aa r X i v : . [ m a t h . G R ] O c t The spectrum of the Chern subring
David J. Green and Ian J. Leary
Dedicated to Charles Thomas, onthe occasion of his 60th birthday
Abstract.
For certain subrings of the mod- p -cohomology of a compact Lie group, wegive a description of the spectrum, analogous to Quillen’s description of the spectrumof the whole cohomology ring. Subrings to which our theorem applies include theChern subring. Corollaries include a characterization of those groups for which theChern subring is F-isomorphic to the cohomology ring.
1. Introduction.
Let G be a compact Lie group ( e . g ., a finite group) and let H ∗ ( G ) = H ∗ ( BG ; F p ) be itsmod- p cohomology ring. This ring is a finitely generated graded-commutative F p -algebra.In [16], D. Quillen studied this ring from the viewpoint of commutative algebra. His resultsmay be stated in terms of the prime ideal spectrum of H ∗ ( G ), but the cleanest statementconcerns the variety, X G ( k ), of algebra homomorphisms from H ∗ ( G ) to an algebraicallyclosed field k of characteristic p . The Chern subring, Ch( G ) ⊆ H ∗ ( G ), is the subringgenerated by Chern classes of unitary representations of G . We give a description of X ′ G ( k ), the variety of algebra homomorphisms from Ch( G ) to k , analogous to Quillen’sdescription of X G ( k ). As corollaries of this result, we classify the minimal prime ideals ofCh( G ), and characterize those groups G for which the natural map from X G ( k ) to X ′ G ( k )is a homeomorphism.In the case when G = E is an elementary abelian p -group, i . e ., a direct productof copies of the cyclic group of order p , X E ( k ) is naturally isomorphic to E ⊗ k , where E is viewed as a vector space over F p and the tensor product is taken over F p . Forgeneral G , Quillen describes X G ( k ) as the colimit of the functor ( − ) ⊗ k over a category A = A ( G ) with objects the elementary abelian subgroups of G , and morphisms thosegroup homomorphisms that are induced by conjugation in G . Our description of X ′ G ( k )is as the colimit of the same functor over a category A ′ . This category has the sameobjects as Quillen’s category, but a morphism in A ′ is a group homomorphism that merelypreserves conjugacy in G . In other words, a group homomorphism f : E → E satisfies: f ∈ A ⇔ ∃ g ∀ e, f ( e ) = g − eg,f ∈ A ′ ⇔ ∀ e ∃ g, f ( e ) = g − eg. This theorem is a corollary of a more general colimit theorem, which says, roughly, thatthe variety for any subring of H ∗ ( G ) that is both ‘large’ and ‘natural’ may be expressedas such a colimit. Other corollaries of this theorem, in the case when G is finite, includea description of the variety for the subring of Ch( G ) generated by Chern classes of rep-resentations realizable over any subfield of C , and a slight variation on the usual proof of1uillen’s theorem in which transfers of Chern classes are used instead of the Evens normmap. Acknowledgements.
This work was started while the first named author held a Max Kade visiting fellowshipat the University of Chicago, and the second named author held a Leibniz fellowshipat the Max Planck Institut f¨ur Mathematik, Bonn. The first named author gratefullyacknowledges the support of the Deutsche Forschungsgemeinschaft and the Max KadeFoundation. The second named author gratefully acknowledges the support of the LeibnizFellowship Programme.The authors thank Dave Benson, Jon Carlson, Hans-Werner Henn, David Kirby andJeremy Rickard for their helpful comments on this work.
2. Representations and the Chern subring.
First we recall some facts concerning Chern classes [3,21]. Let U ( n ) be the group of n × n unitary matrices, and T ( n ) its subgroup of diagonal matrices. Then H ∗ ( T ( n )) is a freepolynomial algebra F p [ x , . . . , x n ] on n generators of degree two. H ∗ ( U ( n )) is isomorphicto F p [ c , . . . , c n ], where c i has degree 2 i . The map from H ∗ ( U ( n )) to H ∗ ( T ( n )) is injectiveand sends c i to the i th elementary symmetric function in the x i .Let G be a compact Lie group. G has faithful finite-dimensional complex representa-tions, and any finite-dimensional representation is equivalent to a unitary representation.If ρ : G → U ( n ) is a unitary representation of G , write ρ : BG → BU ( n ) for the inducedmap of classifying spaces, whose homotopy class depends only on the equivalence classof ρ . The i th Chern class of ρ is defined by c i ( ρ ) = ρ ∗ ( c i ) ∈ H i ( G ). Define c · ( ρ ), the totalChern class of ρ , to be 1 + c ( ρ ) + · · · + c n ( ρ ). Chern classes enjoy the following properties(‘Whitney sum formula’ and ‘naturality’), for any θ : G → U ( m ) and any f : H → G : c · ( ρ ⊕ θ ) = c · ( ρ ) c · ( θ ) c · ( ρ ◦ f ) = f ∗ c · ( ρ ) . There is a unique way to define Chern classes for virtual representations so that theycontinue to enjoy the above properties. Let c ′· = 1 + c ′ + c ′ + · · · be the unique powerseries in F p [[ c , . . . , c n ]] satisfying c ′· c · = 1, and define c i ( − ρ ) = ρ ∗ ( c ′ i ). In general infinitelymany of the c i ( − ρ ) will be non-zero, but note that they are all expressible in terms of the c i ( ρ ). Definition.
The Chern subring Ch( G ) of H ∗ ( G ) is the subring generated by the c i ( ρ )for all i and all virtual representations ρ .By the above remarks, the c i ( ρ ) as ρ ranges over the irreducible representations of G suffice to generate Ch( G ). In the case when G is finite, it follows that Ch( G ) is finitelygenerated, since G has only finitely many inequivalent irreducible representations. Forgeneral G it is also true that Ch( G ) is finitely generated. This is a special case of thefollowing proposition. 2 roposition 2.1. Let G be a compact Lie group, and ρ : G → U ( n ) a faithful unitaryrepresentation of G . If R is a subring of H ∗ ( G ) containing each c i ( ρ ) , then R is finitelygenerated. Proof.
Venkov showed that H ∗ ( G ) is finitely generated by showing that H ∗ ( G ) is finiteover ( i . e ., is a finitely generated module for) H ∗ ( U ( n )) [22,16,5]. Now R is an H ∗ ( U ( n ))-submodule of H ∗ ( G ), so is finitely generated since H ∗ ( U ( n )) is Noetherian. Remark.
For G finite, the finite-generation of H ∗ ( G ) is due independently to Evens andto Venkov by completely different proofs [9,22]. There is another proof (closer to Evens’than to Venkov’s) in [8]. Definition.
A virtual representation ρ of G is said to be p -regular if the virtual dimensionof ρ is strictly positive and for every elementary abelian subgroup E ∼ = ( Z /p ) n of G , therestriction to E of ρ is a direct sum of copies of the regular representation of E . Proposition 2.2.
For each prime p , every compact Lie group G has a p -regular represen-tation. Proof. G has a faithful representation in U ( n ) for some n , and every elementary abeliansubgroup of U ( n ) is conjugate to a subgroup of T ( n ), the torus consisting of diagonalmatrices. Thus it suffices to show that U ( n ) has a virtual representation whose restrictionto ( Z /p ) n ⊆ T ( n ) is the regular representation.Recall that the representation ring R ( T ( n )) of T ( n ) is isomorphic to the Laurentpolynomial ring Z [ τ , τ − , . . . , τ n , τ − n ], where τ i is the 1-dimensional representation τ i : diag( ξ , . . . , ξ n ) ξ i .R ( U ( n )) maps injectively to R ( T ( n )) with image the subring Z [ σ , σ , . . . , σ n , σ − n ], where σ i is the i th elementary symmetric function in the τ j .The polynomial P = n Y i =1 (1 + τ i + · · · + τ p − i )is a symmetric polynomial in the τ j , and so is expressible in terms of σ , . . . , σ n . Thecorresponding ( p n -dimensional) representation of U ( n ) restricts to ( Z /p ) n ⊆ T ( n ) as theregular representation. Remark.
For G finite, the regular representation of G is of course p -regular.Using Quillen’s result that we state as Theorem 5.1, it may be shown that H ∗ ( G )is finite over the subring generated by the Chern classes of any p -regular representation.For genuine (as opposed to virtual) representations, this can be deduced from Venkov’sresult: If ρ : G → U ( n ) is a p -regular representation of G , the kernel of ρ contains noelements of order p , and is therefore a finite group of order coprime to p . It follows that H ∗ ( ρ ( G )) ∼ = H ∗ ( G ) (consider the spectral sequence for the extension ker( ρ ) → G → ρ ( G )),and hence H ∗ ( G ) is finite over the image of ρ ∗ . When p = 2, the representation constructedin the proof of Proposition 2.2 is a genuine representation. David Kirby has shown us anargument to prove that U ( n ) has a p -regular genuine representation if and only if either p = 2, or n = 1, or ( p, n ) = (3 , . Varieties for cohomology. Let k be an algebraically closed field of characteristic p , and let R be a finitely generatedcommutative F p -algebra. Define V R ( k ), the variety for R , to be the set of ring homomor-phisms from R to k , with the Zariski topology, i . e ., the smallest topology in which theset F I = { φ : R → k | ker( φ ) ⊇ I } is closed for each ideal I of R . A ring homomorphism f : R → S gives rise to a continuousmap f ∗ : V S ( k ) → V R ( k ). If S is finite over f ( R ) ( i . e ., S is a finitely generated f ( R )-module) then f ∗ is a closed mapping and has finite fibres, by the ‘going up’ or ‘lying over’theorem [4,6]. If S is finite over f ( R ) and ker( f ) is nilpotent, then f ∗ is surjective.There is a continuous map from V R ( k ) to Spec( R ), the prime ideal spectrum of R ,that sends the map φ : R → k to the ideal ker( φ ). If the transcendence degree of k over F p is sufficiently large (as large as a generating set for R will suffice), then this map issurjective. Thus information about V R ( k ) gives rise to information about Spec( R ).We shall also require the following: Proposition 3.1. (a) Let S be a subring of R containing R p , the subring of p th powersof elements of R . Then the natural map from V R ( k ) to V S ( k ) is a homeomorphism.(b) Let a finite group G act on R , with fixed point subring S = R G . Then the naturalmap V R ( k ) → V S ( k ) induces a homeomorphism V R ( k ) /G → V S ( k ) . Proof.
Each of these claims may be proved by showing that R is finite over S , anddeducing that the map given is continuous, closed, and a bijection. See for example[4,6,16].For p = 2, the ring H ∗ ( G ) is commutative. For p an odd prime, elements of H ∗ ( G ) ofodd degree are nilpotent, so although H ∗ ( G ) is not commutative, the quotient of H ∗ ( G )by its radical, h ∗ ( G ) = H ∗ ( G ) / √
0, is commutative. Any homomorphism from H ∗ ( G ) to k factors through h ∗ ( G ). Define X G ( k ) to be the variety V R ( k ) for R = h ∗ ( G ). By theabove remark, points of X G ( k ) may be viewed as homomorphisms from H ∗ ( G ) to k . Let S be the subring of elements of H ∗ ( G ) of even degree. A homomorphism φ : S → k extendsuniquely to a homomorphism from H ∗ ( G ) to k (if x is in odd degree, then either p is odd,and x = 0, or p = char( k ) = 2, so in either case φ ( x ) is the unique square root of φ ( x )).It follows that the natural map X G ( k ) → V S ( k ) is a homeomorphism, since it is a closed,continuous bijection. Hence X G ( k ) could equally be defined in terms of S .Note that a group homomorphism f : H → G induces a map f ∗ : X H ( k ) → X G ( k ). Wewrite ι GH for f ∗ in the case when f is the inclusion of a subgroup H in G . A theoremof Evens and Venkov [9,22] states that in this case H ∗ ( H ) is finite over H ∗ ( G ). (Todeduce this from the result quoted in the proof of Proposition 2.1, note that a faithfulrepresentation of G restricts to a faithful representation of H .) It follows that ι GH is closedand has finite fibres.Define X ′ G ( k ) to be V Ch( G ) ( k ). By Proposition 2.1 the natural map from X G ( k ) to X ′ G ( k ) is surjective, closed, and has finite fibres. Proposition 3.2.
Let ρ be a representation of G , and let R ( n ) be the subring of H ∗ ( G ) generated by the Chern classes of nρ = ρ ⊕ n = ρ ⊕· · ·⊕ ρ . Then the natural map V R (1) ( k ) → V R ( n ) ( k ) is a homeomorphism. roof. If p does not divide n , then c i ( nρ ) = nc i ( ρ ) + P ( i, n ) , for some expression P ( i, n ) in the c j ( ρ ) for j < i . So in this case R ( n ) = R (1). On theother hand, if n = pm then c · ( nρ ) = c · ( mρ ) p = 1 + c ( mρ ) p + c ( mρ ) p + · · · , so in this case R ( n ) = R ( m ) p , the subring of p th powers of elements of R ( m ), and the map V R ( k ) → V R p ( k ) is a homeomorphism.The methods that we shall use to study the Chern subring apply equally to theStiefel-Whitney subring, defined analogously, in the case when p = 2. (For informationconcerning Stiefel-Whitney classes see [21]). As an alternative, the following propositionmay be applied. Proposition 3.3.
Let p = 2 and let S be the subring of H ∗ ( G ) generated by Stiefel-Whitney classes of real representations of G . Then S ⊆ Ch( G ) ⊆ S, and the natural map from V S ( k ) to X ′ G ( k ) is a homeomorphism. Proof. If θ is an n -dimensional real representation of G , then θ C , the complexificationof θ , is an n -dimensional complex representation of G with c i ( θ C ) = w i ( θ ). Conversely,if ψ is an n -dimensional complex representation of G and ψ R is the same representationviewed as a 2 n -dimensional real representation, then w i ( ψ R ) = 0 for i odd and w i ( ψ R ) = c i ( ψ ). This proves the claimed inclusions. The claimed homeomorphism follows fromProposition 3.1(a).
4. Examples.
In this section we discuss the case of an elementary abelian p -group, and also give anexample to show that the map X G ( k ) → X ′ G ( k ) is not always a homeomorphism. Thisexample was the starting point for the work of this paper.Let E be an elementary abelian p -group of rank n , E ∼ = ( Z /p ) n . Then E may beviewed as a vector space over F p . Write E ∗ for Hom( E, F p ). There is a natural isomorphism E ∗ ∼ = H ( E ). For p = 2, H ∗ ( E ) is isomorphic to the symmetric algebra on H ( E ), orequivalently, the ring of polynomial functions on E viewed as a vector space: H ∗ ( E ) ∼ = S ( E ∗ ) ∼ = F p [ E ] . For p >
2, the Bockstein β : H ( E ) → H ( E ) is injective, and H ∗ ( E ) is isomorphic to thetensor product of the exterior algebra on H ( E ) tensored with the symmetric algebra on B = β ( H ( E )): H ∗ ( E ) ∼ = Λ( E ∗ ) ⊗ S ( B ) ∼ = Λ( E ∗ ) ⊗ F p [ E ] .
5n any case, h ∗ ( E ) is naturally isomorphic to F p [ E ], generated in degree one for p = 2 andin degree two for odd p . It follows that X E ( k ) is naturally isomorphic to E ⊗ k , where E is viewed as a vector space over F p and the tensor product is over F p , so that E ⊗ k ∼ = k n .Irreducible representations of E are 1-dimensional, and the map ρ c ( ρ ) is a naturalbijection between the set of irreducible representations of E and B = β ( H ( E )). (When p = 2, β = Sq , and so β ( x ) = x .) The Chern subring Ch( E ) of H ∗ ( E ) is the subalgebraof H ∗ ( E ) generated by B . For p > h ∗ ( E ), and for p = 2 itmaps onto h ∗ ( E ) , the subring of squares of elements of h ∗ ( E ). In any case, the map from X E ( k ) to X ′ E ( k ) is a homeomorphism. Proposition 4.1.
Let ρ be a direct sum of copies of the regular representation of E , andlet R be the subring of H ∗ ( E ) generated by the Chern classes of ρ . Then the natural mapfrom X E ( k ) to V R ( k ) factors through a homeomorphism k n / GL n ( F p ) ∼ = X E ( k ) / GL( E ) → V R ( k ) . Proof.
By Proposition 3.2 it suffices to consider the case when ρ is the regular represen-tation. Identify Ch( E ) with F p [ E ], generated in degree two. The total Chern class of ρ is c · ( ρ ) = Y x ∈ E ∗ (1 + x ) . This is invariant under the full automorphism group, GL( E ), of E . By a theorem ofDickson, the only i > c i ( ρ ) is non-zero are i = p n − p j , where 0 ≤ j < n =dim F p ( E ). Moreover, these c i ( ρ ) freely generate a polynomial subring of F p [ E ], and thisis the complete ring of GL( E )-invariants in F p [ E ] [6,23]. The claim follows by part (b) ofProposition 3.1. Remark.
Let A be a non-identity element of GL n ( F p ), and let v be an element of k n fixed by A . Then v is in the kernel of I − A , a non-zero matrix with entries in F p , and so v lies in a proper subspace of k n defined over F p ( i . e ., a subspace of the form V ⊗ k forsome proper F p -subspace V of F np ). It follows that GL n ( F p ) acts freely on the complementof all such subspaces. For an elementary abelian group E , let X + E ( k ) = X E ( k ) \ [ F Let q = p n for some n ≥ 2. Let G be the affine transformation group of theline over F q . Then G is expressible as an extension with kernel E = ( F q , +), an elementaryabelian p -group of rank n , and quotient Q = GL ( F q ), cyclic of order q − 1. The conjugationaction of Q on E is transitive on non-identity elements of E . One example of such a groupis the alternating group A ( p = 2, n = 2). An easy transfer argument shows that H ∗ ( G )maps isomorphically to the ring of invariants, H ∗ ( E ) Q , and it follows from Proposition 3.1that X G ( k ) is homeomorphic to X E ( k ) /Q = k n /Q , where Q = GL ( F q ) ≤ GL n ( F p ) ≤ GL n ( k ). 6t is easy to check that G has exactly q distinct irreducible representations. All butone of these are 1-dimensional and restrict to E as the trivial representation. The otherone is ( q − E as the regular representation minus the trivialrepresentation. Hence by Proposition 4.1, X ′ G ( k ) is homeomorphic to X E ( k ) / GL( E ) = k n / GL n ( F p ). Thus the map from X G ( k ) to X ′ G ( k ) is not a homeomorphism. 5. Quillen’s colimit theorem. In [16], Quillen showed that for general G , X G ( k ) is determined by the elementary abeliansubgroups of G . Roughly speaking, he showed that X G ( k ) is equal to the union of theimages of the X E ( k ), where E ranges over the elementary abelian subgroups of G , and thatas little identification takes place between the points of the X E ( k ) as is consistent with thefact that inner automorphisms of G act trivially on H ∗ ( G ). More precisely, let f : E → E be a homomorphism between elementary abelian subgroups of G that is induced by aninner automorphism of G . Then the following diagram commutes. H ∗ ( G ) Id ←− H ∗ ( G ) y Res y Res H ∗ ( E ) f ∗ ←− H ∗ ( E )Consequently the following diagram commutes. X G ( k ) Id −→ X G ( k ) x ι x ι X E ( k ) f ∗ −→ X E ( k )This fact motivates the following definition. Definition. The Quillen category A for a compact Lie group G and a prime p is thecategory whose objects are the elementary abelian p -subgroups of G , with morphismsfrom E to E being those group homomorphisms that are induced by conjugation in G .Any such group homomorphism is of course injective.In general G will have infinitely many elementary abelian p -subgroups. These sub-groups form finitely many conjugacy classes though ([16], lemma 6.3). Thus although theQuillen category for G is infinite in general, it contains only finitely many isomorphismtypes of object (or is ‘skeletally finite’).The morphisms f : E → E in the Quillen category are precisely the maps for whichthe diagram above commutes. It follows that the natural map a E ≤ GE el . ab . X E ( k ) −→ X G ( k )factors through a map α : colim A X E ( k ) → X G ( k ).7 heorem 5.1. (Quillen [16]) The map α : colim A X E ( k ) → X G ( k ) is a homeomorphism. The map α is continuous and is closed because A is skeletally finite. Thus the maincontent of the theorem is that α is a bijection. We shall use only half of this theorem, thestatement that α is surjective, in our main theorem. The surjectivity of α is equivalent tothe statement ‘an element of H ∗ ( G ) is nilpotent if and only if its image in each H ∗ ( E ) isnilpotent’. 6. A new colimit theorem. Motivated by the Quillen category, we define: Definition. A category of elementary abelian subgroups of G is a category whose objectsare (all of) the elementary abelian p -subgroups of G , and whose morphisms from E to E are injective group homomorphisms.The Quillen category, A ( G ), is of course a category of elementary abelian subgroupsof G . Another example is the category C reg ( G ), with the morphism set C reg ( E , E ) equalto the set of all 1-1 group homomorphisms from E to E . Any category of elementaryabelian subgroups of G is a subcategory of C reg ( G ).Any subring R of H ∗ ( G ) gives rise to a category C ( R ) of elementary abelian subgroupsof G , where f : E → E is a morphism in C ( R ) if and only if R Id ←− R y Res y Res h ∗ ( E ) f ∗ ←− h ∗ ( E )commutes. Note that we use h ∗ ( E i ), the cohomology ring modulo its radical, rather than H ∗ ( E i ). Note that if f is an isomorphism of groups and is a morphism in C ( R ), f − is also in C ( R ). Each C ( R ) contains the Quillen category, and hence is skeletally finite.By the argument given in Section 5, the map α : colim A X E ( k ) → X G ( k ) induces a map γ = γ ( R ): colim C ( R ) X E ( k ) → V R ( k ). Definition. Say that a subring of H ∗ ( G ) is large if it contains the Chern classes of some p -regular representation of G . Say that a subring of H ∗ ( G ) is natural if it is generated byhomogeneous elements and is closed under the action of the Steenrod algebra.The new colimit theorem of the title of this section is: Theorem 6.1. Let G be a compact Lie group, and let R be a subring of H ∗ ( G ) that isboth large and natural. Then the map γ : colim C ( R ) X E ( k ) → V R ( k ) is a homeomorphism. It is possible that this theorem could be proved using more general colimit theoremsdue to S. P. Lam, to D. Rector, and to H.-W. Henn, J. Lannes and L. Schwartz [14,18,12].These theorems say, roughly speaking, that the variety for any Noetherian algebra over8he Steenrod algebra should be expressible as a similar sort of colimit. Even with thesetheorems, Quillen’s description of X G ( k ) would still be needed to identify the categoriesthat arise with categories of elementary abelian subgroups of G . The proof given below ismore elementary, in that it relies on no work that is more recent than that of Quillen.The proof of the theorem uses the following lemma. Lemma 6.2. Let S be the subring of H ∗ ( G ) generated by the Chern classes of some p -regular representation of G . Then C ( S ) is equal to the category C reg defined above, andthe map γ ( S ) is a homeomorphism. Proof. Let F be a maximal elementary abelian subgroup of G . Note that the natural mapfrom X F ( k ) (mapping the category with one object and one morphism to C reg ) induces ahomeomorphism X F ( k ) / GL( F ) ∼ = colim C reg X E ( k ) . By Proposition 4.1, the image of ι GF : X F ( k ) → V S ( k ) is homeomorphic to X F ( k ) / GL( F ).If E is any elementary abelian subgroup of G and f : E ֒ → F is any injective group homo-morphism, then Res GE ( ρ ) and f ∗ Res GF ( ρ ) are equal to a sum of (the same number of) copiesof the regular representation of E . Hence C ( S ) is equal to C reg , and Im( γ ) = Im( ι GF ). Itfollows that γ is a homeomorphism onto its image. Finally, by Theorem 5.1, this image isthe whole of V S ( k ). Proof of the theorem. Since R is large, it contains a subring S satisfying the conditionsof Lemma 6.2. Let E , F be two elementary abelian subgroups of G , suppose that therank of E is less than or equal to that of F , and suppose that φ ∈ X E ( k ) and ψ ∈ X F ( k )define the same point of V R ( k ). A fortiori φ and ψ define the same point of V S ( k ),and so by Lemma 6.2 there is an injective group homomorphism f : E ֒ → F such that ψ = φ ◦ f ∗ = f ∗ ( φ ). It suffices to show that such an f is in C ( R ).For any such f : E ֒ → F , let S be the set of subgroups of E such that f restricted to E is a morphism in C ( R ): S = { E ′ ≤ E | ( f | E ′ : E ′ → F ) ∈ C ( R ) } , and define a subset X ( f ) of X E ( k ) by X ( f ) = { φ ∈ X E ( k ) | f ∗ ( φ ) ◦ Res GE ′ | R = ψ ◦ Res GF | R } . From the definitions, X ( f ) ⊇ [ E ′ ∈S ι EE ′ X E ′ ( k ) , and it suffices to show that equality holds. Note that a subgroup E ′ ≤ E is in S if andonly if ιX E ′ ( k ) is a subset of X ( f ). Hence it suffices to show that X ( f ) is equal to someunion of sets of the form ιX E ′ ( k ). Now let I ( f ) be the ideal of H ∗ ( E ) generated by allelements of the form Res GE ( r ) − f ∗ Res GF ( r ), where r ∈ R . The subvariety of X E ( k ) definedby I ( f ) is the set X ( f ) defined above. Since R is natural (in the sense defined above thestatement), the ideal I ( f ) is homogeneous and closed under the action of the Steenrodalgebra. But by a theorem of Serre [19,16], the variety corresponding to any such ideal of H ∗ ( E ) has the required form.Minimal prime ideals of a commutative F p -algebra R correspond to irreducible com-ponents of V R ( k ). Hence one obtains: 9 orollary 6.3. Let R be a large, natural subring of H ∗ ( G ) . The minimal prime ideals of R are in bijective correspondence with the isomorphism types of maximal objects in C ( R ) . An object of a category is called maximal if every map from it is an isomorphism.An isomorphism class of maximal objects in the Quillen category is a conjugacy class ofmaximal elementary abelian subgroups of G . Corollary 6.4. Let R and S be large natural subrings of H ∗ ( G ) , and suppose that R isa subring of S . The natural map V S ( k ) → V R ( k ) is a homeomorphism if and only if thecategories C ( R ) and C ( S ) are equal. Proof. A direct proof could be given at this stage, but it is easier to apply Proposition 9.1,which implies that no subcategory of C reg that strictly contains C ( S ) gives rise to the samevariety. 7. Applications to rings of Chern classes. We start by defining some categories of elementary abelian subgroups of G . Definition. Define categories of elementary abelian subgroups of G , A ′ , A ′ R , A ′ P , andfor each d dividing p − A d , by stipulating that an injective homomorphism f : E → F isin: A ′ if ∀ e , f ( e ) is conjugate (in G ) to e ; A ′ R if ∀ e , f ( e ) is conjugate to e or to e − ; A ′ P if ∀ e , the subgroups h e i and h f ( e ) i are conjugate; A ′ d if ∀ e , f ( e ) is conjugate to ξ ( e ) for some ξ in the order d subgroup of Aut( h e i ).Note that for p = 2, A ′ = A ′ R = A ′ P , and for odd p , A ′ R = A ′ , and A ′ P = A ′ p − . Notealso that the difference between A ′ and the Quillen category A is the difference between‘ ∀ e ∃ gf ( e ) = g − eg ’ and ‘ ∃ g ∀ ef ( e ) = g − eg ’. The reason for introducing these categoriesis the following proposition. Proposition 7.1. Let K be a subfield of C and let | K ( ζ p ): K | = d , where ζ p is a primitive p th root of 1. Let G be a compact Lie group, and in cases (c) and (d) suppose that G isfinite. Let R be the subring of H ∗ ( G ) generated by Chern classes of:(a) All representations of G ;(b) Representations of G realisable over the reals;(c) Permutation representations of G ;(d) Representations of G realisable over K .In each case, the variety V R ( k ) is homeomorphic to colim C ( R ) X E ( k ) . The category C ( R ) is: (a) A ′ , (b) A ′ R , (c) A ′ P , (d) A ′ d . Proof. In each case, the morphisms in the category given are precisely those group ho-momorphisms for which χ ( e ) = χ ( f ( e )) for all characters χ coming from representationsof the given type. (See [20] Chapter 12 for case (d), and for case (c) note that if e , e ′ areelements of G of order p , then the permutation action of e on G/ h e ′ i has a fixed pointif and only if h e i is conjugate to h e ′ i .) The proposition therefore follows from the lemmabelow. 10 emma 7.2. Let A be an additive subgroup of the representation ring of G , generated bygenuine representations, and containing a p -regular representation. Let R be the subringof H ∗ ( G ) generated by the Chern classes of all elements of A . Then R is large and natural,and hence by Theorem 6.1 γ : colim C ( R ) X E ( k ) → V R ( k ) is a homeomorphism. Furthermore, f : E ֒ → F is a morphism in C ( R ) if and only if for all e ∈ E , and all characters χ of elements of A , χ ( e ) = χ ( f ( e )) . Proof. First, suppose that A is generated by a single representation ρ . The image of ρ ∗ : H ∗ ( U ( n )) → H ∗ ( G ) is natural since H ∗ ( U ( n )) is graded and acted upon by the Steen-rod algebra. The general case follows from the Cartan formula. By hypothesis, R is large.The claimed homeomorphism follows from Theorem 6.1, and it only remains to describe C ( R ).A representation is determined up to equivalence by its character. Hence for any f as in the statement and ρ a generator of A , f ∗ Res GF c · ( ρ ) = Res GE c · ( ρ ), and so any such f is in C ( R ). For the converse, note that since Ch( E ) is a unique factorization domain, arepresentation of E is determined up to equivalence by its dimension and its total Chernclass. Thus if f : E ֒ → F is a homomorphism for which there exists e and χ with χ ( f ( e )) = χ ( e ), there exists i and ρ , a generator of A , such that Res GE ( c i ( ρ )) − f ∗ Res GF ( c i ( ρ )) = 0.Hence f is not in C ( R ).Quillen’s description of X G ( k ) (Theorem 5.1, and Theorem 8.1), Corollary 6.4 andProposition 7.1 together yield: Corollary 7.3. The natural map X G ( k ) → X ′ G ( k ) is a homeomorphism if and only if thecategories A ( G ) and A ′ ( G ) are equal. Example. (A p -group G for which the map X G ( k ) → X ′ G ( k ) is not a homeomorphism.)Let E be the additive group F np for some n > 2, and let U ≤ GL( E ) = GL n ( F p ) be theSylow p -subgroup of GL( E ) consisting of upper triangular matrices. Let Q be the subgroupof U consisting of all matrices ( a i,j ) that are constant along diagonals, i . e ., a i,j = a i +1 ,j +1 whenever 1 ≤ i < m and 1 ≤ j < m . Finally, let G be the split extension with kernel E and quotient Q . The group E is a maximal elementary abelian subgroup of G . Easymatrix calculations show that the orbits of the action of Q on elements of E are equal tothe orbits of the action of U , and that any element of GL( E ) that preserves the U -orbitsin E is fact an element of U . It follows that the image of X E ( k ) in X ′ G ( k ) is X E ( k ) /U ,whereas the image of X E ( k ) in X G ( k ) is of course X E ( k ) /Q . Thus G is a p -group such thatthe fibres of the map X G ( k ) → X ′ G ( k ) above general points of one irreducible componenthave order | U : Q | = p ( n − n − / . Example. (A group G for which X ′ G ( k ) has fewer irreducible components than X G ( k ).)Let G be GL ( F p ). There are two distinct Jordan forms for elements of order p in G (resp.one if p = 2), and hence G has two conjugacy classes (resp. one conjugacy class if p = 2) ofelements of order p . All maximal elementary abelian subgroups of G have rank two. The11ubgroups E = ∗ ∗ and E = ∗ ∗ are maximal elementary abelian subgroups, and are not conjugate, although every non-identity element of E is conjugate to every non-identity element of E . It follows thatthe images of X E ( k ) and X E ( k ) in X G ( k ) are distinct irreducible components of X G ( k ),whereas their images in X ′ G ( k ) are equal. 8. Transfers of Chern classes. Throughout this section, G shall be a finite group. Following Moselle [15], we consider the‘Mackey closure’ of Ch( G ), or in other words the smallest natural subring of H ∗ ( G ) thatcontains Ch( G ) and is closed under transfers. More formally, we make the following: Definition. Let G be a finite group. Define Ch( G ) recursively as the subring of H ∗ ( G )generated by Ch( G ) and the image of Ch( H ) under the transfer Cor GH for each propersubgroup H < G .We shall prove: Theorem 8.1. Let G be a finite group, with Quillen category A , and let R = Ch( G ) .The map α induces a homeomorphism α : colim A X E ( k ) → V R ( k ) . We do not use the injectivity of Quillen’s map in proving Theorem 8.1, so two imme-diate corollaries of Theorem 8.1 are: Corollary 8.2. (Quillen) For a finite group G , the map α : colim A X E ( k ) → X G ( k ) isinjective. Corollary 8.3. For a finite group G , the inclusion of Ch( G ) in H ∗ ( G ) induces a homeo-morphism of varieties. Proof of the theorem. The transfer map Cor GH commutes with the action of the Steenrodalgebra, by either a topological argument [1] or an algebraic one [10]. It follows that R = Ch( G ) is a large natural subring of H ∗ ( G ). By Theorem 6.1, it suffices to show that C ( R ) = A .First we show that if E , F are elementary abelian subgroups of G such that E is notconjugate to a subgroup of F , then there is no map in C ( R ) from E to F . Since any mapis the composite of an isomorphism followed by the inclusion of a subgroup, it suffices toconsider the case when E and F have the same rank. (Note that if f is any map in C ( R )that is an isomorphism of elementary abelian groups, then the inverse of f is also in C ( R ),so f is an isomorphism in C ( R ).)Let N = N G ( F ) be the normalizer of F in G , let θ = C F − | F | − F given by the difference of the regular representation andthe trivial representation, and let ρ = Ind NF ( θ ) be the induced representation of N . Equiv-alently, ρ is the regular representation of N minus the permutation representation on thecosets N/F of F . Note that ρ is a genuine representation of N of dimension | N | − | N : F | .12ow let F ′ be any elementary abelian subgroup of N . The regular representation of N restricts to F ′ as a sum of | N : F ′ | copies of the regular representation of F ′ . The number oforbits of F ′ on the cosets N/F , or equivalently the number of trivial F ′ -summands of thepermutation module C N/F , is equal to | N : F ′ F | . It follows that Res NF ′ ( ρ ), the restrictionto F ′ of ρ , contains the trivial F ′ -module as a direct summand if and only if F ′ F = F ′ , i . e ., if and only if F is not a subgroup of F ′ .Let n be the dimension of ρ , so that n = | N | (1 − / | F | ), and note that since therestriction to F of ρ does not contain the trivial representation, Res NF ( c n ( ρ )) is non-zero.Let x F = Cor GN ( c n ( ρ )). For E an elementary abelian subgroup of G of the same rank as F , the Mackey formula affords a calculation of Res GE ( x F ):Res GE ( x F ) = X EgN Cor EE ∩ gNg − c ∗ g Res Ng − Eg ∩ N ( c n ( ρ )) , where for any subgroup K of G , c g is the homomorphism k g − kg , and the sum isover a set of double coset representatives for E \ G/N . The restriction map from E to anysubgroup is surjective in cohomology, and Cor EE ′ Res EE ′ is equal to multiplication by | E : E ′ | .Hence the transfer Cor EE ′ is zero for any proper subgroup E ′ of E . Thus the only non-zerocontributions to the above sum come from terms in which g − Eg ≤ N . On the other hand,we know that the restriction of c n ( ρ ) to an elementary abelian subgroup of N is non-zeroif and only if that subgroup contains F . Since we are assuming that E has the same rankas F , it follows that the only non-zero terms will come from g such that F = g − Eg . If F = g − Eg = h − Eh , then g − hF h − g = F , so g − h ∈ N , and so EgN = EhN . Itfollows that, for E and F of the same rank, Res GE ( x F ) = c ∗ g ( c n ( ρ )) for any g such that g − Eg = F , and is zero if there is no such g , i . e ., if E and F are not conjugate in G . SinceRes NF c n ( ρ ) is non-zero in h ∗ ( F ), it follows that when E and F have the same rank, thereare morphisms from E to F in C ( R ) only if E is conjugate to F .It remains to show that the automorphisms of F in the category C ( R ) are preciselythe maps induced by conjugation in G . Let C = C G ( F ) be the centralizer of F in G ,and suppose that | C : F | = p m r , for some r coprime to p . For any representation λ of F , Res CF Ind CF ( λ ) = p m rλ . It follows that the image of Ch( C ) in Ch( F ) contains thesubring of p m th powers. In F p [ F ] = Ch( F ), there is a homogeneous element y suchthat the F p GL( F )-submodule generated by y is free (see [2], pp. 45–46). The element y = y p m also has this property, and y = Res CF ( y ′ ) for some y ′ ∈ Ch( C ). Now let y ′ = y ′ Res NC ( c n ( ρ )), where ρ and n are as in the previous paragraph. Then y ′ is anelement of Ch( C ) whose restriction to an elementary abelian subgroup of C is non-zeroonly if that subgroup contains F . Moreover, the restriction to F of the representation ρ is invariant under GL( F ), and so y = Res CF ( y ′ ) generates a free F p GL( F )-submodule ofCh( F ). Now define z F ∈ H ∗ ( G ) by z F = Cor GC ( y ′ ). The Mackey formula givesRes GF ( z F ) = X F gC Cor FF ∩ gCg − c ∗ g Res Cg − F g ∩ C ( y ′ ) . Now Res CF ′ ( y ′ ) = 0 unless F ′ contains F , and so only those terms for which g − F g = F can be non-zero. Thus Res GF ( z F ) = X g ∈ N/C c ∗ g ( y ) , C = C G ( F ) in N = N G ( F ). Since y generates a free F p GL( F )-summand of Ch( F ), an automorphism f of F satisfies f ∗ Res GF ( z F ) = Res GF ( z F )if and only if f is equal to conjugation by some element of N . Remark. The first part of this proof is very similar to Quillen’s second proof of thisstatement, using the Evens norm map [17,11,5]. The second part is less similar however.Our argument is complicated by the weaker technique that we are using to constructelements, but is simplified by our use of Theorem 6.1 which means that we do not need toconstruct as many elements as are needed in [17].Corollary 8.3 seems to be fairly well-known, although we have been unable to findit stated in the literature. Our first proof of Corollary 8.3 was essentially independentof the rest of this paper, but used a comparatively recent theorem of Carlson ([7], ortheorem 10.2.1 of [11]): For G a p -group, with centre Z , the radical of ker(Res GZ ) is equalto the radical of the ideal generated by the images of Cor GH , where H ranges over all propersubgroups of G .To prove Corollary 8.3 directly, note that if G p is a Sylow p -subgroup of G , the transferfrom H ∗ ( G p ) to H ∗ ( G ) is surjective. Thus it suffices to consider the case when G is a p -group. Let G be a p -group, with centre Z . Using representations induced from Z up to G it may be shown that the image of Ch( G ) in H ∗ ( Z ) contains the subring of p m th powersfor sufficiently large m (as in the proof of Theorem 8.1). Thus if y ∈ H ∗ ( G ), there exists m and x ∈ Ch( G ) such that y = y p m − x is in the kernel of Res GZ . By Carlson’s theorem,there exists n , subgroups H (1) , . . . , H ( l ) of G and x ′ i ∈ H ∗ ( H ( i )) such that y = y p n = X i Cor GH ( i ) ( x ′ i ) . By induction on the order of G , there exists N such that, for each i , ( x ′ i ) p N ∈ Ch( H ( i )).Noting that for x of degree 2 i ,Cor GH ( x p ) = Cor GH ( P i x ) = P i Cor GH ( x ) = Cor GH ( x ) p , it follows that y p N = X i Cor GH ( i ) (( x ′ i )) p N = X i Cor GH ( i ) (( x ′ i ) p N ) ∈ Ch( G ) . 9. A closure operation.Definition. Let C be a category of elementary abelian subgroups of a group G . Define C , the closure of C , to be the smallest subcategory of C reg such that:1. C is contained in C ;2. if f : E → E is in C , and F i ≤ E i with f ( F ) ≤ F , then f : F → F is in C ;3. if f : E → E is in C and is an isomorphism of groups, then f − : E → E is in C .Say that C is closed if C = C . Note that the categories A , A ′ , and C ( R ) for any R ≤ H ∗ ( G ) are closed. 14 roposition 9.1. For any C containing the Quillen category A , the category C is theunique largest subcategory of C reg such that the natural map colim C X E ( k ) → colim C X E ( k ) is a homeomorphism. Proof. Let D be the subcategory of C reg whose morphisms f : E → F are those grouphomomorphisms that make the diagram X E ( k ) f −→ X F ( k ) y ι y ι colim C X E ( k ) Id −→ colim C X E ( k )commute. Then D has the property claimed, and it suffices to show that C = D . Notealso that C is contained in D and that D is closed. Let f : E → E be a morphism in D .Since f : E → E is in D (resp. in C ) if and only if f : E → f ( E ) is, it may be assumedthat f is a group isomorphism. Let φ be an element of X + E ( k ), i . e ., an element of X E ( k )not contained in X F ( k ) for any proper subgroup F of E . Since GL( E ) acts freely on X + E ( k ), it follows that f : E → E is uniquely determined by ψ = f ∗ ( φ ).By definition of D , ψ and φ have the same image in colim C X E ( k ). Since C is skeletallyfinite (because it contains A ), there are chains ( F , . . . , F m ) of objects of C and ( f , . . . , f m )of morphisms in C , where f i : F i − ǫ ( i ) → F i − ǫ ( i ) for some ǫ ( i ) ∈ { , } , and ( ψ , . . . , ψ m ), ψ i ∈ X F i ( k ), with F = E , F m = E , ψ = φ, ψ m = ψ, f i ∗ ( ψ i − ǫ ( i ) ) = ψ i − ǫ ( i ) . Let F ′ i be the unique subgroup of F i such that ψ i ∈ X + F ′ i ( k ). Then F ′ i has the same rank as E and f i restricts to an isomorphism f ′ i from F ′ i − ǫ ( i ) to F ′ i − ǫ ( i ) . Letting δ ( i ) = 1 − ǫ ( i ), f ′ δ ( i ) i is a morphism in C from F ′ i − to F ′ i , and the composite f ′ = f ′ δ ( m ) m ◦ · · · ◦ f ′ δ (1)1 is a morphism in C from E to E such that f ′ ( φ ) = ψ . Hence f ′ = f , and f is a morphismin C as claimed.For any category C of elementary abelian subgroups of a group G , one may define asubring R ( C ) of H ∗ ( G ) as the inverse image of lim C H ∗ ( E ). This subring is large and isnatural because lim C H ∗ ( E ) is. Proposition 9.2. For C any category of elementary abelian subgroups of G containingthe Quillen category A , C ( R ( C )) = C . Proof. Clearly, C ( R ( C )) contains C , and is closed. Hence it suffices to show that theinduced map of varieties is a homeomorphism. Quillen showed that the map from H ∗ ( G )15o lim A H ∗ ( E ) contains the subring of p n th powers for some n (in fact this is equivalentto the injectivity of the map colim A X E ( k ) → X G ( k )) [16]. Let S = lim C H ∗ ( E ), and notethat if x is any element of S , the p n th power of x is in the image of R = R ( C ). It followsthat the map V S ( k ) → V R ( k ) is a homeomorphism, and it suffices to show that the naturalmap colim C X E ( k ) → V S ( k )is a homeomorphism. But this is a special case of lemma 8.11 of [16].The proposition shows that there is a sort of ‘Galois correspondence’ between largenatural subrings of H ∗ ( G ) and categories of elementary abelian subgroups of G . 10. Some other categories. For each n ≥ 0, define a category A ( n ) of elementary abelian subgroups of a group G bydeclaring that the morphism f : E ֒ → F is in A ( n ) if and only if for all e , . . . , e n ∈ E , thereexists g ∈ G such that f ( e i ) = g − eg .Note that A (0) is the category C reg of Section 6, and A (1) is the category A ′ . Foreach n , A ( n ) ⊇ A ( n +1) , and when n is greater than or equal to the p -rank of G , A ( n ) isequal to Quillen’s category A . This suggests that A ( ∞ ) should be defined to be A . Eachof the categories A ( n ) is closed in the sense of section 9, and the subrings R ( n ) = R ( A ( n ) )form a natural filtration of H ∗ ( G ) = R ( ∞ ) .The categories A ( n ) ( G ) are related to the generalized characters of G due to Hopkins,Kuhn and Ravenel [13] in the same way that the category A ′ is related to ordinary char-acters. It seems possible that there should be a description of the variety for the subringof elements of H ∗ ( G ) coming from E ( BG ), where E is a generalized cohomology theoryto which Hopkins-Kuhn-Ravenel’s work applies, in the same way that Chern classes areelements of H ∗ ( G ) coming from K ( BG ). We shall not make a precise conjecture, butshall give examples to show that the categories A ( n ) can be distinct from each other. Proposition 10.1. For each n ≥ and each prime p , there is a p -group G for which A ( n ) ( G ) = A ( n +1) ( G ) . Proof. For n = 0, the cyclic group of order p (for p odd), or the elementary abeliangroup of order four (for p = 2), will suffice. Hence we may assume that n > 0. Let C be a cyclic group of order p , let E be a faithful F p C -module of F p -dimension n + 1, andlet c ∈ GL( E ) represent the action on E of a generator for C . Now let Z be a vectorspace over F p with basis { z M } indexed by the maximal F p -subspaces of E , so that Z hasdimension ( p n +1 − / ( p − M of E , pick a linear map ψ M : E → Z , with kernel M and image generated by z M . For each M , define b M ∈ GL( E ⊕ Z ) by the equation b M ( e, z ) = ( c ( e ) , z + ψ M ( e )) . Let A be the subgroup of GL( E ⊕ Z ) generated by the b M , and let G be the semidirectproduct ( E ⊕ Z ): A .The subgroup Z is left invariant by A , so is central in G . Let φ be the homomorphismsending A ≤ GL( E ⊕ Z ) to GL(( E ⊕ Z ) /Z ) ∼ = GL( E ), and let B = ker( φ ) ≤ A . Note that16lements of B act trivially on Z and on ( E ⊕ Z ) /Z , and so B may be identified with asubgroup of the elementary abelian p -group Hom( E, Z ).We claim that the automorphism c of E is a morphism in A ( n ) ( G ), but is not amorphism in A ( n +1) ( G ). If M is any rank- n subgroup of E , then the element b M ∈ G actson M in the same way as c . On the other hand, if c were a morphism in A ( n +1) ( G ), therewould have to be an element d of G , acting on E ⊕ Z as d ( e, z ) = ( c ( e ) , z ). But then, forany M , d ′ = d − b M would be an element of ( E ⊕ Z ) : B acting as d ′ ( e, z ) = ( e, z + ψ M ( e )).To complete the proof, it suffices to show that there can be no such element d ′ .Let R be the image of F p C in the ring End( E ). Since F p C is a commutative localring, it follows that R is too. In particular, the non-units in R form an ideal. Fix M , amaximal subgroup of E . The group B ≤ A ≤ GL( E ) is generated as a normal subgroupby the elements b pM , and b − M b N , where N ranges over all other maximal subgroups of E .The action of these generators on E ⊕ Z is given by: b pM ( e, z ) = c p ( e ) , z + p − X i =0 ψ M ( c i ( e )) ! = ( e, ψ M (¯ re ) + z ) , where ¯ r is the image of ¯ c = P p − i =0 c i in R = End( E ), and b − M b N ( e, z ) = ( e, z + ψ N ( e ) − ψ M ( e )) . We therefore have to show that the element d ′ described above does not lie in the subgroupof GL( E ⊕ Z ) generated by( e, z ) ( e, z + ψ M (¯ re )) and ( e, z ) ( e, z + ψ N ( c i e ) − ψ M ( c i e )) , for all N = M and 0 ≤ i ≤ p − d ′ : ( e, z ) ( e, z + ψ M ( e )) is in the subgroup B , there are µ, λ N ∈ R ≤ End( E ) suchthat Im λ N ⊆ ker( ψ N ) = N for all N = M , andIm − µ ¯ r + X N = M λ N ⊆ ker( ψ M ) = M. From the first family of equations, it follows that each λ N is a non-unit in R , and of course¯ r is a non-unit. 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