1 Introduction
The Dade group and endo-permutation modules are important invariants of block
theory of finite group algebras [6]. For instance, they occur in the description of source algebras of blocks (see e.g. [9, §50] or [5, §50]), or as sources of simple modules for p-soluble groups (see e.g. [9, §30]). They also play an important role in the description of equivalences between block algebras, such as derived equivalences in the sense of Rickard or Morita equivalences (see e.g. the recent papers [7, 1]). The final classification of endo-permutation modules was obtained by Bouc in [2], but one last question about the structure of the Dade group remained open, namely whether lifting endo-permutation modules from positive characteristic to characteristic zero can be turned into a group homomorphism. The aim of this note is to fill this gap.
Throughout p denotes a prime number, P a finite p-group, and 𝒪 a complete discrete valuation ring of characteristic 0 containing a root of unity of
order exp(P), the exponent of P, with a residue field k:=𝒪/𝔭 of characteristic p, where 𝔭=J(𝒪) is the unique maximal ideal of 𝒪.
We let R∈{𝒪,k}.
All modules considered are assumed to be finitely generated left modules, and we will consider 𝒪P-lattices only, that is 𝒪P-modules which are free as 𝒪-modules. For an 𝒪P-lattice L, the reduction modulo 𝔭 of L is the kP-module L/𝔭L, and a kP-module M is said to be liftable if there exists an 𝒪P-lattice M^ such that M≅M^/𝔭M^.
Very few classes of modules are known to be liftable from k to 𝒪 in general. However, it is known that any endo-permutation kP-module can be lifted to an endo-permutation 𝒪P-lattice.
This nontrivial result is a consequence of their classification, due to Bouc [2].
Let us fix some more precise notation.
Let DR(P) denote the group of endo-permutation RP-lattices (i.e. the Dade group of P).
The reduction homomorphism modulo 𝔭
πp:D𝒪(P)→Dk(P)
maps the equivalence class of an endo-permutation 𝒪P-lattice to the equivalence class of its reduction modulo 𝔭.
By an important result of Bouc [2, Corollary 8.5], the map πp is surjective.
Moreover, its kernel is isomorphic to the group X(P) of one-dimensional 𝒪P-lattices (see Lemma 2.1).
The aim of this note is to prove that this reduction map always admits a section which is a group homomorphism.
Theorem 1.1.
Let P be a finite p-group.
The group homomorphism πp:D𝒪(P)→Dk(P) has a group-theoretic section.
By the above, it is clear that (b) follows from (a), so we only have to prove (a). In other words, we have to show how to choose the lifts of all capped endo-permutation kP-module in a suitable fashion. When p is odd, the result is easy and does not require any other deep result about endo-permutation modules. We will briefly recall this construction in Lemma 3.3. Thus the main question is to deal with the case of 2-groups in characteristic 2. Furthermore, we will explain in Remark 2.2 that, as a consequence of the surjectivity of πp, the result is equivalent to another result in terms of Dade P-algebras mentioned without proof in [9, Remark 29.6].
In fact, our aim is not only to prove that πp always admits a group-theoretic section, but also more accurately to describe how to define a section in a natural way on a set of generators of Dk(P).
2 Endo-permutation lattices and the Dade group
We start by recalling some basic facts about endo-permutation modules and the Dade group. We refer to the survey [10] for more details and suitable references.
An RP-module M is said to be endo-permutation if its endomorphism algebra EndR(M) is a permutation RP-module,
where EndR(M) is endowed with its natural RP-module structure via the action of P by conjugation:
ϕg(m)=g⋅ϕ(g-1⋅m) for all g∈P, all ϕ∈EndR(L) and all m∈M.
Notice that, if R=𝒪, then it is easy to see that any endo-permutation 𝒪P-module is necessarily free as an 𝒪-module, i.e. an 𝒪P-lattice, because 𝒪 is a PID.
Hence, in the sequel, we consider RP-lattices only.
In particular, the dimensiondimM of an RP-lattice M is the rank of M viewed as a free R-module.
Moreover, writing M*=HomR(M,R) for the dual of the RP-lattice M, we have
EndR(M)≅M⊗RM*
as RP-lattices.
An endo-permutation RP-lattice M is said to be capped if it has at least one indecomposable direct summand with vertex P, and in this case there is in fact a unique isomorphism class of indecomposable direct summands of M with vertex P, called the cap of M and denoted by Cap(M). Unfortunately, Cap(M) may appear with a multiplicity as a direct summand of M and we shall need to avoid this.
Following [8, Definition 5.3], we say that an endo-permutation RP-lattice M is strongly capped if it is capped and if Cap(M) has multiplicity one as a direct summand of M.
We note that the class of strongly capped endo-permutation RP-lattices is closed under taking duals and tensor products (see [8, Lemma 5.4]).
A strongly capped endo-permutation RP-lattice M has dimension dimR(M) prime to p, because dimR(Cap(M)) is prime to p
(see [9, Corollary 28.11]) and any other direct summand has dimension divisible by p since its vertex is strictly smaller than P.
This is the reason why we shall use strongly capped endo-permutation RP-lattices rather than capped endo-permutation RP-lattices.
Two capped endo-permutation RP-lattices M and N are equivalent if there exist two capped permutation RP-lattices S and T
such that M⊗RS≅N⊗RT. This happens if and only if Cap(M)≅Cap(N) and this defines an equivalence relation.
The Dade group is usually defined as the set of equivalence classes [M] of capped endo-permutation RP-lattices M.
Any class contains, up to isomorphism, a unique indecomposable endo-permutation RP-lattice, namely the cap of any element of the class.
This indecomposable endo-permutation RP-lattice is of course strongly capped, so that we can define the Dade group by restricting each class and using only strongly capped endo-permutation RP-lattices.
We do so and define the Dade group DR(P) as the set of equivalence classes [M] of strongly capped endo-permutation RP-lattices M,
endowed with the product [M]⋅[N]=[M⊗RN] induced by the tensor product over R. (This definition coincides with [8, Corollary–Definition 5.5].)
We note that two strongly capped endo-permutation RP-lattices M and N are equivalent if there exist two strongly capped permutation RP-lattices S and T such that M⊗RS≅N⊗RT.
The identity element is the class [R] of the trivial RP-lattice R and [R] consists of all strongly capped permutation RP-lattices (i.e. permutation RP-lattices RX, where X is a basis of RX permuted by the action of P, having a trivial direct summand R with multiplicity one, corresponding to a unique fixed point in X).
If L is an indecomposable endo-permutation RP-lattice, it is the cap of any element of its class and an arbitrary element of the class [L] has the form Cap(L)⊗RRX where RX is a strongly capped permutation RP-lattice.
An RP-lattice M is called endo-trivial if EndR(M)≅R⊕Q, where Q is a projective RP-lattice (or equivalently a free RP-lattice because P is a p-group).
Clearly any endo-trivial RP-lattice M is endo-permutation and moreover
TR(P):={[M]∈DR(P)∣Cap(M) is endo-trivial}
is a subgroup of DR(P). For simplicity, TR(P) is called here the group of endo-trivial RP-lattices (but it is only isomorphic to the usual group of endo-trivial RP-lattices).
We recall that the Dade group is known to be a finitely generated abelian group, hence a product of cyclic groups. The most important examples of indecomposable endo-permutation RP-lattices are given by the relative Heller translates of the trivial RP-lattice, which we denote by ΩP/Qm(R) (m∈ℤ, Q≤P) as usual, and simply write Ωm(R) when Q=1.
In fact, when p is odd, Dk(P) is generated by such modules, by [2, Theorem 7.7].
The group homomorphism πp:D𝒪(P)→Dk(P) was defined in the introduction and we recall that it is surjective by [2, Corollary 8.5].
Lemma 2.1.
The kernel of πp:DO(P)→Dk(P) is isomorphic to the group X(P) of one-dimensional OP-lattices.
Proof.
Suppose that L is an indecomposable endo-permutation 𝒪P-lattice with [L]∈Ker(πp). Then dimk(L/𝔭L)=1, hence dim𝒪(L)=1.
If, conversely, dim𝒪(L)=1, then the one-dimensional kP-module L/𝔭L
must be trivial since there are no nontrivial pn-th roots of unity in the field k of characteristic p. Therefore [L]∈Ker(πp).
∎
Remark 2.2.
The endomorphism algebra EndR(M) of an endo-permutation RP-lattice M is naturally endowed with the structure of a so-called Dade P-algebra, that is, an 𝒪-simple permutation P-algebra whose Brauer quotient with respect to P is nonzero. Furthermore, there exists also a version of the Dade group, denoted by DRalg(P), obtained by defining an equivalence relation on the class of all Dade P-algebras rather than capped endo-permutation RP-lattices, where multiplication is given by the tensor product over R. We refer to [9, §28–29] for this construction. This induces a canonical homomorphism
dR:DR(P)→DRalg(P),[M]↦[EndR(M)],
which is surjective by [9, Proposition 28.12]. The identity element of DRalg(P) being the class of the trivial P-algebra R, it follows that the kernel of dR is isomorphic to X(P) when R=𝒪, whereas it is trivial when R=k.
Now, reduction modulo 𝔭 also induces a group homomorphism
πpalg:D𝒪alg(P)→Dkalg(P),[A]↦[A/𝔭A].
Because End𝒪(M)/𝔭End𝒪(M)≅Endk(M/𝔭M) for any 𝒪P-lattice, it follows that we have a commutative diagram with exact rows and columns:
The injectivity of πpalg follows from the commutativity of the bottom-right square because
Ker(πpalgd𝒪)=Ker(dkπp)=Ker(πp)=X(P)
and its image under d𝒪 yields Ker(πpalg)=d𝒪(X(P))={1}.
The surjectivity of πp implies that πpalg is also surjective, hence an isomorphism, so that
Dk(P)≅Dkalg(P)≅D𝒪alg(P).
Therefore, finding a group-theoretic section of πp is equivalent to finding a group-theoretic section of d𝒪.
3 Determinant
Given an 𝒪P-lattice L, we may consider the composition of the underlying representation of P with the determinant homomorphism det:GL(L)→𝒪×. This is a linear character of P and is called the determinant of L. Given g∈P, we write det(g,L) for the determinant of the action of g on L. If det(g,L)=1 for every g∈P, that is, if the determinant of L is the trivial character, then we say that L is an 𝒪P-lattice of determinant 1.
Lemma 3.1.
Let L and N be OP-lattices of determinant 1.
L* is an 𝒪P-lattice of determinant
1.
L⊗𝒪N is an 𝒪P-lattice of determinant
1.
Proof.
(a) Since the action of g∈P on φ∈L* is given by (g⋅φ)(x)=φ(g-1x) for all x∈L, we have clearly
det(g,L*)=det(g-1,L)=det(g,L)-1.
Since L has determinant 1, so has L*.
(b) The determinant of a tensor product satisfies the well-known property
det(g,L⊗𝒪N)=det(g,L)dimN⋅det(g,N)dimL.
Since both determinants are 1, we obtain det(g,L⊗𝒪N)=1.
∎
Among the lifts of a strongly capped endo-permutation kP-module M, there always exists one which has determinant 1, by [9, Lemma 28.1], using our assumption that there are enough roots of unity in 𝒪 and the fact that dimk(M) is prime to p because M is strongly capped.
This lift of determinant 1 is unique up to isomorphism and will be written ΦM.
Lemma 3.2.
Let M and N be strongly capped endo-permutation kP-modules. Then:
ΦM⊗kM* is a permutation 𝒪P-lattice lifting the permutation kP-module M⊗kM*.
Proof.
(a) It is clear that ΦM* lifts M*.
Since ΦM has determinant 1, so has ΦM* by Lemma 3.1, and therefore ΦM*≅ΦM*.
(b) The lattice ΦM⊗𝒪ΦN has determinant 1 by Lemma 3.1 and is therefore isomorphic to ΦM⊗kN.
(c) We have ΦM⊗kM*≅ΦM⊗𝒪ΦM* by (b).
Using (a), it follows that
[ΦM⊗kM*]=[ΦM⊗𝒪ΦM*]
=[ΦM]⋅[ΦM*]=[ΦM]⋅[ΦM*]=[𝒪],
which is the class consisting of all strongly capped permutation 𝒪P-lattices. Therefore ΦM⊗kM* is a permutation 𝒪P-lattice.
∎
These properties of the determinant allow us to prove Theorem 1.1 (a) in the odd characteristic case.
It is briefly mentioned without proof in [9, end of Remark 29.6] that the map d𝒪:D𝒪(P)→D𝒪alg(P) always has a group-theoretic section when p>2, which is equivalent to Theorem 1.1 (a) thanks to Remark 2.2.
For completeness, we provide a proof of this result in terms of lifts of modules.
Lemma 3.3.
Suppose that p is an odd prime.
Any permutation 𝒪P-lattice has determinant
1.
Let [L]∈D𝒪(P). If Cap(L) has determinant
1
, then any element of the class [L] also has determinant
1.
Let M1 and M2 be two indecomposable endo-permutation kP-modules and let N be the cap of M1⊗kM2.
Then the cap of ΦM1⊗𝒪ΦM2 is isomorphic to ΦN.
The map
Dk(P)→D𝒪(P),[M]→[ΦM]
is a well-defined group homomorphism which is a section of πp.
Proof.
(a) Let L=𝒪X be a permutation 𝒪P-lattice, where X is a basis of L permuted under the action of P.
For any g∈P, the permutation action of g on X decomposes as a product of cycles of odd length, because the order of g is odd.
Any such cycle is an even permutation, so the determinant of the action of g on L is 1.
(b) By the definition of the Dade group, an arbitrary element of the class [L] has the form Cap(L)⊗𝒪𝒪X, where 𝒪X is a strongly capped permutation 𝒪P-lattice.
Since both L and 𝒪X have determinant 1, so has their tensor product by part (b) of Lemma 3.1.
(c) Again by Lemma 3.1 (b), the determinant 1 is preserved by tensor product. Hence the claim follows from (b).
(d) This follows from (b) and (c).
∎
From now on, unless otherwise stated, we assume that P is a finite 2-group and k has characteristic 2.
It turns out that Lemma 3.3 fails when p=2 in general.
It is clear that a (strongly capped) permutation kP-module always lifts in a unique way to a (strongly capped) permutation 𝒪P-lattice.
However, we emphasise that this lift may be different from the lift of determinant 1.
It follows that two strongly capped endo-permutation 𝒪P-lattices in the same class in D𝒪(P) need not have the same determinant.
The problem is made clear through the following two results.
Lemma 3.4.
Let g∈P∖{1} and let C=〈g〉.
If C=P, then det(g,𝒪P)=-1.
If C<P, then det(g,𝒪P)=1.
Proof.
(a) Since P=〈g〉, the action by permutation of g on P is given by a cycle of even length, hence an odd permutation.
Therefore det(g,𝒪P)=-1.
(b) Viewed by restriction as an 𝒪C-lattice, 𝒪P is isomorphic to a direct sum of |P:C| copies of 𝒪C.
Since C<P, the index |P:C| is even.
Therefore, using (a), we obtain
det(g,𝒪P)=det(g,(𝒪C)|P:C|)=det(g,𝒪C)|P:C|=(-1)|P:C|=1,
as was to be proved.
∎
Corollary 3.5.
Let g∈P∖{1} and let C=〈g〉.
If C=P, then det(g,Ω1(𝒪))=-1.
If C<P, then det(g,Ωm(𝒪))=1 for any m∈ℤ.
Proof.
(a) From the short exact sequence
0→Ω1(𝒪)→𝒪P→𝒪→0,
we obtain
det(g,Ω1(𝒪))det(g,𝒪)=det(g,𝒪P)=-1,
by Lemma 3.4. Since det(g,𝒪)=1, the result follows.
(b) Recall that Ω-m(𝒪)≅Ωm(𝒪)*.
By Lemma 3.1, passing to the dual preserves the property that the determinant is 1.
Therefore we may assume that m>0 and we proceed by induction on m.
Since P is a 2-group, every projective 𝒪P-lattice is free, so there is a short exact sequence of the form
0→Ωm(𝒪)→(𝒪P)r→Ωm-1(𝒪)→0
for some integer r.
It follows that
det(g,Ωm(𝒪))det(g,Ωm-1(𝒪))=det(g,(𝒪P)r)=det(g,𝒪P)r=1,
by Lemma 3.4.
Starting from the obvious equality det(g,𝒪)=1, we conclude by induction that det(g,Ωm(𝒪))=1.
∎
If P=C2n, we note that there are two natural lifts for Ω1(k)=ΩC2n/11(k).
One of them is Ω1(𝒪), but it does not have determinant 1, by Corollary 3.5.
The other one is ΦΩ1(k), which turns out to be isomorphic to 𝒪-⊗𝒪Ω1(𝒪),
where 𝒪- denotes the one-dimensional module with the generator of C2n acting by -1.
Corollary 3.5 together with an induction on n show that the same holds for the other generators, namely
ΦΩC2n/Q1(k)≅𝒪-⊗𝒪ΩC2n/Q1(𝒪)
for every Q≤C2n of index at least 4.
4 Lifting from characteristic 2
Our aim is to construct a section for πp using a set of generators for the Dade group Dk(P).
Since the assignment M↦ΦM has good multiplicative behaviour by Lemma 3.2 (b), one might expect that the multiplicative order in the Dade group is preserved, but this is in fact not at all straightforward. If [M] has order n in Dk(P), then M⊗n is isomorphic to a permutation module kX, where X is a basis of kX permuted by the action of P. Then we obtain
(ΦM)⊗n≅ΦM⊗n≅ΦkX,
but ΦkX may not be a permutation 𝒪P-lattice. The only obvious thing is that [ΦkX] lies in Ker(πp)≅X(P).
However, we now show that a much better result holds when the order n is a power of 2.
Lemma 4.1.
Let M be a strongly capped endo-permutation kP-module.
If [M] has order
2
in Dk(P), then [ΦM] has order
2
in D𝒪(P).
If [M] has order
4
in Dk(P), then [ΦM] has order
4
in D𝒪(P).
Proof.
(a) Since [M] has order 2 in Dk(P), we have M≅M*. By Lemma 3.2 (a), we obtain
ΦM≅ΦM*≅ΦM*.
Hence [ΦM]=[ΦM]-1 in D𝒪(P) and the result follows.
(b) Let N=M⊗kM. Since [M] has order 4 in Dk(P), it follows that [N] has order 2 and so [ΦN] has order 2 in D𝒪(P) by (a).
Therefore, using Lemma 3.2, we obtain
[ΦM]4=[(ΦM)⊗4]=[ΦM⊗4]
=[ΦN⊗2]=[ΦN⊗𝒪ΦN]=[ΦN]2=1,
as required.
∎
We can now prove Theorem 1.1 in the general case.
Proof of Theorem 1.1.
(a) First of all, the case p≥3 is proved in Lemma 3.3. Hence, we may assume that p=2.
We rely on the fact that
Dk(P)≅(ℤ/2ℤ)a×(ℤ/4ℤ)b×ℤc
for some non-negative integers a,b,c (see [10, Theorem 14.1]).
This follows either from Bouc’s classification of endo-permutation modules [2, Section 8], or from the detection theorems of [4]
which do not depend on the full classification. In any case, the result uses a reduction to the cases of cyclic 2-groups, semi-dihedral 2-groups and generalized quaternion 2-groups, and in these cases the torsion subgroup of the Dade group over k only contains elements of order 2 or 4 (see [3]).
Now we choose generators for each factor ℤ/2ℤ, ℤ/4ℤ, or ℤ. Each of them can be lifted to an element of D𝒪(P) of the same order, by Lemma 4.1.
This procedure for the generators then extends obviously to a group homomorphism Dk(P)→D𝒪(P) which is a group-theoretic section of the surjection πp.
(b) By Lemma 2.1, Ker(πp) is isomorphic to the group X(P) of all one-dimensional 𝒪P-lattices.
Thus the isomorphism D𝒪(P)≅X(P)×Dk(P) follows from (a).
∎
Remark 4.2 (Non-uniqueness of the section).
It is clear that the group-theoretic section of πp obtained through Lemma 3.3 in odd characteristic and in the proof of Theorem 1.1 in characteristic 2 is not unique in general, because it suffices to send generators of Dk(P) to lifts of the same order in D𝒪(P). Thus the number of group-theoretic sections of πp depends on the structure of the kernel Ker(πp)≅X(P).
However, in case p is odd, we note that the section is uniquely determined on the torsion part Dkt(P) of Dk(P), because Dkt(P) is a 2-group and X(P) a p-group.
Acknowledgements
The authors thank the referee for pointing out a mistake in an earlier version of this paper and proposing a correction.
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and
Jacques Thévenaz
