H}^{π}(σ)| = |H_{0}| · |(N_{H}(σ)/H_{0})_{π}|,  (9.1.b)

where G_{π} denotes the set of π-elements in an arbitrary group G.

Our aim is to replace under suitable conditions, which will arise along the way, the factor |H_{0}| in Formula (8.4.a) with the bigger factor |N_{H}^{π}(σ)| without changing the result of the alternating sum; thereby obtaining in the case, where π is the set of all primes, an integral expression after passing to a sum over H-orbits in Δ(M(H)).

For H ≤ G and π as above we enlarge Δ(M(H)) by defining

Δ̃^{π}(M(H)) := {(s, σ) ∈ H × Δ(M(H)) | s ∈ N_{H}^{π}(σ)}

and we give a partition

Δ̃^{π}(M(H)) = Δ̃_{0}^{π}(M(H)) ∪ Δ̃_{1}^{π}(M(H)) ∪ Δ̃_{2}^{π}(M(H)) ∪ Δ̃_{3}^{π}(M(H)) (9.1.c)

of Δ̃^{π}(M(H)) into four disjoint subsets by defining for (s, σ) ∈ Δ̃^{π}(M(H)) with σ as in (9.1.a):

(s, σ) ∈ Δ̃_{0}^{π}(M(H)) : ⟺ s ∈ H_{0},
(s, σ) ∈ Δ̃_{1}^{π}(M(H)) : ⟺ s ∉ H_{n},
(s, σ) ∈ Δ̃_{2}^{π}(M(H)) : ⟺ s ∈ H_{i+1} ∖ H_{i} and H_{i}(s) < H_{i+1}
for some i ∈ {0, …, n − 1},
(s, σ) ∈ Δ̃_{3}^{π}(M(H)) : ⟺ s ∈ H_{i+1} ∖ H_{i} and H_{i}(s) = H_{i+1}
for some i ∈ {0, …, n − 1},

where H_{i}(s) denotes the subgroup of G generated by H_{i} and s (s normalizes H_{i}, since s ∈ N_{H}(σ)). Note that the index i in the definition of Δ̃_{2}^{π}(M(H)) and Δ̃_{3}^{π}(M(H)) is uniquely determined by the condition s ∈ H_{i+1} ∖ H_{i}.

Next we consider the map

f: Δ̃_{3}^{π}(M(H)) → Δ̃_{1}^{π}(M(H)) ∪ Δ̃_{2}^{π}(M(H)),
(s, σ) ↦ (s, (H_{0}, φ_{0}) < ⋯ < { \atop (H_{i+1}, φ_{i+1})} < ⋯ < (H_{n}, φ_{n})),

for σ as in (9.1.a), where i ∈ {0, …, n − 1} is given by s ∈ H_{i+1} ∖ H_{i}, and the pair (H_{i+1}, φ_{i+1}) = (H_{i}(s), φ_{i+1}) is ommitted from σ.

For (s, σ) ∈ Δ̃_{1}^{π}(M(H)) with σ as in (9.1.a) we have

f^{−1}((s, σ)) = {(s, (H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n}) < (H_{n}(s), φ))|
φ ∈ B(H_{n}(s)), m_{(H_{n},φ_{n})}^{(H_{n}(s),φ)} > 0},

and for (s, σ) ∈ Δ̃_{2}^{π}(M(H)) with σ as in (9.1.a) and i ∈ {0, …, n − 1} with s ∈ H_{i+1} ∖ H_{i}, and hence H_{i}(s) < H_{i+1}, we have

f^{−1}((s, σ)) = {(s, (H_{0}, φ_{0}) < ⋯ < (H_{i}, φ_{i}) < (H_{i}(s), φ) < ⋯ < (H_{n}, φ_{n}))|
φ ∈ B(H_{i}(s)), m_{(H_{i},φ_{i})}^{(H_{i}(s),φ)}m_{(H_{i}(s),φ)}^{(H_{i+1},φ_{i+1})} ≠ 0}.

9.2 Introducing for H ≤ G, σ as in (9.1.a), and χ ∈ M(H) the abbreviation

g(χ, σ) := (−1)^{|σ|}m_{σ}m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H} ∈ A_{+}(H),  (9.2.a)

we can rewrite Equation (8.4.a) as

|H| · a_{H}^{M,A,p}(χ) = ∑_{(s,σ)∈Δ̃_{0}^{π}(M(H))} g(χ, σ),

where the factor |H_{0}| in Equation (8.4.a) is replaced by summing over pairs (s, σ) with s ∈ H_{0}, where σ is as in (9.1.a). Using the partition in (9.1.c) and abbreviating Δ̃^{π}(M(H)) (resp. Δ̃_{i}^{π}(M(H)) for i = 0, 1, 2, 3) by Δ̃^{π} (resp. Δ̃_{i}^{π}), we may continue with

|H| · a_{H}^{M,A,p}(χ) = ∑_{(s,σ)∈Δ̃^{π}} g(χ, σ) − ∑_{(s,σ)∈Δ̃_{1}^{π}∪Δ̃_{2}^{π}∪Δ̃_{3}^{π}} g(χ, σ).  (9.2.b)

We examine the last sum in (9.2.b) further using the function f from 9.1:

∑_{(s,σ)∈Δ̃_{1}^{π}∪Δ̃_{2}^{π}∪Δ̃_{3}^{π}} g(χ, σ) = ∑_{(s,σ)∈Δ̃_{1}^{π}}(g(χ, σ) + ∑_{(s′,σ′)∈f^{−1}((s,σ))} g(χ, σ′)) (9.2.c)

+ ∑_{(s,σ)∈Δ̃_{2}^{π}}(g(χ, σ) + ∑_{(s′,σ′)∈f^{−1}((s,σ))} g(χ, σ′)).

For (s, σ) ∈ Δ̃_{1}^{π} with σ as in (9.1.a), i.e. s ∉ H_{n}, we have

g(χ, σ) + ∑_{(s′,σ′)∈f^{−1}((s,σ))} g(χ, σ′) =  (9.2.d)

(−1)^{|σ|}m_{σ}(m_{φ_{n}}(res_{H_{n}}^{H}(χ)) − ∑_{φ∈B(H_{n}(s))} m_{(H_{n},φ_{n})}^{(H_{n}(s),φ)}m_{φ}(res_{H_{n}(s)}^{H}(χ)))[H_{0}, φ_{0}]_{H},

and for (s, σ) ∈ Δ̃_{2}^{π} with σ as in (9.1.a) and i ∈ {0, …, n−1} with s ∈ H_{i+1}∖H_{i}, hence H_{i}(s) < H_{i+1}, we have

g(χ, σ)+ ∑_{(s′,σ′)∈f^{−1}((s,σ))} g(χ, σ′)  (9.2.e)

= (−1)^{|σ|} \frac{m_{σ}}{m_{(H_{i},φ_{i})}^{(H_{i+1},φ_{i+1})}} m_{φ_{n}}(res_{H_{n}}^{H}(χ)) ×

× (m_{(H_{i},φ_{i})}^{(H_{i+1},φ_{i+1})} − ∑_{φ∈B(H_{i}(s))} m_{(H_{i},φ_{i})}^{(H_{i}(s),φ)}m_{(H_{i}(s),φ)}^{(H_{i+1},φ_{i+1})})[H_{0}, φ_{0}]_{H}

= 0,

since

m_{(H_{i},φ_{i})}^{(H_{i+1},φ_{i+1})} = ∑_{φ∈B(H_{i}(s))} m_{(H_{i},φ_{i})}^{(H_{i}(s),φ)}m_{(H_{i}(s),φ)}^{(H_{i+1},φ_{i+1})}

by transitivity of restriction.

Finally substituting (9.2.d) and (9.2.e) in (9.2.c) we can rewrite (9.2.b) as

|H| · a_{H}^{M,A,p}(χ)  (9.2.f)
= ∑_{(s,σ)∈Δ̃^{π}} g(χ, σ) − ∑_{(s,σ=((H_{0},φ_{0})<···<(H_{n},φ_{n})))∈Δ̃_{1}^{π}}(−1)^{|σ|}m_{σ}h(χ, s, σ)[H_{0}, φ_{0}]_{H}

for H ≤ G and χ ∈ M(H), with

h(χ, s, σ) = m_{φ_{n}}(res_{H_{n}}^{H}(χ)) − ∑_{φ∈B(H_{n}(s))} m_{(H_{n},φ_{n})}^{(H_{n}(s),φ)}m_{φ}(res_{H_{n}(s)}^{H}(χ)). (9.2.g)

9.3 Theorem Let H ≤ G and χ ∈ M(H). Assume that for a set π of primes the following condition holds:

(∗_{π})

For all T ⊴ U ≤ H such that U/T is a cyclic π-group and all ψ ∈ B(T) which are fixed under U (i.e. ^{u}ψ = ψ for all u ∈ U), the coefficients of ψ in the two elements p_{T} (res_{T}^{U}(ϑ)) and res_{T}^{U}(p_{U}(ϑ)) in A(T) with respect to the basis B(T), where ϑ := res_{U}^{H}(χ), coincide, i.e.

m_{ψ}(res_{T}^{U}(ϑ)) = ∑_{φ∈B(U)} m_{(T,ψ)}^{(U,φ)} · m_{φ}(ϑ).  (9.3.a)

Then one has

a_{H}^{M,A,p}(χ) = \frac{1}{|H|} ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈Δ(M(H))}(−1)^{|σ|}|N_{H}^{π}(σ)|m_{σ} × (9.3.b)
× m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H}

= ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈H\Δ(M(H))}(−1)^{|σ|} \frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|}m_{σ} ×

× m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H},

where the second sum runs over a set of representatives for the H-orbits of Δ(M(H)).

Proof We consider the term h(χ, s, σ) in (9.2.g). With U = H_{n}(s), T = H_{n}, and ψ = φ_{n}, condition (∗_{π}) implies that h(χ, s, σ) = 0. Therefore we obtain from (9.2.f):

a_{H}^{M,A,p}(χ) = \frac{1}{|H|} ∑_{(s,σ)∈Δ̃^{π}(M(H))} g(χ, σ)

= \frac{1}{|H|} ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈Δ(M(H))}(−1)^{|σ|}|N_{H}^{π}(σ)|m_{σ} ×

× m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H}.

This is the first equation of the theorem. Since the summands of the above sum are constant on H-orbits of Δ(M(H)), we may collect them and obtain further:

a_{H}^{M,A,p}(χ) = \frac{1}{|H|} ∑_{{σ=((H_{0},φ_{0})<···<(Hn,φn)) \atop ∈H\Δ(M(H))}}(−1)^{|σ|} \frac{|H|}{|N_{H}(σ)|}|N_{H}^{π}(σ)|m_{σ} ×

× m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H}.

Now the second equation of the theorem follows from Equation (9.1.b):

\frac{|N_{H}^{π}(σ)|}{|N_{H}(σ)|} = \frac{|H_{0}| · |(N_{H}(σ)/H_{0})_{π}|}{|H_{0}| · |N_{H}(σ)/H_{0}|} = \frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|}.

9.4 Corollary Let H ≤ G, χ ∈ M(H), and assume that condition (∗_{π}) holds for some set π of primes. Then

|H|_{π′} · a_{H}^{M,A,p}(χ) ∈ A_{+}(H),

i.e. in a_{H}^{M,A,p}(χ) occur only π′-numbers as denominators.

Proof It is well-known (cf. [17, V.19.14]) that |G_{π}| is a multiple of |G|_{π} for any finite group G. Therefore, the factor |(N_{H}(σ)/H_{0})_{π}|/|N_{H}(σ)/H_{0}| in the second sum in (9.3.b) splits into an integer and a rational number whose denominator divides |H|_{π′}:

\frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|} = \frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|_{π}} · \frac{1}{|N_{H}(σ)/H_{0}|_{π′}} .

9.5 Corollary If (∗_{π}) holds for the set π of all primes and for all H ≤ G and χ ∈ M(H), then we have

a_{H}^{M,A,p}(χ)  (9.5.a)
= ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈H\Δ(M(H))}(−1)^{n}m_{σ}m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H}

for all H ≤ G and all χ ∈ M(H); in particular, a^{M,A,p} is integral.

If additionally A(H) = M(H) and p_{H} = id_{M(H)} for all H ∈ C(M), then b^{M,A} ∘ a^{M,A,p} = id_{M}, i.e. a^{M,A,p} is an integral canonical induction formula.

Proof Formula (9.5.a) is an immediate consequence of Formula (9.3.b). Since M(H) is free for all H ≤ G, we have C(M) = C(ℚ ⊗ M). Therefore, Corollary 6.6 yields the last statement.

9.6 Remark Comparing the general formula for a_{H}^{M,A,p}(χ) in Lemma 8.4 to the second one in (9.3.b), one realizes that under the hypothesis (∗_{π}) one can replace the factor |H_{0}| with the bigger factor |N_{H}^{π}(σ)| without changing the whole alternating sum. In the standard example, i.e. M = R, A = R^{ab}, and p_{H}(χ) = χ for linear χ ∈ Irr(H) and p_{H}(χ) = 0 for non-linear χ ∈ Irr(H), this was observed in [3] for the set π of all primes by interpreting both alternating sums as Euler characteristics of certain chain complexes which could be proven to be homotopy equivalent. Theorem 9.3 will be the main tool for integrality proofs of all the canonical induction formulae in Example 6.13.

9.7 Example We show that for M = R, A = R^{ab}, B, and p as in Example 6.13 (a) and Remark 7.3, condition (∗_{π}) is satisfied for all H ≤ G and all χ ∈ R(H), where π is the set of all primes. Then Corollary 9.5 implies that a^{M,A,p} is an integral canonical induction formula, and that the explicit formula (9.5.a) holds. Note that in this example m_{σ} = 1 for all chains σ ∈ M(G). This formula is the same as the one obtained in [2] with a different proof for the integrality.

So let U be a finite group and T ⊴ U such that U/T is cyclic. Let furthermore ϑ ∈ Irr(U) and ψ ∈ T̂ such that ψ: T → ℂ^{×} is U-stable. Then we have to show that the multiplicities of ψ in p_{T} (res_{T}^{U}(ϑ)) and res_{T}^{U}(p_{U}(ϑ)) coincide. If ϑ is linear, then p_{T} (res_{T}^{U}(ϑ)) = res_{T}^{U}(ϑ) = res_{T}^{U}(p_{U}(ϑ)) and we are done. If ϑ(1) > 1, then res_{T}^{U}(p_{U}(ϑ)) = 0 and we have to show that ψ does not occur in p_{T} (res_{T}^{U}(ϑ)) which is equivalent to

0 = (ψ, res_{T}^{U}(ϑ))_{T} = (ind_{T}^{U}(ψ), ϑ)_{U}

by Frobenius reciprocity. Since ψ is U-stable, the cyclic subgroup T/ker(ψ) is central in U/ker(ψ), and since U/T is cyclic, U/ker(ψ) is abelian. Therefore, ind_{T}^{U}(ψ), being a character which comes via inflation from a character of the abelian group U/ker(ψ), splits into linear characters, and hence (ind_{T}^{U}(ψ), ϑ)_{U} = 0.

Note that by exactly the same proofs we obtain for any field K of characteristic zero containing a primitive exp(G)-th root of unity, an equivalent canonical induction formula for R_{K} ≅ R from R_{K}^{ab} ≅ R^{ab}, R_{K}(H) being the Grothendieck group of KH−mod or equivalently the ring of virtual Kcharacters, and R_{K}^{ab}(H) ⊆ R_{K}(H) the span of Ĥ(K) for H ≤ G.

9.8 Example We show that for M = R_{F}, A = R_{F}^{ab}, B, and p as in Example 6.13 (b) and Remark 7.3, condition (∗_{π}) is satisfied for all H ≤ G and all [V] ∈ R_{F} (H), V ∈ F H−mod, where π is the set of all primes. Again Corollary 9.5 applies, showing that a^{R_{F} ,R_{F}^{ab},p} is an integral canonical induction formula. Note that in Equation (9.5.a) we have m_{σ} = 1 for all σ ∈ Δ(M(G)), and that m_{φ}([V]) counts the multiplicity of F_{φ} as a composition factor in V ∈ F H−mod for H ≤ G and φ ∈ Ĥ(F).

So let T ⊴ U be as in the previous example, let S ∈ F U−mod be irreducible, and let ψ ∈ T̂(F) be U-stable. We have to show that the multiplicities of [F_{ψ}] in p_{T} (res_{T}^{U}([S])) and in res_{T}^{U}(p_{U}([S])) coincide. If dim_{F} S = 1, this holds trivially as in Example 9.7. If dim_{F} S > 1, then res_{T}^{U}(p_{U}([S])) = 0, and we have to show that F_{ψ} is not a composition factor of res_{T}^{U}([S]). But since T is normal in U, res_{T}^{U}(S) is semisimple and we have to show that

0 = Hom_{F T} (F_{ψ}, res_{T}^{U}(S)) ≅ Hom_{F U}(ind_{T}^{U}(F_{ψ}), S).

As in the previous example, U/ker(ψ) is abelian, and ind_{T}^{U}(F_{ψ}) is the inflation of an F U/ker(ψ)-module. Since U/ker(ψ) is abelian, ind_{T}^{U}(F_{ψ}) has only onedimensional composition factors, and therefore, Hom_{F U}(ind_{T}^{U}(F_{ψ}), S) = 0 as required.

Let d: R → R_{F} be the decomposition map (cf. [22, 15.2]) which is a morphism of Mackey functors on G with d(R^{ab}) ⊆ R_{F}^{ab}. We remark that the diagram

R ⟶^{a} R_{+}^{ab}
d↓  ↓^{d_{+}}
R_{F} ⟶^{ã} R_{F +}^{ab}

with the canonical induction formulae a from Example 9.7 and ã from the present example, is not commutative. In fact, there can’t be any such canonical induction formula ã: R_{F} → R_{F +}^{ab}, because by Proposition 6.11 we needed

R ⟶^{p} R^{ab}
d↓  ↓^{d}
R_{F} ⟶^{p̃} R_{F}^{ab}

to commute for some p̃ ∈ ℤ−Con(G)(R_{F} , R_{F}^{ab}). But for G = S_{4}, ε the sign character, χ_{2} the unique irreducible character of degree 2, and l = char(F) = 3, the virtual character χ = 1 + ε − χ_{2} is in the kernel of d_{G}, and d_{G}(p_{G}(χ)) = d_{G}(1 + ε) ≠ 0.

9.9 Example Let M = P_{O}, A = P_{O}^{ab}, B, and p be as in Example 6.13 (c) and Remark 7.3, i.e. B(H) = Ĥ(O)for H ≤ G an l′-group and B(H) = ∅ otherwise. We will show that a^{P_{O},P_{O}^{ab},p} is integral. First we show that, with π = l′ being the set of all primes distinct from l = char(O/rad(O)), condition (∗_{l′}) is satisfied for all H ≤ G and [V] ∈ P_{O}(H), V ∈ OH−proj.

Let T ⊴ U ≤ G with U/T a cyclic l′-group, ψ: T → O^{×} a U-invariant homomorphism such that O_{ψ} is projective. Then in particular, T is an l′-group, and with T also U. Let V ∈ OU−proj be indecomposable. We have to show that the multiplicites of [O_{ψ}] in p_{T} (res_{T}^{U}([V])) and res_{T}^{U}(p_{U}([V])) coincide. But since U is an l′-group, we have P_{O}(U) ≅ R(U), P_{O}^{ab}(U) = R^{ab}(U), and the proof of Example 9.7 can be repeated.

Hence, from Theorem 9.3 we have

a_{H}^{P_{O},P_{O}^{ab},p}([V]) =  ∑_{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈H\Δ(M(H))}(−1)^{n} \frac{|(N_{H}(σ)/H_{0})_{l′}|}{|N_{H}(σ)/H_{0}|}m_{σ} ×
× m_{φ_{n}}(res_{H_{n}}^{H}([V]))[H_{0}, [O_{φ_{0}}]]_{H}.

Since all occurring groups H_{i} are l′-groups, we have |(N_{H}(σ)/H_{0})_{l′}| = |N_{H}(σ)_{l′}|· |H_{0}| and it suffices to show that

\frac{|N_{H}(σ)_{l′}|}{|N_{H}(σ)|_{l′}} and \frac{m_{φ_{n}}(res_{H_{n}}^{H}([V]))}{|N_{H}(σ)|_{l}}

are natural numbers. However, |N_{H}(σ)|_{l′} divides |N_{H}(σ)_{l′}| by [17, V.19.14], and |N_{H}(σ)|_{l} divides m_{φ_{n}}(res_{H_{n}}^{H}([V])) by the following argument. Let Q be a Sylow l-subgroup of N_{H}(σ), H′ := QH_{n}, and let W be an indecomposable summand of res_{H′}^{H}(V). It suffices to show that |Q| divides the multiplicity m_{φ_{n}}(res_{H_{n}}^{H′}([W])), which we may assume to be non-zero. Since W is projective, W is a direct summand of ind_{1}^{H′}(O) ≅ ind_{H_{n}}^{H′}(ind_{1}^{H_{n}}(O)). Since H′/H_{n} ≅ Q is an l-group, we have W ≅ ind_{H_{n}}^{H′}(X) for some indecomposable OH_{n}-module X by Green’s indecomposability theorem. Mackey’s decomposition formula then yields

res_{H_{n}}^{H′}(W) ≅ ⨁_{s∈H_{n}\H′/H_{n}} ind_{H_{n}∩^{s}H_{n}}^{H_{n}}(res_{H_{n}∩^{s}H_{n}}^{^{s}H_{n}}(^{s}X)) ≅ ⨁_{s∈H′/H_{n}} ^{s}X.

Since O_{φ_{n}} is a direct summand of res_{H_{n}}^{H′}(W), and since φ_{n} is stable under H′ ≤ N_{H}(σ), we have res_{H_{n}}^{H′}(W) ≅ ⨁_{i=1}^{|H′/H_{n}|} O_{φ_{n}}, and therefore m_{φ_{n}}(res_{H_{n}}^{H′}([W])) = |H′/H_{n}| = |Q|.

10 A global point of view

10.1 Often it is possible to define a Mackey functor on any finite group not only on the subgroups of a given group. So let us assume that we have a free abelian group M(G) for each G ∈ gr, the category of finite groups, such that we have induction and restriction maps whenever there is a subgroup inclusion H ≤ G, and that there are conjugation maps c_{g,H}: M(H) → M(^{g}H), whenever H ≤ G and g ∈ G. Moreover let us assume that the axioms in the definition of a Mackey functor are satisfied. We also assume that with respect to restrictions, we can extend M to a contravariant functor M: gr → Ab to the category of abelian groups, i.e. that we now have a map res_{f}: M(G′) → M(G) for every group homomorphism f: G → G′. We assume that in the situation H ≤ G, g ∈ G, for f: ^{g}H → H, h′ ↦ g^{−1}hg, the map res_{f}: M(H) → M(^{g}H) coincides with the conjugation map c_{g,H}. Let us furthermore assume that we are given a subfunctor A ⊆ M: gr → Ab, such that A(G) has a stable ℤ-basis B(G) for every G ∈ gr with respect to group isomorphisms and such that the restriction of a basis element in B(G) to a subgroup H ≤ G is a ℤ-linear combination of B(H) with non-negative coefficients. Finally suppose we have for each G ∈ gr a morphism p_{G}: M(G) → A(G) commuting with res_{f} for each group isomorphism f. To this situation we will refer in the sequel as the global standard situation. When restricting our attention to subgroups of a given finite group G we are brought back to the situation of Hypothesis 8.1. Note that we can consider each of the Examples in 1.8 (a), (b), (d), (e) as a global standard situation; in part (d) and (e) we can choose O as the ring of integers of the maximal unramified extension of ℚ_{l} in an algebraic closure.

10.2 Assume that A ⊆ M: gr → Ab is given as in 10.1. Then each group homomorphism f: G → G′ induces a map

res_{+}_{f}: A_{+}(G′) → A_{+}(G), [H′, a]_{G′} ↦ ∑_{g′∈f(G)\G′/H′}[f^{−1}(^{g′}H′), res_{f_{g′,H′}} (a)]_{G}, (10.2.a)

where f_{g′,H′}: f^{−1}(^{g′}H′) → ^{g′}H′ → H′ is the restriction of f to f^{−1}(^{g′}H′) followed by the conjugation map h̃ ↦ g′^{−1}h̃g′ for h̃ ∈ ^{g′}H′. It is not difficult to see that res_{f} is a ring homomorphism, if A is a ring valued functor, and that res_{+}_{f} ∘ res_{+}_{f′} = res_{+}_{(f′∘f)}, if f′: G′ → G′′ is another group homomorphism, so that A_{+} is again a contravariant functor gr → Ab and −_{+} is a functor from the contravariant functor category F^{∘}(gr, Ab), resp. F^{∘}(gr, Ri) for the category Ri of rings, to itself. Moreover it is a straight forward calculation that b^{M,A}: A_{+} → M is a functorial morphism. Assuming M, A, B, and p as in the global standard situation we can define a_{G}^{M,A,p}: ℚ ⊗ M(G) → ℚ ⊗ A_{+}(G) exactly as in Section 8 for each G ∈ gr, and all our previous results which hold under Hypothesis 8.1 are still valid.

10.3 Proposition Let M, A, B, and p be given as in the global standard situation 10.1, and assume that p_{G}|_{A(G)} = id_{A(G)} for all G ∈ gr and that res_{ν}(ker(p_{G/N})) ⊆ ker(p_{G}) for all canonical epimorphisms ν: G → G/N, N ⊴ G. Then the diagram

ℚ ⊗ M(G′) ⟶^{a_{G′}^{M,A,p}} ℚ ⊗ A_{+}(G′)
res_{f} ↓  ↓^{res_{+}}^{f}
ℚ ⊗ M(G) ⟶^{a_{G}^{M,A,p}} ℚ ⊗ A_{+}(G)

commutes for all group homomorphisms f: G → G′, i.e. a^{M,A,p} is a functorial morphism from ℚ ⊗ M to ℚ ⊗ A_{+}, considered as functors gr → ℚ−Mod.

Proof Since f can be written as a composition G⟶^{f}f(G) ≤ G′ of an epimorphism and an inclusion we may assume that f: G → G′ is surjective. Note that, since p commutes with isomorphisms, res_{f} maps ker(p_{G′}) to ker(p_{G}). We will write a_{U} instead of a_{U}^{M,A,p} for U ∈ gr, and proceed by induction on |G|. If |G| = 1, then the result holds trivially. So let |G| > 1. By hypothesis we have M(G′) = A(G′) ⊕ ker(p_{G′}). If χ′ ∈ A(G′), then by Proposition 6.12 we have a_{G′}(χ′) = [G′, χ′]_{G′} and also a_{G}(res_{f}(χ′)) = [G, res_{f}(χ′)]_{G} which is equal to res_{+}_{f}([G′, χ′]_{G′}) = res_{+}_{f}(a_{G′}(χ′)) by (10.2.a).

Now let χ′ ∈ ker(p_{G′}). Since ρ_{G}^{A} is injective, it suffices to show that

(π_{H}^{A} ∘ res_{+}_{H}^{G} ∘ res_{+}_{f} ∘ a_{G′})(χ′) = (π_{H}^{A} ∘ res_{+}_{H}^{G} ∘ a_{G} ∘ res_{f})(χ′)  (10.3.a)

for all H ≤ G. We first assume that H < G. The right hand side of (10.3.a) is equal to (π_{H}^{A} ∘ a_{H} ∘ res_{H}^{G} ∘ res_{f})(χ′) = (p_{H} ∘ res_{H}^{G} ∘ res_{f})(χ′). The left hand side is equal to (π_{H}^{A} ∘ res_{+}_{f: H→f(H)} ∘ res_{+}_{f(H)}^{G′} ∘ a_{G})(χ′) = (π_{H}^{A} ∘ res_{+}_{f: H→f(H)} ∘ a_{f(H)} ∘ res_{f(H)}^{G′})(χ′) which is equal to (π_{H}^{A} ∘ a_{H} ∘ res_{f: H→f(H)} ∘ res_{f(H)}^{G′})(χ) = (p_{H} ∘ res_{H}^{G} ∘ res_{f})(χ′) by induction and (10.3.a) holds in this case.

Now let H = G. The right hand side of (10.3.a) is equal to p_{G}(res_{f}(χ′)) which is 0, since res_{f} maps ker(p_{G′}) to ker(p_{G}). On the other hand, if we write

a_{G′}(χ′) = ∑_{(H′,φ′)∈G′\M(G′)} α_{(H′,φ′)}^{G′}(χ′)[H′, φ′]_{G′}

with α_{(H′,φ′)}^{G′}(χ′) ∈ ℚ as a linear combination of the basis elements [H′, φ′]_{G′}, we obtain for the left hand side of (10.3.a),

(π_{G}^{A} ∘ res_{+}_{f} ∘ a_{G′})(χ′) = ∑_{φ′∈B(G′)} α_{(G′,φ′)}^{G′}(χ′)res_{f}(φ′),

since (π_{G}^{A} ∘ res_{+}_{f})([H′, φ′]_{G′}) = 0 for (H′, φ′) ∈ M(G′) with H′ < G′. But looking at the explicit formula (6.1.a) we see that α_{(G′,φ′)}^{G′}(χ′) = 0 for all φ′ ∈ B(G′), since p_{G′}(χ′) = 0, and the proof is complete.

10.4 Remark In the present treatise we avoided to place ourselves into the framework of global Mackey functors as they are defined for example by Webb in [30] or by Bouc in [6], because it would require even more elaborate technicalities, and because we would like to apply the theory also to the absolute Galois group G of a field K and work with all subgroups H ≤ G of finite index. It is possible to establish the notion of a Mackey functor as in Definition 1.1 also in this context.

However, the reader who prefers to work with global Mackey functors should have no difficulties to translate our results into the language of global Mackey functors, where one has also induction maps along arbitrary group homomorphisms.

11 Computation of canonical induction formulae

Throughout this section let M, A, B, and p: M → A be given as in Hypothesis 8.1. Let furthermore a := a^{M,A,p}: ℚ ⊗ M → ℚ ⊗ A_{+} be the associated morphism of ℚ-restriction functors on G.

11.1 For each H ≤ G, χ ∈ M(H), and (K, ψ) ∈ M(H) we denote by α_{(K,ψ)}^{H}(χ) the coefficient of a_{H}(χ) at the basis element [K, ψ]_{H}, i.e.

a_{H}^{M,A,p}(χ) = ∑_{(K,ψ)∈H\M(H)} α_{(K,ψ)}^{H}(χ)[K, ψ]_{H}.  (11.1.a)

The aim of this section is to describe how to compute the coefficients α_{(K,ψ)}^{H}(χ). Usually, even for small groups it takes too long to use the explicit formula (6.1.a). More efficient is the use of the commutative diagram (6.1.b). Note that ℚ ⊗ A^{+}(H) has a ℚ-basis c_{(L,λ)}^{(H)} ∈ (∏_{K≤H} A(K))^{H}, (L, λ) ∈ H\M(H), with c_{(L,λ)}^{(H)} having entry ∑_{n∈N_{H}(L)/N_{H}(L,λ)} ^{n}λ at the component L, the appropriate conjugate entries at all components which are H-conjugate to L, and zero entries everywhere else. This means that each H-conjugate of λ occurs exactly once in c_{(L,λ)}^{(H)}. Obviously, c_{(K,ψ)}^{(H)} = c_{(L,λ)}^{(H)}, if and only if (K, ψ) =_{H} (L, λ). Let

γ_{(L,λ),(K,ψ)}^{H} ∈ ℤ denote the coefficient of ρ_{H}^{A}([K, ψ]_{H}) at the basis element c_{(L,λ)}^{(H)} for (K, ψ), (L, λ) ∈ M(H). Then γ_{(L,λ),(K,ψ)}^{H} depends only on the H-orbits of (L, λ) and (K, ψ) in M(H), and we denote by Γ_{M(H)} the square matrix (γ_{(L,λ),(K,ψ)}^{H}) indexed by (L, λ), (K, ψ) ∈ H\M(H), i.e. representatives of the H-orbits of M(H). Then Diagram (6.1.b) translates into the matrix equation

Γ_{M(H)} · (α_{(K,ψ)}^{H}(χ))_{(K,ψ)}= (m_{ψ}(res_{K}^{H}(χ)))_{(K,ψ)},  (11.1.b)

where (K, ψ) runs through H\M(H). For (L, λ), (K, ψ) ∈ M(H) we write (L, λ) ≤_{H} (K, ψ), if (L, λ) ≤ ^{h}(K, ψ) for some h ∈ H.

11.2 Proposition With the notation of 11.1, the coefficients γ_{(L,λ),(K,ψ)}^{H} for H ≤ G and (L, λ), (K, ψ) ∈ M(H) are given by

γ_{(L,λ),(K,ψ)}^{H} = ∑_{h∈L\H/K} m_{(L,λ)}^{^{h}(K,ψ)} = ∑_{h∈H/K} m_{(L,λ)}^{^{h}(K,ψ)}.  (11.2.a)

In particular, γ_{(L,λ),(K,ψ)}^{H} = 0 unless (L, λ) ≤_{H} (K, ψ), and after a suitable ordering of H\M(H), the matrix Γ_{M(H)} is upper triangular with diagonal entry |N_{H}(K, ψ)/K| at position (K, ψ).

Proof By the definitions of ρ_{H}^{A} and c_{(L,λ)}^{(H)}, the number γ_{(L,λ),(K,ψ)}^{H} is the coefficient of λ with respect to B(L) in

π_{L}^{A}(res_{+}_{L}^{H}([K, ψ]_{H})) = ∑_{h∈L\H/K} π_{L}([L ∩ ^{h}K, res_{L∩^{h}K}^{^{h}K}(^{h}ψ)]_{L})

= ∑_{{h∈L\H/K \atop L∩^{h}K=L}} res_{L}^{^{h}K}(^{h}ψ) = ∑_{{h∈L\H/K \atop L≤^{h}K}} ∑_{μ∈B(L)} m_{(L,μ)}^{^{h}(K,ψ)} · μ.

Now the first equation in 11.2.a follows, since L ≤ ^{h}K whenever m_{(L,λ)}^{^{h}(K,ψ)} ≠ 0, and the second equation follows, since if L ≤ ^{h}K then LhK = hK.

11.3 Lemma With the notation from 11.1 we have

α_{(K,ψ)}^{H}(χ) = \frac{|K|}{|N_{H}(K, ψ)|} ∑_{{σ=((H_{0},φ_{0})<···<(Hn,φn))∈Δ(M(H)) \atop (H_{0},φ_{0})=(K,ψ)}}(−1)^{n}m_{σ}m_{φ_{n}}(res_{H_{n}}^{H}(χ)) (11.3.a)

for all H ≤ G and all χ ∈ M(H).

Proof By (8.4.a) we have

|H|α_{(K,ψ)}^{H}(χ) = ∑_{{σ=((H_{0},φ_{0})<···<(Hn,φn))∈Δ(M(H)) \atop (H_{0},φ_{0})=_{H} (K,ψ)}}(−1)^{n}|H_{0}|m_{σ}m_{φ_{n}}(res_{H_{n}}^{H}(χ)).

Since in the H-orbit of σ = ((H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n})) ∈ Δ(M(H)) with (H_{0}, φ_{0}) =_{H} (K, ψ) there are precisely |N_{H}(K, ψ)/N_{H}(σ)| chains among its |H/N_{H}(σ)| elements which start with (K, ψ), the result follows.

There is one general result about the vanishing of some of the coefficients α_{(K,ψ)}^{H}(χ).

11.4 Proposition Assume the notation of 11.1 and assume further that there is a stable ℤ-basis B̃(H) of M(H) for all H ≤ G, containing B(H), such that res_{K}^{H}(χ) is a ℤ-linear combination of B̃(K) with non-negative coefficients for each K ≤ H and χ ∈ B̃(H), and that p_{H} is the identity on B(H) and zero on B̃(H) ∖ B(H) for all H ≤ G.

Then for each H ≤ G and χ ∈ B̃(H) the following holds: If (K, ψ) ∈ M(H) such that m_{ψ}(res_{K}^{H}(χ)) = 0, (i.e. ψ does not occur in res_{K}^{H}(χ)), then α_{(K,ψ)}^{H}(χ) = 0.

Proof We can extend the definitions from 8.1 and 8.3 to multiplicities m_{(K,ϑ)}^{(H,χ)} ∈ ℕ_{0} for K ≤ H ≤ G, χ ∈ B̃(H), ϑ ∈ B̃(K), and m_{ϑ}(ζ) ∈ ℕ_{0} for

ζ ∈ M(K) with respect to B̃(K). For ϑ ∈ B(K) this definition of m_{ϑ}(ζ) coincides with the old one by the hypothesis on p_{K}.

We show that each summand in (11.3.a) is zero, if m_{ψ}(res_{K}^{H}(χ)) = 0. Let σ = ((H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n})) ∈ Δ(M(H)) with (H_{0}, φ_{0}) = (K, ψ). Then

0 = m_{ψ}(res_{K}^{H}(χ)) = ∑_{ϑ∈B̃(H_{n})} m_{(K,ψ)}^{(H_{n},ϑ)}m_{ϑ}(res_{H_{n}}^{H}(χ))

by transitivity of restriction. Since all the factors in the sum are non-negative, and since m_{(K,ψ)}^{(H_{n},φ_{n})} ≠ 0, we have m_{φ_{n}}(res_{H_{n}}^{H}(χ)) = 0, and the summand in (11.3.a) associated to σ vanishes.

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