A cancellation theorem for modules over integral group rings
aa r X i v : . [ m a t h . K T ] J u l A CANCELLATION THEOREM FOR MODULES OVERINTEGRAL GROUP RINGS
J. K. NICHOLSON
Abstract.
A long standing problem, which has it roots in low-dimensionalhomotopy theory, is to classify all finite groups G for which Z [ G ] has stablyfree cancellation (SFC). We extend results of R. G. Swan by giving a conditionfor SFC and use this to show that Z [ G ] has SFC provided at most one copyof the quaternions H occurs in the Wedderburn decomposition of R [ G ]. Thisgeneralises the Eichler condition in the case of integral group rings. Introduction
A ring R is said to have stably free cancellation (SFC) if P ⊕ R n ∼ = R n + m implies P ∼ = R m for finitely generated projective modules P or, equivalently, if R has nonon-trivial stably free modules. In this paper, we will be interested in the problemof determining which finite groups G have the property that Z [ G ] has SFC.Fix a finite group G once and for all, and recall that the real group ring R [ G ] issemisimple and so admits a Wedderburn decomposition R [ G ] ∼ = M n ( D ) × · · · × M n r ( D r )where n , · · · , n r are integers ≥ D i is one of the real division algebras R , C or H . We say that Z [ G ] satisfies the Eichler condition if D i = H whenever n i = 1. By the Jacobinski cancellation theorem [12], this is a sufficient conditionfor Z [ G ] to have SFC.It is well-known, and is discussed in §
1, that Z [ G ] fails the Eichler conditionprecisely when G has a quotient which is a binary polyhedral group, i.e. a non-cyclic finite subgroup of H × . They are the generalised quaternion groups Q n for n ≥ e T , e O, e I , the binary tetrahedral, octahedral and icosahedral groups.It was shown by Swan [13] that, if G is a binary polyhedral group, then Z [ G ] hasSFC if and only if G is one of the seven groups(*) Q , Q , Q , Q , e T , e O, e I. Building on work of A. Fr¨ohlich [4], we show in § H = G/N and Z [ G ] hasSFC, then Z [ H ] has SFC also. In particular, Z [ G ] fails to have SFC whenever G hasa quotient which is Q n for n ≥
6. Note that this does not yet characterise whichgroups have SFC; it remains to determine SFC for Z [ G ] when G has a quotient inone of the groups in ( ∗ ) but none in Q n for n ≥ G is one of the groups in ( ∗ ) and H satisfies the Eichler condition, then Z [ G × H ] Mathematics Subject Classification.
Primary 20C05; Secondary 19B28, 19A13.
Key words and phrases.
Stably free modules, binary polyhedral groups, cancellation. has SFC provided H has no normal subgroup of index 2. He also proved that Z [ e T n × e I m ] has SFC for all n, m ≥ Z [ Q × C ] fails to have SFC.The main aim of this paper is to extend these results to a large class of newgroups, including many which do not split as direct products. Theorem A.
Let N be a normal subgroup of G which is contained in all normalsubgroups N ′ for which G/N ′ is binary polyhedral. If H = G/N , and the map Z [ H ] × → K ( Z [ H ]) is surjective, then Z [ G ] has SFC if and only if Z [ H ] has SFC. This gives a relation between SFC and the classical problem of determining when K ( Z [ H ]) is represented by units [5]. By results of B. Magurn, R. Oliver and L.Vaserstein (Theorem 7.16 and Corollary 7.17 in [7]) this is true when H is one ofthe groups in ( ∗ ) . Theorem B.
Let N be a normal subgroup of G which is contained in all normalsubgroups N ′ for which G/N ′ is binary polyhedral. If H = G/N is one of the groupsin ( ∗ ) , then Z [ G ] has SFC. This recovers Swan’s result for direct products and also proves SFC for infinitelymany new groups such as C ⋊ C , the double cover of Q . In §
2, we prove thefollowing generalisation of the Eichler condition to integral group rings. Define the H -multiplicity m H ( G ) of a finite group G to be the number of copies of H in theWedderburn decomposition of R [ G ]. Theorem C. If m H ( G ) ≤ , then Z [ G ] has SFC. In §
4, we conclude by showing how close these results bring us to a full classifi-cation of the finite groups G for which Z [ G ] has SFC. In particular, we formulatea precise question regarding the complexity that the final classification might take. Remark . Following Swan [13], one can also consider cancellation in the stableclasses of other projectives, i.e. when P ⊕ Z [ G ] ∼ = Q ⊕ Z [ G ] implies P ∼ = Q forprojectives P and Q . In this case, unit representation in K ( Z [ G ]) is replaced bysubtle questions about the maps Aut( P ) → K ( Z [ G ]) for rank 1 projectives P .This involves extensive computations of Aut( P ) for the various modules that ariseand will be left to a later article.The question of determining the finite groups G for which Z [ G ] has SFC orig-inated with Swan [13], [14] in the 1980s and has since found applications in low-dimensional homotopy theory in the work of F. R. Beyl, M. P. Latiolais and N.Waller [1], and F. E. A. Johnson [6]. More recently, Beyl and Waller showed how,starting with an explicit description for a non-trivial stably free module over Z [ Q ],a cohomolically 2-dimensional finite 3-complex X (3) with π ( X (3) ) = Q can beconstructed which, they propose, may not be homotopic to a finite 2-complex, i.e. X (3) is a potential counterexample to the D(2)-problem [6] of C. T. C Wall.In this context, our results place large restrictions of the possible fundamentalgroups of “exotic” 3-complexes X (3) which could be constructed in this way. Foran excellent recent reference for this material, the reader is encouraged to readChapter 1 of [8]. This is not the case for all binary polyhedral groups; for example, this fails for Q . CANCELLATION THEOREM FOR MODULES OVER INTEGRAL GROUP RINGS 3 Preliminaries
Let R be a ring and let C ( R ), often written f K ( R ), denote its projective classgroup, i.e. the equivalence classes of finitely generated projective modules where P ∼ P if P ⊕ R n ∼ = P ⊕ R m for some n, m ≥ LF n ( R ) denote the set of isomorphism classes of locally-free modules of rank n . Note that this coincides with the rank n projective modules when R = Z [ G ] byTheorems 2.21 and 4.2 in [12], and extends naturally to rings R = Z [ G ] /I for anideal I in Z [ G ]. We will now assume that R is of this form. The map P P ⊕ R induces a sequence LF ( R ) LF ( R ) LF ( R ) · · · C ( R ) ∼ ∼ ∼ where all the maps are surjections by Serre’s Theorem and all but the first mapare isomorphisms by Bass’ Cancellation Theorem (see § LF ( R ) ։ C ( R ) is bijective precisely when R has cancellation in all projectiveclasses. Furthermore, R has SFC precisely when the fibre over 1 ∈ C ( R ) is trivial.We now make the following refinement of Fr¨ohlich’s result mentioned in theintroduction, relying on results from [13]. This gives one direction of Theorem A. Theorem 1.1.
Let H = G/N . If Z [ G ] has SFC, then Z [ H ] has SFC.Proof. Since the map Z [ G ] ։ Z [ H ] of Z -orders induces a surjection Q [ G ] ։ Q [ H ],we can apply Theorem A10 in [13]. In particular, this shows that LF ( Z [ G ]) LF ( Z [ H ]) C ( Z [ G ]) C ( Z [ H ])is a weak pullback in the sense that that compatible elements in the corners havenot-necessarily-unique lifts to LF ( Z [ G ]). It follows that the fibre in LF ( Z [ G ])over 1 ∈ C ( Z [ G ]) maps onto the fibre in LF ( Z [ H ]) over 1 ∈ C ( Z [ H ]). (cid:3) We now state a more general version of the Jacobinski cancellation theoremwhich we will need in the proof of Theorem A. Let A be a semisimple separable Q -algebra which is finite-dimensional over Q and let Λ be a Z -order in A , i.e. afinitely-generated subring of A such that Q · Λ = A . For example, if Λ = Z [ G ]and A = Q [ G ] for G a finite group. Since Λ R = Λ ⊗ R has a real Wedderburndecomposition, the Eichler condition generalises to Z -orders Λ in the natural way. Theorem 1.2 (Jacobinski, Theorem 9.3 [14]) . If Λ satisfies the Eichler condition,then Λ has SFC. Finally we include a proof of the following result mentioned in the introduction.This reduces the Eichler condition to a purely group-theoretic property.
Proposition 1.3. Z [ G ] fails the Eichler condition if and only if G has a quotientwhich is a binary polyhedral group.Proof. If G fails the Eichler condition, the Wedderburn decomposition gives a map G → H × , i.e. a quaternionic representation. Since G is an R -basis for G , the imagemust contain an R -basis for H . Since H is non-commutative, the image must benon-abelian and so a binary polyhedral group. Conversely, a quotient of G into abinary polyhedral group gives a representation G → H × which does not split over J. K. NICHOLSON R or C since the image is non-abelian. Hence the representation is irreducible andso represents a term in the Wedderburn decomposition. (cid:3) Quaternionic Respresentations
Recall that we defined the H -multiplicity m H ( G ) of a finite group G to be thenumber of copies of H in the Wedderburn decomposition of R [ G ], i.e. the numberof irreducible one-dimensional quaternionic representations. We can reinterpret thehypothesis of Theorem A in terms of the H -multiplicity. The following is similar toLemma 3.4 in [1]. Lemma 2.1.
Let N be a normal subgroup of G and let H = G/N . Then N iscontained in all normal subgroups N ′ for which G/N ′ is binary polyhedral if andonly if m H ( G ) = m H ( H ) .Proof. Firstly note that m H ( G ) ≥ m H ( H ) holds in general by lifting quaternionicrepresentations. By looking at the real Wedderburn decomposition, every one-dimensional quaternionic representation of G corresponds to a map ϕ : G → H × such that the image contains an R -basis for H . In particular, Im ϕ is a non-abelianfinite subgroup of H × and so is a binary polyhedral group.Hence quaternionic representations of G precisely correspond to lifts of repre-sentations from binary polyhedral groups. The result follows immediately. (cid:3) It is straightforward to compute [6] that m H ( Q n ) = ⌊ n/ ⌋ for n ≥
2, and m H ( e T ) = 1, m H ( e O ) = 2 and m H ( e I ) = 2 can be deduced from the character tablesfor these groups along with their Frobenius-Schur indicators. Remark . This gives a way to restate Swan’s determination of SFC over binarypolyhedral groups in an alternate form: if G is a binary polyhedral group, then Z [ G ] has SFC if and only if m H ( G ) ≤ H = Q , Q , e T are the only binary polyhedral groups with m H ( H ) = 1.By lifting quaternionic representations, every group with m H ( G ) = 1 must maponto one of these groups H and have m H ( G ) = m H ( H ). This shows that TheoremC follows from Theorem B.We conclude this section by making a few brief comments about this result.Firstly, this is the best possible result of this form in the sense that, for every n ≥
2, there are groups G ( n ) and H ( n ) with m H ( G ( n )) = m H ( H ( n )) = n forwhich Z [ G ( n )] has SFC and Z [ H ( n )] fails to have SFC. In particular, we can take G ( n ) = e T n for n ≥ H (2) = Q × C and H ( n ) = Q n for n ≥ Z [ G ] × ։ K ( Z [ G ]) for G = Q , Q , e T . To see this, consider SK ( Z [ G ]) = ker( K ( Z [ G ]) → K ( Q [ G ])) . Theorem 14.2 and Example 14.4 in [10] show that SK ( Z [ G ]) = 1 for G = Q , Q , e T .If Wh( G ) = K ( Z [ G ]) / ± G denotes the Whitehead group, then [10] we havetors(Wh( G )) = SK ( Z [ G ]) and rank(Wh( G )) = r R ( G ) − r Q ( G ) , where r F ( G ) is the number of irreducible F -representations of G . Looking at therational and real Wedderburn decompositions of these groups shows that r R ( G ) = r Q ( G ) in each case. Hence Wh( G ) = 1 for G = Q , Q , e T and we get the strongerresult that ± G ։ K ( Z [ G ]), i.e. that K ( Z [ G ]) is represented by the trivial units. CANCELLATION THEOREM FOR MODULES OVER INTEGRAL GROUP RINGS 5 Proof of Theorem A
Let N be a normal subgroup of G and let H = G/N . Suppose m H ( G ) = m H ( H )and that Z [ H ] has SFC. Since the other direction was proven in the precedingsection, it will suffice to prove that Z [ G ] has SFC subject to the conditions ofTheorem A. Consider the following pullback diagram for Z [ G ] induced by N . Z [ G ] Λ Z [ H ] ( Z /n Z )[ H ]where Λ = Z [ G ] b N , b N = P g ∈ N g and n = | N | . This is the standard pullback con-struction for the ring Z [ G ] and trivially intersecting ideals I = ker( Z [ G ] → Z [ H ]) = ∆( N ) · G and J = b N · Z [ G ] , where ∆( N ) = ker( Z [ N ] → Z ) is the augmentation ideal (see Example 42.3 in [3]). Proposition 3.1. Λ satisfies the Eichler condition.Proof. Since R is flat, we can apply R ⊗ − to the diagram to get another pullbackdiagram which induces R [ G ] ∼ = R [ H ] × Λ R . Hence m H ( G ) = m H ( H ) + m H (Λ R )which implies that m H (Λ R ) = 0 by the hypotheses on G and H . (cid:3) Now note that Λ is a Z -order in Λ Q , which is a semisimple separable Q -algebraof finite dimension over Q since Q [ G ] ∼ = Q [ H ] × Λ Q . Hence we can apply Theorem1.2 to get that Λ has SFC. In particular, the maps LF (Λ) → C (Λ) and LF ( Z [ H ]) → C ( Z [ H ])have trivial fibres over 1 ∈ C (Λ) and 1 ∈ C ( Z [ H ]) respectively.Consider the following diagram induced by the maps on projective modules. LF ( Z [ G ]) LF ( Z [ H ]) × LF (Λ) C ( Z [ G ]) C ( Z [ H ]) × C (Λ) ϕ ϕ Proving Z [ G ] has SFC now amounts to proving that the fibres of ϕ , ϕ over (1 , Z [ G ] above is a Milnorsquare [9] and so induces an exact sequence K ( Z [ H ]) × K (Λ) K (( Z /n Z )[ H ]) C ( Z [ G ]) C ( Z [ H ]) × C (Λ) ϕ which is part of the Mayer-Vietoris sequence for the diagram. Here K ( R ) = GL( R ) ab where GL( R ) = S n GL n ( R ) with respect to the inclusions GL n ( R ) ֒ → GL n +1 ( R ).This sequence extends ϕ and so the fibres of ϕ are in correspondence withIm( K (( Z /n Z )[ H ]) → C ( Z [ G ])) ∼ = K (( Z /n Z )[ H ]) K ( Z [ H ]) × K (Λ) . Furthermore, by Theorem 8.1 of [11], we know that P ⊗ ( Z /n Z )[ H ] ∼ = ( Z /n Z )[ H ] ∼ = Q ⊗ ( Z /n Z )[ H ] . J. K. NICHOLSON
Hence, by the more general construction of projectives modules over Milnor squares(see, for example, Proposition 4.1 in [14]) the fibre ϕ − (1 ,
1) is in correspondencewith the double coset Z [ H ] × \ ( Z /n Z )[ H ] × / Λ × where we have used that Aut ( Z /n Z )[ H ] (( Z /n Z )[ H ]) = ( Z /n Z )[ H ] × .One result which helps towards comparing these fibres is due to Swan [14]. Recallthe notation used earlier where A is a semisimple separable Q -algebra which is finite-dimensional over Q and Λ is a Z -order in A , i.e. a finitely-generated subring of A such that Q · Λ = A . Suppose that I is a two-sided ideal of finite index in Λ. Theorem 3.2 (Swan, Corollary 10.5 [14]) . Suppose Λ satisfies the Eichler condi-tion. Then, with respect to the quotient map Λ → Λ /I , we have Λ × E (Λ /I ) × andthe map (Λ /I ) × → K (Λ /I ) induces (Λ /I ) × Λ × ∼ = K (Λ /I ) K (Λ) . Note that the statement that (Λ /I ) × → K (Λ /I ) induces this isomorphism isnot stated explicitly in [14]. To see this, recall that the proof uses Swan’s adaptationof the Strong Approximation Theorem (Theorem 8.1) to comparecoker(Λ × → (Λ /I ) × )to an appropriate quotient of ν ( b Λ × ), where here ν denotes the reduced norm and b Λ = b Z ⊗ Z Λ is the adelic completion. This is then identified withcoker(GL n (Λ) → GL n (Λ /I ))by showing that ν ( b Λ × ) = ν (GL n ( b Λ)). Since ν is compatible with the inclusionsGL n ( b Λ) ֒ → GL n +1 ( b Λ), the map between cokernels is induced by the inclusion ofunits. The result then follows by letting n → ∞ .Applying this result to the case Λ = Z [ G ] b N implies Λ × E ( Z /n Z )[ H ] × and that( Z /n Z )[ H ] × Λ × ∼ = K (( Z /n Z )[ H ]) K (Λ) . Let A denote this common abelian group. Then the fibres Z [ H ] × \ ( Z /n Z )[ H ] × / Λ × and K (( Z /n Z )[ H ]) K ( Z [ H ]) × K (Λ)are in correspondence if and only if the maps Z [ H ] × → A and K ( Z [ H ]) → A havethe same images. Now consider the following commutative diagram. Z [ H ] × ( Z /n Z )[ H ] × ( Z /n Z )[ H ] × Λ × ∼ = AK ( Z [ H ]) K (( Z /n Z )[ H ]) K (( Z /n Z )[ H ]) K (Λ) ∼ = A ϕ If ψ : Z [ H ] × → A denotes the map along the top row and ψ : K ( Z [ H ]) → A denotes the map along the bottom row, then commutativity shows that ψ = ψ ◦ ϕ .Since ϕ is surjective, Im ψ = Im ψ and so Z [ G ] has SFC. This completes the proofof Theorem A. CANCELLATION THEOREM FOR MODULES OVER INTEGRAL GROUP RINGS 7 Groups with Stably Free Cancellation
We now return to the question of determining the finite groups G for which Z [ G ]has SFC. By Theorem 1.1, this is equivalent to determining the set Q of all finitegroups G for which Z [ G ] fails to have SFC but such that Z [ H ] has SFC for all non-trivial quotients H = G/N . The classification would then be that Z [ G ] has SFCif and only if G has no quotient in Q . Note that Q is infinite since it contain aninfinite subset of the groups Q n for n ≥
6. Apart from this, Q must also contain Q × C but it is not yet clear how many other groups Q contains. Question 4.1.
Does Q contain only finitely many groups not of the form Q n for n ≥ ? Fix a finite group G which has no quotient of the form Q n for n ≥
6, but issuch that Z [ G ] fails to have SFC. By Swan [13] and Theorem 1.1, G has a normalsubgroup N for which G/N is one of the groups in ( ∗ ). We may also assume G hasanother normal subgroup N ′ with N N ′ , N ′ N and G/N ′ a binary polyhedralgroup. If not, Z [ G ] would have SFC by Theorem B.Hence G maps onto G = G/ ( N ∩ N ′ ), which has disjoint quotients onto G/N and
G/N ′ in that the corresponding normal subgroups N/ ( N ∩ N ′ ) and N ′ / ( N ∩ N ′ )are disjoint. By the second isomorphism theorem, note that N/ ( N ∩ N ′ ) ∼ = ( N · N ′ ) /N ′ E G/N ′ and so G is an extension of a group G/N in ( ∗ ) by a normal subgroup of a groupin ( ∗ ), and similarly for the map G → G/N ′ . Since there are only seven groups in( ∗ ), there are only finitely many possible groups G which can arise.It is therefore of interest to determine the finite set Q of finite groups G whichadmit two disjoint quotients onto groups in ( ∗ ), and then to try to determine all G ∈ Q for which Z [ G ] has SFC. The simplest examples are the groups G × C for G one of the groups in ( ∗ ). This fails to have SFC when G = Q by the results ofSwan [13]. Remark . If every group of in Q failed to have SFC, then every element in Q not of the form Q n for n ≥ Q and so Question 4.1 would beanswered in the affirmative. However this is false: e T gives an example of a groupof this form which has SFC.To see a more interesting family of a group in Q , let ϕ : C → Aut( C ) beinduced by sending the generator of C to the map ( x, y ) ( x − , y − ). Then thesemidirect product C ⋊ ϕ C has six disjoint quotients onto Q ∼ = C ⋊ C induced by the six subgroups C ≤ C ,and can be shown to have no further quotients which are binary polyhedral groups.A full classification of this family of groups can be carried out using tables of groupextensions, the details of which we leave for a later article.More generally, one can consider the set Q n of finite groups G which admit n mutually disjoint quotients onto groups in ( ∗ ), i.e. for which there are normalsubgroups N , · · · , N n such that G/N i is a group in ( ∗ ) for all i , N i N j for all i = j and N ∩ · · · ∩ N n = { } . Analogously to the case of Q , groups in Q n musthave a normal series of length n whose factor groups are normal subgroups of groups It does not contain all the groups of the form Q n for n ≥ Q ։ Q . J. K. NICHOLSON in ( ∗ ). Hence Q n is finite and, for given n , can be determined computationally bysolving the extension problems which arise.For any G as considered above, we can choose a maximal collection of normalsubgroups N , · · · , N n such that G/N i is a group in ( ∗ ) for all i and N i N j forall i = j . Then G = G/ ( N ∩ · · · N n ) ∈ Q n and, since any quotient to a binarypolyhedral group must factor through the map G → G , we can make the followingobservation. Theorem 4.3.
If the map Z [ G ] × → K ( Z [ G ]) is surjective for every G ∈ S n ≥ Q n such that Z [ G ] has SFC, then Q ⊆ { G ∈ [ n ≥ Q n : Z [ G ] fails to have SFC } ∪ { Q n : n ≥ } . Remark . In particular, Question 4.1 would be answered in the affirmative ifthere are only finitely many G ∈ S n ≥ Q n for which Z [ G ] has SFC, and K ( Z [ G ])has unit representation for all such G . This seems unlikely to be true, though acounterexample is yet to be found.The above can be improved upon significantly by noting that, if n > m ≥ G ∈ Q n has a quotient H ∈ Q m . If Z [ H ] fails to have SFC, then weknow already that G
6∈ Q and so can neglect to consider it in the inclusion above.By applying techniques from [10], unit representation can be proven for manygroups in S n ≥ Q n . We will explore this, and other questions, in a later article. References [1] F. R. Beyl, M. P. Latiolais, N. Waller,
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