On p-Monomial Modules over Local Domains^{∗}

Robert Boltje^{†}and Adam Glesser^{†} Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A.

             boltje@math.ucsc.edu and aglesser@math.ucsc.edu

December 15, 2004

Abstract

We consider direct summands of monomial modules for group algebras over local rings of positive residue characteristic p. If the base ring is a local domain that contains a root of unity of sufficiently large order, the Krull Schmidt theorem holds for these modules and their representation rings over two such base rings are canonically isomorphic.

Introduction

The class of p-monomial modules (direct summands of monomial modules) was studied in [Bo98] and [BK00] under the name of linear source modules as a natural generalization of the well-understood p-permutation modules, also known as trivial source modules (direct summands of permutation modules). The base ring R used there was assumed to be a complete discrete valuation ring with residue field of characteristic p > 0 (or a field of characteristic p). The reason is that one wants to have available the Krull-Schmidt Theorem, Green’s theory of vertices and sources, and the Green correspondence. Here we consider the above mentioned class of modules over an arbitrary local ring R with residue field of characteristic p. We show in Theorems 1.6 and 1.9 that the Krull-Schmidt theorem and Green’s theory hold in this general case provided that R is an integral domain that contains a root of unity of order exp(G) in the case that R has characteristic 0, or exp(G)_{p′} in the case that R has characteristic p.

There are two reasons we are interested in such a result. The first is to make the ring R as small as possible in order to carry out calculations more effectively, whether by hand or by computer. Trivial source modules are of importance in Rickard’s version (cf. [Ri96]) of Broué’s abelian defect group conjecture. The conjectured chain complexes should already arise over such small R (for example, the localization ℤ[ζ] with respect to a prime ideal containing p, where ζ is a complex root of unity of order exp(G)).

∗MR Subject Classification 20C11, 20C20, 19A22

†Research supported by the NSF, DMS-0200592 and 0128969

The other reason has to do with considering the representation ring L_{R}(G) of p-monomial RG-modules as a functor in G. In order to have a ring R which is large enough for every finite group G to ensure the Krull-Schmidt Theorem, and Green’s theory, we need to have all roots of unity in R. Let K denote the extension of ℚ in ℂ generated by all roots of unity and let R denote the localization of the ring of integral elements of K with respect to any maximal ideal containing p. Then R is good enough for our purposes.

We are very grateful to the referee of a previous version of this paper (that dealt only with the proof of Theorem 1.4) to bring to our attention Green’s result on the radical quotients of the endomorphism rings of Green correspondents.

1 Notation and Statement of Results

1.1 Notation Throughout, R denotes a commutative local ring, 𝔭 its maximal ideal, F := R/𝔭 its residue field, and R^{×} its group of units. We assume that the characteristic of F is a prime number p.

For every ring Λ we denote by J(Λ) its Jacobson radial. The radical of a Λmodule M is denoted by rad_{Λ}(M). All modules will be left modules. If M and N are Λ-modules, we indicate by M | N that M is isomorphic to a direct summand of N. If Λ is an R-algebra, we call a Λ-module absolutely indecomposable (in the sense of Huppert, cf. [Hu75]) if End_{Λ}(M)/J(End_{Λ}(M)) ≅ F as F -algebras.

Throughout, G denotes a finite group and Syl_{p}(G) the set of its Sylow psubgroups. The exponent of G will be denoted by exp(G). We write H ⩽ G (resp. H < G) to indicate that H is a subgroup (resp. a proper subgroup) of G. The index of H in G is denoted by [G : H]. For H ⩽ G and g ∈ G we set ^{g}H := gHg^{−1}. By RG we denote the group algebra of G over R. If φ ∈ Hom(H, R^{×}) is a group homomorphism, we denote by R_{φ} the RH-module with underlying R-module R and H-action given by hα := φ(h)α for h ∈ H and α ∈ R. If φ = 1 is the trivial homomorphism we usually write R instead of R_{φ}. If φ ∈ Hom(H, R^{×}) and g ∈ G we denote by ^{g}φ ∈ Hom(^{g}H, R^{×}) the homomorphism which maps an element x ∈ ^{g}H to φ(g^{−1}xg). If g ∈ G and M is an RH-module, we denote by ^{g}M the R[^{g}H]-module with underlying R-module M and ^{g}H-action given by xm := g^{−1}xgm for x ∈ ^{g}H and m ∈ M. We call ^{g}M the g-conjugate of M. We denote by Res_{H}^{G} and Ind_{H}^{G} the usual restriction and induction functors between the categories of RG-modules and RH-modules.

1.2 As usual, an RG-module M is called a permutation RG-module if it has a finite G-invariant R-basis. Equivalently, M is isomorphic to a finite direct sum of RG-modules of the form Ind_{H}^{G}(R) with H ⩽ G.

More generally, M is called a monomial RG-module if it is isomorphic to a finite direct sum of RG-modules of the form Ind_{H}^{G}(R_{φ}) for some H ⩽ G and φ ∈ Hom(H, R^{×}). Note that conjugations, restrictions, and inductions of permutation (resp. monomial) modules are again permutation (resp. monomial) modules, since the Mackey decomposition formula holds for any base ring.

An RG-module M is called a p-monomial (resp. p-permutation) RG-module if it is isomorphic to a direct summand of a monomial (resp. permutation) RG-module. Note that a p-permutation RG-module is also a p-monomial RGmodule and that every p-monomial RG-module is finitely generated and free as R-module. The terminology is justified by the characterization of p-permutation and p-monomial modules below. A proof in the case where R is a complete discrete valuation ring can be found in [Br85] and [Bo98].

1.3 Proposition Let R be a commutative local ring. For an RG-module M the following are equivalent:

(i) M is a p-monomial (resp. p-permutation) RG-module.

(ii) There exists P ∈ Syl_{p}(G) such that Res_{P}^{G}(M) is a monomial (resp. permutation) RP -module.

(iii) For every p-subgroup P of G, the module Res_{P}^{G}(M) is a monomial (resp. permutation) RP -module.

Note that conjugations, restrictions, inductions, direct summands, finite direct sums, and tensor products of p-monomial (resp. p-permutation) modules are again p-monomial (resp. p-permutation) modules.

If R is a complete discrete valuation ring it is shown in [Bo98] that an indecomposable p-monomial RG-module has a source of the form R_{ψ}. If H ⩽ G and φ ∈ Hom(H, R^{×}) we denote by N_{G}(H, φ) the set of all elements g ∈ G satisfying ^{g}H = H and ^{g}φ = φ. If n is a natural number, n_{p′} denotes the largest factor of n not divisible by p. One has the following result on the absolute indecomposability of p-monomial RG-modules.

1.4 Theorem Assume that R is a complete discrete valuation ring or a field (i.e., 𝔭 = 0). Let M be an indecomposable p-monomial RG-module with vertex P ⩽ G and source R_{ψ}, ψ ∈ Hom(P, R^{×}), and set E := End_{RG}(M). Assume that R contains a root of unity of order [N_{G}(P, ψ) : P]_{p′}. Then E/J(E) ≅ F as F -algebras.

1.5 (a) We say that the Krull-Schmidt-Theorem holds for p-monomial RGmodules, if every p-monomial RG-module M has the following two properties:

(i) M has a finite decomposition M = U_{1} ⊕ ⋯ ⊕ U_{r} into indecomposable RG-submodules.

(ii) For any two decompositions M = U_{1} ⊕ ⋯ ⊕ U_{r} = V_{1} ⊕ ⋯ ⊕ V_{s} as in (i) one has r = s and there exists a permutation σ of degree r such that U_{i} = V_{σ(i)} for all i = 1, …, r.

Note that (i) always holds for our rings R, since M is a finitely generated free R-module and R is local. Moreover, recall that (ii) holds whenever End_{RG}(U_{i}) is a local ring for all i = 1, …, r, of some decomposition of M as in (i), cf. [NT89, Theorem I.6.1] (b) In the sequel we will need the following additional hypotheses on R:

(i) R is a local domain.

(ii) If R has characteristic 0, then R contains a root of unity of order exp(G), and if R has characteristic p, then R contains a root of unity of order exp(G)_{p′}.

(c) Note that, under the hypothesis 1.5(b), F is a splitting field for F G and that RG/J(RG) ≅ F G/J(F G) is a product of matrix rings over F, cf. [CR81, Theorem 17.1].

1.6 Theorem Assume that R satisfies the hypotheses in 1.5(b).

(a) Let M be an indecomposable p-monomial RG-module and let E := End_{RG}(M). Then E/J(E) ≅ F as R-algebras.

(b) The Krull-Schmidt Theorem holds for p-monomial RG-modules.

1.7 The representation ring L_{R}(G) of p-monomial RG-modules is defined as follows. It is the free abelian group on the isomorphism classes {M} of pmonomial RG-modules M modulo the subgroup generated by all elements of the form {M ⊕ N} − {M} − {N}. The class of the element {M} is denoted by [M] ∈ L_{R}(G). If R satisfies the hypotheses in 1.5(b), it is an easy consequence of the Krull-Schmidt Theorem (cf. Theorem 1.6(b)) that the abelian group L_{R}(G) has a canonical ℤ-basis consisting of the elements [M], where M runs through a set of representatives of the isomorphism classes of indecomposable p-monomial RG-modules. The ring structure on L_{R}(G) is given by [M] · [N] = [M ⊗_{R} N] for arbitrary p-monomial RG-modules M and N.

Similarly, one can define the representation ring T_{R}(G) of p-permutation RGmodules. If the hypotheses in 1.5(b) are satisfied, T_{R}(G) can be considered as the subring of L_{R}(G) which, as an abelian group, is generated by the canonical basis elements [M], where M is an indecomposable p-permutation RG-module.

Note that if R ⊆ S and S is a local ring with residue field of characteristic p, then the functor S⊗_{R}− from the category of RG-modules to the category of SGmodules maps p-monomial (resp. p-permutation) RG-modules to p-monomial (resp. p-permutation) SG-modules. It induces ring homomorphisms L_{R}(G) → L_{S}(G) and T_{R}(G) → T_{S}(G).

1.8 Remark Assume again that the hypotheses in 1.5(b) on R hold. Then, with the Krull-Schmidt Theorem in place, Green’s theory of vertices and sources, and the Green correspondence are valid for p-monomial RG-modules. If M is an indecomposable p-monomial RG-module with vertex P, its source must be of the form R_{ψ} with ψ ∈ Hom(P, R^{×}), and we call (P, ψ) a defect pair of M. Note that M is a p-permutation RG-module if and only if ψ = 1. Note also that the defect pairs (P, ψ) of M are uniquely determined up to G-conjugacy as the maximal pairs with the property R_{ψ} | Res_{P}^{G}(M). Here we use the obvious partial order on such pairs: (Q, λ) ⩽ (P, ψ) if and only if Q ⩽ P and λ = ψ|_{Q}.

Under the hypotheses in 1.5(b), it follows easily from Theorem 1.6(b) that an RG-module M is a p-monomial (resp. p-permutation) RG-module if and only if every indecomposable direct summand has a linear (resp. trivial) source, i.e., a source of the form R_{ψ} (resp. R). In the case that R is a complete discrete valuation ring, these modules have been called linear (resp. trivial) source modules, a name now justified for rings R satisfying the hypotheses in 1.5(b).

1.9 Theorem Assume that R satisfies the hypotheses in 1.5(b), that S is a local domain which contains R, and that the maximal ideal 𝔓 of S satisfies 𝔭 = R∩𝔓. Then the functor S⊗_{R}− induces a bijection between the isomorphism classes of indecomposable p-monomial (resp. p-permutation) RG-modules and the isomorphism classes of indecomposable p-monomial (resp. p-permutation) SG-modules. This bijection preserves defect pairs and is compatible with the Green correspondence. In particular, the induced ring homomorphisms

L_{R}(G) → L_{S}(G) and T_{R}(G) → T_{S}(G)

are isomorphisms.

2 Proofs

We continue to assume the notations from 1.1. Before we prove Proposition 1.3 and Theorems 1.4, 1.6, and 1.9, we recall and establish a few useful facts for the reader’s convenience.

2.1 Lemma Let Λ be an R-algebra which is finitely generated as R-module and let M be a Λ-module.

(a) One has 𝔭Λ ⊆ J(Λ) and 𝔭M ⊆ rad_{Λ}(M).

(b) Assume that M and N are finitely generated Λ-modules which are free as R-modules and assume that f ∈ Hom_{Λ}(M, N) has the property that the induced map \overline{f}: M/𝔭M → N/𝔭N is an isomorphism. Then f is an isomorphism.

(c) Assume that M is finitely generated and free as R-module and that f ∈ End_{Λ}(M) satisfies f(M) ⊆ rad_{Λ}(M). Then f is contained in J(End_{Λ}(M)).

(d) Assume that S is a local ring containing R as a unitary subring and that the maximal ideal 𝔓 of S satisfies 𝔭 = R ∩ 𝔓. Furthermore, set \overline{S} := S/𝔓 and \overline{Λ} := Λ/J(Λ). Then

(S ⊗_{R} Λ)/J(S ⊗_{R} Λ) ≅ (\overline{S} ⊗_{F} \overline{Λ})/J(\overline{S} ⊗_{F} \overline{Λ})

as \overline{S}-algebras. In particular, if \overline{Λ} ≅ ∏_{i=1}^{l} Mat_{n_{i}}(F), then (S ⊗_{R} Λ)/J(S ⊗_{R} Λ) ≅ ∏_{i=1}^{l} Mat_{n_{i}}(\overline{S}).

(e) Assume that Λ = RG and that P ∈ Syl_{p}(G). Then M | Ind_{P}^{G}(Res_{P}^{G}(M)).

(f) Assume that Λ = RP for a finite p-group P, that Q ⩽ P, and that M = Ind_{Q}^{P}(R_{ψ}) for some ψ ∈ Hom(Q, R^{×}). Then End_{Λ}(M)/J(End_{Λ}(M)) ≅ F. In particular, Ind_{Q}^{P}(R_{ψ}) is absolutely indecomposable and the Krull-Schmidt Theorem holds for p-monomial RP -modules, cf. 1.5(a).

Proof (a) Every simple Λ-module is finitely generated as R-module and therefore, by Nakayama’s Lemma, annihilated by 𝔭 and 𝔭Λ. Thus, 𝔭Λ ⊆ J(Λ). Moreover, for an arbitrary Λ-module M on has 𝔭M = 𝔭ΛM ⊆ J(Λ)M ⊆ rad_{Λ}(M).

(b) This is well-known. See for instance [La95, Proposition X.4.5].

(c) It suffices to show that id_{M} − fg is an isomorphism for every g ∈ End_{Λ}(M). It is easy to see that f(M) ⊆ rad_{Λ}(M) implies that id_{M} − fg is surjective. Therefore, the F -linear map \overline{id_{M} − fg}: M/𝔭M → M/𝔭M is surjective, and also bijective, since M/𝔭M is a finite-dimensional F -vector space. Now, Part (b) implies the result.

(d) Let i: 𝔓 → S and j: J(Λ) → Λ denote the inclusion maps. By Part (a) we have (i ⊗ id)(𝔓 ⊗_{R} Λ) = 𝔓(S ⊗_{R} Λ) ⊆ J(S ⊗_{R} Λ). We claim that also (id ⊗ j)(S ⊗_{R} J(Λ)) ⊆ J(S ⊗_{R} Λ). In fact, since 𝔭Λ ⊆ J(Λ) and since Λ/𝔭Λ is a finite-dimensional F -algebra, J(Λ)/𝔭Λ = J(Λ/𝔭Λ) is a nilpotent ideal of Λ/𝔭Λ. Choose n ∈ ℕ such that J(Λ)^{n} ⊆ 𝔭Λ. Then

[(id ⊗ j)(S ⊗_{R} J(Λ)]^{n} ⊆ (id ⊗ j)(S ⊗_{R} 𝔭Λ) = (i ⊗ id)(𝔭S ⊗_{R} Λ)
⊆ (i ⊗ id)(𝔓 ⊗_{R} Λ) ⊆ J(S ⊗_{R} Λ) .

This implies the inclusion claimed above. Therefore, the kernel of the natural surjection S ⊗_{R} Λ → \overline{S} ⊗_{R} \overline{Λ} ≅ \overline{S} ⊗_{F} \overline{Λ} which is equal to (i ⊗ id)(𝔓 ⊗_{R} Λ) + (id ⊗ j)(S ⊗_{R} J(Λ)) is contained in J(S ⊗_{R} Λ). This implies the result.

(e) It is easy to check that the maps f: M → RG⊗_{RP} M, m ↦ ∑_{g∈G/P} g ⊗ g^{−1}m, and h: RG ⊗_{RP} M → M, g ⊗ m ↦ [G : P]^{−1}gm, are RG-module homomorphisms satisfying h ∘ f = id_{M}.

(f) By Part (a), the trivial F P -module F is the only simple RP -module. Thus, rad_{RP} (M) is equal to the intersection of the kernels of all RP -module homomorphisms M = Ind_{Q}^{P}(R_{ψ}) → F. Obviously, the kernel of each such homomorphism contains the RP -submodule {∑_{g∈P/Q} g ⊗ α_{g} ∈ RP ⊗_{RQ} R_{ψ} | ∑_{g∈P/Q} α_{g} ∈ 𝔭}. On the other hand, this submodule is maximal, since the factor module is isomorphic to F. Thus the above submodule equals rad_{RP} (M) and M/rad_{RP} (M) ≅ F.

By Part (c), the kernel of the R-module homomorphism End_{RP} (M) → M/rad_{RP} (M)(≅ F), f ↦ f(1 ⊗ 1) + rad_{RP} (M) is contained in J(End_{RP} (M)). This implies that the kernel is equal to J(End_{RP} (M)). Thus, we have End_{RP} (M)/J(End_{RP} (M)) ≅ F.

2.2 Proof of Proposition 1.3. We prove only the version for p-monomial RGmodules. The version for p-permutation RG-modules is proved similarly.

(i)⇒(ii): Let P ∈ Syl_{p}(G). If M is a p-monomial RG-module, then Res_{P}^{G}(M) is a p-monomial RP -module. Now, Lemma 2.1(f) implies that Res_{P}^{G}(M) is a monomial RP -module.

(ii)⇒(iii): If Res_{P}^{G}(M) is monomial for some P ∈ Syl_{p}(G), then, by conjugation, for all P ∈ Syl_{p}(G). Now, (iii) follows from the transitivity of restriction.

(iii)⇒(i): By Lemma 2.1(e), we have M | Ind_{P}^{G}(Res_{P}^{G}(M)) for each P ∈ Syl_{p}(G). Since Res_{P}^{G}(M) is a monomial RP -module, M is a p-monomial RGmodule.

2.3 Lemma Asume that R is a local domain, let H, I ⩽ G, φ ∈ Hom(H, R^{×}), ψ ∈ Hom(I, R^{×}), and set M := Ind_{H}^{G}(R_{φ}) and N := Ind_{I}^{G}(R_{ψ}).

(a) For g ∈ G with the property that φ|_{H∩^{g}I} = (^{g}ψ)|_{H∩^{g}I}, let f_{g} ∈ Hom_{RG}(M, N) be defined by

f_{g}(x ⊗ 1) := ∑_{h∈H/H∩^{g}I} φ(h)^{−1}xhg ⊗ 1

for x ∈ G. If g runs through a set of representatives of the double cosets H\G/I satisfying φ|_{H∩^{g}I} = (^{g}ψ)|_{H∩^{g}I}, then the elements f_{g} form an R-basis of Hom_{RG}(M, N).

(b) If H is a normal subgroup of G, then End_{RG}(M) ≅ R[N_{G}(H, φ)/H].

Proof (a) One can verify easily that f_{g} is a well-defined RG-module homomorphism which is (up to a unit in R) independent of the choice of g in HgI. The rest follows from [BK00, 1.6,1.7], since R is a domain.

(b) One verifies easily, that f_{g} ∘ f_{g′} = f_{g′g}. Thus, End_{RG}(M) is isomorphic to the opposite ring of the group algebra R[N_{G}(H, φ)/H], which in turn is isomorphic to the group algebra by mapping a group element to its inverse.

2.4 Corollary Let S be a local domain containing R as a subring and let M be a p-monomial RG-module.

(a) The canonical map

S ⊗_{R} End_{RG}(M) ↦ End_{SG}(S ⊗_{R} M) , s ⊗ f ↦ (t ⊗ m ↦ st ⊗ f(m)) ,

of S-algebras is an isomorphism.

(b) Let 𝔓 := J(S) and assume that 𝔭 = R ∩ 𝔓. Moreover, set \overline{S} := S/𝔓 and \overline{E} := End_{RG}(M)/J(End_{RG}(M)). Then one has an isomorphism

End_{SG}(S ⊗_{R} M)/J(End_{SG}(S ⊗_{R} M) ≅ (\overline{S} ⊗_{F} \overline{E})/J(\overline{S} ⊗_{F} \overline{E})

of \overline{S}-algebras. In particular, if \overline{E} is isomorphic to F as F -algebras, then End_{SG}(S ⊗_{R} M)/J(End_{SG}(S ⊗_{R} M) is isomorphic to \overline{S} as \overline{S}-algebras.

Proof (a) The functors S ⊗_{R} Hom_{RG}(−, −) and Hom_{SG}(S ⊗_{R} −, S ⊗_{R} −) on the direct product of the category of RG-modules with itself to the category of S-modules are additive and the maps

λ_{X,Y} : S ⊗ Hom_{RG}(X, Y) → Hom_{SG}(S ⊗_{R} X, S ⊗_{R} Y) ,
s ⊗ f ↦ (t ⊗ x ↦ st ⊗ f(x)) ,

for RG-modules X and Y, form a natural transformation. By Lemma 2.3(a), λ_{X,Y} is an isomorphism in the case that X = Ind_{H}^{G}(R_{φ}) and Y = Ind_{I}^{G}(R_{ψ}) with H, I, φ, ψ as in Lemma 2.3(a), since the S-basis elements 1 ⊗ f_{g} with f_{g} ∈ Hom_{RG}(X, Y) are mapped to the corresponding S-basis elements of Hom_{SG}(S ⊗_{R} X, S ⊗_{R} Y), since S ⊗_{R} X ≅ Ind_{H}^{G}(S_{φ}) and S ⊗_{R} Y ≅ Ind_{I}^{G}(S_{ψ}) in a canonical way. Now the naturality of the maps λ_{X,Y} extends this property to monomial RG-modules and then to p-monomial RG-modules.

(b) This is an immediate consequence of Part (a) and Lemma 2.1(d).

2.5 Proof of Theorem 1.4. Recall from the hypothesis that M is an indecomposable p-monomial RG-module with defect pair (P, ψ). Let H := N_{G}(P) and let N be the Green correspondent of M. Then N | Ind_{P}^{H}(R_{ψ}) =: X and we assume that N is a submodule of X. For the endomorphism rings E := End_{RG}(M) and E′ := End_{RH}(N) one has an F -algebra isomorphism E/J(E) ≅ E′/J(E′). This is an easy consequence of a Theorem of Green, cf. [NT89, Theorem IV.5.4]. Let e ∈ EndRH(X) be an idempotent with e(X) = N. Then, by Lemma 2.3(b) and Lemma 2.1(a), one has an F -algebra isomorphism

End_{RH}(X)/J(End_{RH}(X)) ≅ F [N_{G}(P, ψ)/P]/J(F [N(P, ψ)/P] .

By 1.5(c), the latter F -algebra is isomorphic to a direct product of matrix algebras over F. Moreover

E′ ≅ eEnd_{RH}(X)e and E′/J(E′) ≅ \overline{e}[End_{RH}(X)/J(End_{RH}(X))]\overline{e} ,

where \overline{e} denotes the class of e modulo J(End_{RH}(X)). This implies that E′/J(E′) is again a direct product of matrix algebras over F. On the other hand it is a division ring, by [CR81, Proposition 6.10]. Thus, E/J(E) ≅ E′/J(E′) ≅ F as F -algebras.

2.6 Proof of Theorem 1.6. Note that (b) is an immediate consequence of (a), cf. 1.5(a).

In order to prove Part (a) we distinguish the two possible cases for the characteristic of R, namely 0 or p.

Assume first that R has characteristic 0. We proceed in two steps. First we prove the result for a special ring R_{0} and then use this to prove the general case. Let ζ ∈ ℂ be a root of unity of order exp(G).

Step 1. Assume that R = R_{0} is the localization of the ring ℤ[ζ] with respect to some prime ideal of ℤ[ζ] containing p. We will show that Part (a) holds for R_{0}. Let 𝔭_{0} denote the maximal ideal of R_{0}, R̂_{0} its 𝔭_{0}-adic completion with maximal ideal 𝔭̂_{0}. Then the natural ring homomorphism F_{0} := R_{0}/𝔭_{0} → R̂_{0}/𝔭̂_{0} is an isomorphism. By [CR81, Theorem 30.18(ii)] we know that with M also R̂_{0}⊗_{R_{0}} M is an indecomposable p-monomial R̂G-module. Since End_{R̂_{0}G}(R̂_{0}⊗_{R_{0}} M)/J(End_{R̂_{0}G}(R̂_{0} ⊗_{R_{0}} M)) ≅ R̂_{0}/𝔭̂_{0} ≅ F_{0} by Theorem 1.4, it follows from Corollary 2.4(b) that \overline{E} ≅ F_{0}. Thus, Part (a) holds for R = R_{0}.

Step 2. For our general ring R, there exists an injective ring homomorphism i: ℤ[ζ] → R and i^{−1}(𝔭) is a prime ideal of ℤ[ζ] containing p. Thus, i can be extended to a ring homomorphism R_{0} → R with a ring R_{0} as considered in Step 1. We identify R_{0} with its image under i. If M is an indecomposable p-monomial R_{0}G-module, then End_{R_{0}G}(M)/J(End_{R_{0}G}(M)) ≅ F_{0} by Step 1 and

End_{RG}(R ⊗_{R_{0}} M)/J(End_{RG}(R ⊗_{R_{0}} M)) ≅ (F ⊗_{F_{0}} F_{0})/J((F ⊗_{F_{0}} F_{0}) ≅ F

by Corollary 2.4(b). Therefore, R ⊗_{R_{0}} M is again indecomposable. For every subgroup H ⩽ G one has a bijection between Hom(H, R_{0}^{×}) and Hom(H, R^{×}), by composition with the inclusion R_{0} → R, since ζ is already contained in R_{0}. Moreover, R ⊗_{R_{0}} Ind_{H}^{G}((R_{0})_{φ}) ≅ Ind_{H}^{G}(R_{φ}) for φ ∈ Hom(H, R_{0}^{×}). Thus, if Ind_{H}^{G}((R_{0})_{φ}) = M_{1} ⊕⋯⊕M_{n} is a decomposition into indecomposable submodules, then Ind_{H}^{G}(R_{φ}) ≅ (R ⊗_{R_{0}} M_{1}) ⊕ ⋯ ⊕ (R ⊗_{R_{0}} M_{n}) is again a decomposition into indecomposable modules with local endomorphism rings. This implies that the Krull-Schmidt Theorem holds for p-monomial RG-modules, cf. 1.5(a). Moreover, it shows that every indecomposable p-monomial RG-module is of the form R ⊗_{R_{0}} M for an indecomposable p-monomial R_{0}G-module M. As shown above, such a module has an endomorphism ring with radical quotient isomorphic to F. This completes the proof if R has characteristic 0.

Now assume that R has characteristic p, let ζ ∈ R be a root of unity of order exp(G)_{p′} and let R_{0} be the smallest subring of R containing ζ. Then R_{0} is a finite field. Therefore, Part (a) holds for R_{0} by Theorem 1.4. Now we proceed as in Step 2 above to obtain the result for R.

2.7 Proof of Theorem 1.9. The Krull-Schmidt Theorem holds for p-monomial SG-modules and RG-modules by Theorem 1.6 and we can identify Hom(H, R^{×}) and Hom(H, S^{×}) for each H ⩽ G, since R satisfies the hypotheses in 1.5(b). Moreover, Ind_{H}^{G}(S_{φ}) ≅ S ⊗_{R} Ind_{H}^{G}(R_{φ}) for every φ ∈ Hom(H, R^{×}). Thus, Theorem 1.6 and Corollary 2.4(b) imply that the functor S ⊗_{R} − induces a surjective map between the isomorphism classes of indecomposable p-monomial RG-modules and those of indecomposable p-monomial SG-modules.

If (P, ψ) is a defect pair of an indecomposable p-monomial RG-module M, then, by the characterization of defect pairs in Remark 1.8, there exists a defect pair (Q, φ) of S ⊗_{R} M satisfying (P, ψ) ⩽ (Q, φ). On the other hand, with M also S ⊗_{R} M is relatively P -projective. This implies (P, ψ) = (Q, φ). Moreover, if we set H := N_{G}(P) and if the p-monomial RH-module N is the Green correspondent of M, then also N has (P, ψ) as a defect pair. By the above, also S ⊗_{R} M and S ⊗_{R} N have (P, ψ) as a defect pair. It is now easy to see that S ⊗_{R} N is the Green correspondent of S ⊗ M.

We still have to show that the surjective map between the isomorphism classes of indecomposable p-monomial RG-modules and those of indecomposable p-monomial SG-modules is injective. Since defect pairs are preserved, it suffices to show that the number of isomorphism classes of indecomposable p-monomial RG- and SG-modules with a fixed defect pair (P, ψ) coincide. Moreover, since the Green correspondence is respected, we may assume that G = N_{G}(P). In this case, by Lemma 2.3(b), we have End_{RG}(Ind_{P}^{G}(R_{ψ})) ≅ RN as R-algebras and End_{RG}(Ind_{P}^{G}(S_{ψ})) ≅ SN as S-algebras, with N := N_{G}(P, ψ)/P. If

1 = ∑_{i=1}^{l} ∑_{j=1}^{n_{i}} e_{ij}  (∗)

is a primitive idempotent decomposition of 1 in RN satisfying

e_{ij} is (RN)^{×}-conjugate to e_{i′j′} ⟺ i = i′ ,  (∗∗_{RN})

then the number of isomorphism classes of indecomposable p-monomial RGmodules with defect pair (P, ψ) is equal to l. Let R_{0} ⊆ R be defined as in 2.6, depending on the characteristic of R. If R has characteristic p, then R_{0} is a finite field and many of the following arguments can be shortened. We denote the residue field of R_{0} by F_{0}.

Now choose a primitive idempotent decomposition of R_{0}N as in (∗) satisfying (∗∗_{R_{0}N}). Let R̂_{0} denote the completion of R_{0} with respect to F_{0}. In the case that R has characteristic p we have R̂_{0} = R_{0}. In general we have a commutative diagram

[figure: R̂_{0}N; RN; R̂_{0}N/J(R̂_{0}N); R_{0}N; RN/J(RN); R_{0}N/J(R_{0}N)]

where all maps are the canonical ones. By [CR81, 30.17] applied to the projective indecomposable R_{0}N-modules and by 1.5(c) applied to R_{0} together with Lemma 2.1(d) applied to Λ = R_{0}N and R_{0} ⊆ R̂_{0}, the decomposition (∗) is still a primitive idempotent decomposition in R̂_{0}N satisfying (∗∗_{R̂_{0}N}). By standard theorems on lifting idempotents over the base ring R̂_{0}, we obtain that the image of (∗) in R̂_{0}N/J(R̂_{0}N) is again a primitive idempotent decomposition satisfying (∗∗_{R̂_{0}N/J(R̂_{0}N)}). The same must therefore hold for any algebra through which the canonical map R_{0}N → R̂_{0}N/J(R̂_{0}N) factors, including R_{0}N/J(R_{0}N). Again by 1.5(c) applied to R_{0} together with Lemma 2.1(d) applied to Λ = R_{0}N and R_{0} ⊆ R, the image of (∗) in RN/J(RN) is again a primitive idempotent decomposition in RN/J(RN) satisfying (∗∗_{RN/J(RN)}). Since the natural map R_{0}N → RN/J(RN) factors through RN, the decomposition (∗) is also a primitive idempotent decomposition in RN satisfying (∗∗_{RN}). Altogether, we have shown that the number of isomorphism classes of indecomposable p-monomial RG-modules with defect pair (P, ψ) is equal to l, a number that only depends on R_{0}. Since S leads to the same ring R_{0}, the bijectivity of the map between isomorphism classes of indecomposable p-monomial RG- and SG-modules is now proved.

As noted in Remark 1.8, an indecomposable p-monomial RG-module is a p-permutation RG-module if and only if its defect pairs are of the form (P, 1). Thus, the invariance of defect pairs implies that the map between isomorphism classes of indecomposable p-monomial modules restricts to a bijection between the isomorphism classes of indecomposable p-permutation modules over R and S.

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