A theorem of Hertweck on -adic conjugacy of -torsion units in group rings
A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck.
Let be a commutative ring, a finite group, a normal -subgroup of and denote by the group ring of over . It is shown that a torsion unit in mapping to the identity under the natural homomorphism is conjugate in the unit group of to an element in . Here denotes the -adic integers. The result is achieved proving a result in the context of the so-called double action formalism for group rings over -adic rings. This widely generalizes a theorem of Hertweck and a related theorem by Caicedo-Margolis-del Río and has consequences for the study of the Zassenhaus Conjecture for integral group rings.
Key words and phrases:Unit Group, Group Ring, Zassenhaus Conjecture, -adic Conjugacy
2010 Mathematics Subject Classification:16S34, 16U60, 20C11, 20C05
A long standing conjecture by Zassenhaus states that for a finite group and a torsion unit in the integral group ring there exists a unit in the rational group algebra of conjugating onto an element of the form for some .
A main achievement in the study of this conjecture has been Weiss’ proof for nilpotent groups [Wei91]. Weiss in fact showed that the unit conjugating onto can even be taken in the -adic group ring [Wei91, Corollary on page 184]. Though one can in general not hope that the conjugation will always take place in the -adic group ring, the smallest non-abelian case provides a well-known counterexample [Her06, Example 3.4], in some instances this is true, as for Frobenius groups and units which map to the identity when factoring out the Frobenius kernel [Her12].
The idea that studying -adic conjugacy of units in can give insight into the Zassenhaus Conjecture, i.e. into conjugacy questions in the rational group algebra, was introduced by Hertweck in [Her06] and was a main ingredient in the results of [Her08, CMdR13]. The result presented here greatly increases the knowledge on this topic and first applications of it to the Zassenhaus Conjecture will be published soon.
Before stating the result whose proof will occupy most of this note, we need to introduce the double-action module. Let be a commutative ring, a finite group and a cyclic group generated by an element . For a torsion unit of order dividing the order of define the -module . As -module this is just and the action of is defined by
This formalism allows to compare modules associated to different torsion units in . When speaking of a -adic ring we will mean a complete discrete valuation ring of characteristic with residue class field of characteristic . We can now state the results.
Let be a -adic ring with quotient field and let be a torsion unit of -power order in . Assume that is conjugate to an element of in the unit group of . Then is conjugate to in the unit group of if and only if is isomorphic to a direct summand of a direct sum of copies of .
Using previous results of Hertweck relying on the work of Weiss [Wei88] this implies the following. Here denotes the semilocalisation of at the primes dividing the order of the finite group .
Let be a finite group and a normal -subgroup of . Then any torsion unit which maps to the identity under the natural homomorphism is conjugate to an element of in the unit group of and thus also in the unit group of .
This also emphasises the difference between studying torsion units or torsion subgroups in , since for subgroup it is known that such a result does not hold [Her02, Example 4.1].
The following section introduces the basic concepts which connect torsion units and bimodules. After that we study the idempotents in -adic group rings and their relation with partial augmentations and relative projectivity before proving Proposition 1 and Theorem 1 in the last section. Many of the results in the second and fourth paragraph are well known, see [The95], but we include proofs for completeness and to avoid too many assumptions on the ground ring .
Remark on this note: This preprint is not intended for peer-reviewed publication, since this is not my own result. The results were publicly presented, with a sketch of the proof, by Hertweck in the Algebra Seminar of the University of Stuttgart on January 29th 2013, cf. the abstract of the talk [Her13]. A physical copy of a preprint proving the results was given to me by Hertweck in late March 2013. It is my understanding that Hertweck was planning a generalisation of these results, but I have no knowledge if this was achieved. Since Hertweck has not been known to be mathematically active since late 2013 and this result is of major interest in the study of torsion units of group rings, apart from having also a beautiful proof, I feel this way of publication as appropriate.
The proof presented here follows very closely the proof of Hertweck. The wording has been changed, a few details added, some notation changed, at places results have been made more general or more special, but all ideas and the basic structure remain.
2. Notation and basic facts
Let be a commutative ring and a finite group. Denote by the group of units of augmentation in , i.e. the units whose coefficients sum up to . Let be a group homomorphism from a finite group to . Define a right -module which is as -module and where the action of is given by
Such a module is sometimes called a double-action module. Note that this module may also be viewed as an -bimodule. Moreover the module corresponds in this notation to where maps to .
The connection between double-action modules and conjugacy of subgroups in the unit group of is given by the concept of -equivalence of homomorphisms, cf. [Wei91], [RS87, Section (1.2)]. If is another homomorphism then and are called -equivalent if there exists a unit in such that for all . Then and are -equivalent if and only if . More precisely such an isomorphism between and is given by right multiplication by the unit in and this unit is the conjugating unit in the definition of -equivalence.
Denote by the standard anti-involution on , i.e. the -linear extension of for to . Then for and we have . Moreover if is an idempotent in then the direct summand of is projective. The group homomorphism also yields a conjugation action of on . Denote the set of -fixed points by
Then is a subring of with the same identity. The -fixed points can also be defined for -invariant subspaces of .
We first collect some basic facts about the direct summands of .
Let be a commutative ring, a finite group and a group homomorphism. Set .
Any direct summand of is of the form for some idempotent in .
A direct indecomposable summand of is of the form for a primitive idempotent of .
Let be an idempotent in and view as a direct summand of . Then
In particular .
Let be a direct summand of and a submodule of such that . Let with and . Then and are orthogonal idempotents since
where and . So from we conclude . Moreover for we have
So implies and hence .
This follows from a).
The isomorphism is given by mapping to . Now let and . Then from
we get that .
Let be a complete noetherian local ring, e.g. a complete discrete valuation ring.
Let be an idempotent in . Then is primitive if and only if is local.
For idempotents and in let be an isomorphism of -modules. Then is given as the right multiplication with a unit in and .
There is a bijection between the isomorphism classes of indecomposable direct summands of and conjugacy classes of primitive idempotents in . This conjugacy action is understood to be taking place in the units of .
This is a direct consequence of b) and Lemma 1.
A fundamental notion when studying conjugacy of units are the so called partial augmentations. Let be an element in and denote by the conjugacy class of an element in . Then is called the partial augmentation of at . Then is a trace function, i.e. for any [Seh93, Lemma (7.2)]. Note that is frequently denoted , but we prefer the first notation since we will be distinguishing between conjugacy in and subgroups of .
We will also need the character associated to the double-action module . Since is free as -module we can associate to it a character . Fixing as a basis of an easy calculation [Seh93, Lemma (38.12)] then shows that for any we have
For an idempotent in let be the character associated to the direct summand of . Since annihilates while acting as on we obtain
In particular for we obtain the characters of the right projective -lattices:
3. Idempotents and partial augmentations
For a commutative ring define to be the additive commutator of , i.e. the -linear span of all elements of the form for . Then consists exactly of those elements of satisfying for all [Seh93, Lemma (7.2)]. Note that for of characteristic an element satisfies for any ordinary character of . Moreover for denote by the support of , i.e. the set of elements of having non-vanishing coefficient in .
We first consider linear independence modulo of idempotents in .
Let be a -modular system and let be representatives of the conjugacy classes of primitive idempotents in . If are elements in such that then .
Let be the irreducible -characters of with corresponding primitive idempotents in . Let be the homomorphism between the Grothendieck groups of projective -modules and -modules, as defined in [CR90, § 18]. Thus assigns to a projective -module, with respect to the basis , its composition factors with respect to the basis . So for each we have Now for and so
From the assumption that we know for any and so
But by [CR90, (21.20) Theorem] the map is a split injection and thus the above equality implies . ∎
We next proof a generalization of an observation of Külshammer [Kül94, Proposition 3] using a result of Swan. For a Dedekind ring of characteristic an element is called -singular, if the order of is not a unit in . For a prime denote by the -linear span of -regular elements in , i.e. those elements whose order is not divisible by .
Let be a Dedekind ring of characteristic and an idempotent in . Then for any -singular element . In particular for a prime not invertible we have .
We need to establish one more connection between idempotents and partial augmentations and the following elementary observation will be quite useful.
Let be a non-trivial -element. Set and let . Then for any .
Moreover let and assume that for
Then for we have
If then clearly . If on the other hand and are elements in which are conjugate in , where , then they have the same -part, namely . Thus conjugation between and takes place already in and so .
For the second statement observe that since lies in the centre of multiplication by is permuting the conjugacy classes in and so we have . Thus and so we can assume . So by the first statement our assumption implies
and this implies the second statement of the lemma. ∎
Let be a -adic ring and a primitive idempotent in . Then the multiplicity of is the number of conjugates of appearing in an orthogonal primitive decomposition of in . Here conjugation is understood to take place in the unit group of .
Let be a -adic ring, a non-trivial -element and set . Let be representatives of conjugacy classes of primitive idempotents in with multiplicity respectively. Let be integers such that for any we have
Then for all .
4. Relative Projectivity and Idempotents
This section is devoted to the proof of Lemma 7 which, in the words of Hertweck, ”seems to be more or less known” and information on it can be found e.g. in Thevenaz’s book [The95]. Part d) of the lemma seems however to be less well known. We will first introduce the necessary notation. Let be a commutative ring and and subgroups of such that . Denote by a transversal of the right cosets of in . Then and we can define the relative trace map
Abusing notation we will write the image of as . We will need the following result which follows by lifting idempotents.
Let be a -adic ring, a non-trivial -element and set . Furthermore set
For a set of representatives of the conjugacy classes of primitive idempotents in there exists a set of representatives of conjugacy classes of primitive idempotents in where and for . Moreover for the difference is an idempotent of which is orthogonal to .
is an ideal of and . Moreover for we have , so . Thus we obtain
Set . Let be a primitive orthogonal idempotent decomposition in for a certain index set . By the assumption on a lifting theorem is available [The95, (3.1) Theorem] showing that is a primitive orthogonal idempotent decomposition in . For each let be a primitive orthogonal idempotent decomposition in for certain . Then is a primitive orthogonal idempotent decomposition in . Every primitive idempotent in is conjugate to one of the .
Fix some . Since is primitive all are except one, say . In particular lies in for . Set , so is an idempotent in orthogonal to . Then . Finally if is another primitive idempotent in such that lies in then and are conjugate by [The95, (3.2) Theorem (d)]. ∎
Remark: In the proof we used [The95, Theorems 3.1, 3.2] where it is assumed that the residue class field of is algebraically closed. This is however not needed in the proof of these theorems, so we can apply them in our setting.
We will also need the following variation of Green’s indecomposability theorem. Denote by the centre of .
Let be a -adic ring. Let be a subgroup of of -power index such that . If is an indecomposable -module then is an indecomposable -module.
Arguing by induction we can assume . Set and denote and . We will imitate the arguments of [Dad71, Proposition 12.10]. We view as an -subalgebra of . Let be a -element such that . Denote by the endomorphism of mapping to . Then is a central element of and by Frobenius reciprocity we have
as -modules. In particular is generated by and . Denote by and the radical of and respectively. Denoting by the image of in we find that generates a nilpotent ideal in , since the latter is a ring of characteristic and of -power order. Setting we thus get an isomorphism
By the indecomposability of we know that is a division ring and thus so is its quotient . Hence so is . So is local and is indecomposable. ∎
We are now ready to prove the main lemma of this paragraph connecting all the concepts developed so far. For a summand of the double-action module associated to a torsion unit in we will write sometimes .
Let be a -adic ring, a non-trivial -element and in a primitive idempotent. View as an -module where is a cyclic group of the same order as . The following are equivalent.
is projective relative to .
There exists a primitive idempotent such that
is an orthogonal decomposition of .
Set and let .
a) b): By [CR90, (19.2) Theorem (iv)] is a direct summand of the module . So, since is indecomposable, there exists a direct summand of such that is a direct summand of . But is indecomposable by Lemma 6 and so
Hence as -modules we have
where each is a direct summand of isomorphic to and for all . Here indices are understood to be taken modulo . i.e. . Let
with . Then the are primitive and pairwise orthogonal idempotents in and . Since is centralized by we obtain
where . So for and b) holds with .
b) c) is clear.
c) a): Let such that . Since is centralized by we have
For an element denote by the map from to given by right multiplication with . Then is an endomorphism of . Moreover is the identity map on and
So is projective relative to by Higman’s criterion [CR90, (19.2) Theorem (iii)].
b) d): By assumption as -modules we have . So the character corresponding to the -module is induced from a character of an -module, namely . Since for any the element is not conjugate to any element in this implies . But by (1) . So for all and this is equivalent to .
d) c): By assumption we have for all . We will argue by contradiction, so assume . Setting we know by Lemma 5 that there exist a primitive idempotent in and an idempotent in such that . Moreover is the sum of primitive idempotents in . So by applying the implication c) b) d) to each primitive summand of we obtain for all . Moreover there exists some such that . Since in central in this implies and so by Lemma 4. Altogether we get
contradicting our assumption. ∎
We will record one more fact following from the proof of a) b) in the preceding lemma.
Let be a -adic ring, a non-trivial -element and a primitive idempotent. If is an indecomposable direct summand of then .
Proof of Proposition 1: If and are conjugate in then the -modules and are isomorphic. So assume that is isomorphic to a direct summmand of a direct sum of copies of . Set and .
Let be conjugacy classes of primitive idempotents in as described in Lemma 5. So for and for we have an idempotent in and an idempotent in orthogonal to such that . Moreover the are representatives of the conjugacy classes of primitive idempotents in .
The indecomposable summands of are of the form by Lemma 1. So by our assumption that is a direct summand of a direct sum of copies of there exist non-negative integers such that
Here all summands on the right hand side are indecomposable and different indices correspond to non-isomorphic modules by Lemma 2.
Let be representatives of the conjugacy classes of primitive idempotents in and let be the multiplicity of in for . So
Here also the summands on the right hand side are indecomposable and different indices correspond to non-isomorphic modules.
for every . By Lemma 2 an isomorphism is given by right multiplication with a unit in and . Since is an isomorphism of -modules we get
Let be the multiplicity of in for . Since the lie in the -module is a direct summand of by Lemma 1 and from we obtain
Note that the summands here are not necessarily indecomposable, since the might be not primitive in . So for we get
Since for any the idempotents and are orthogonal we have