Prime Ideals in the Burnside Ring

Prime Ideals in the Burnside Ring

Daniel Ferrand June 2013

This note is an attempt to clarify the remarkable concise paper byA. Dress, entitledA characteri- sation of Solvable Groups [D 69], where he gives a description of the spectrum of the Burnside ring which allows him to prove that this spectrum is connected if and only if the group is solvable.

The Burnside ring Burn(G) of a finite groupGis the Grothendieck ring of finiteG-sets. A setT with a transitive action of G may be identified, after the choice of an element t ∈ T, with the quotient set G/H, where H is the stabilizer of the point t. But the passage from the subgroup H to the quotient T =G/His, in a sense, contravariant ; because it is repeatedly used byDress, it may get the reader, like me, lost in a kind of counter-intuitive hesitation. So I chose, in this paper, to use the language of G-sets alone (except for some specific proofs). Another obvious motive for doing so is a view toward a future study of a “Burnside ring” for the finite ´etale coverings of a given scheme (hopefully more accessible than the Burnside ring of the fundamental group of the scheme).

In what follows,Gdenotes afinite group, and all theG-sets are finite.

These restrictions will no more be specified.

1. Connected G-sets

A (finite) setT endowed with a transitive action ofGwill be calledconnected G-sets¹. By definition, a connected G-set is non empty. We usually reserve the letters T, U, S for the connected G-sets. The transitivity property is equivalent to the map

G ×T −→ T×T, (g, t)7−→(gt, t) being surjective.

For twoG-setsX andY, we will use the term morphism to indicate a map f :X −→ Y of G-sets, that is a map such that f(gx) =gf(x) for all g∈Gand allx∈X.

A morphism X −→ T where X is non empty and T connected, is surjective ; a surjective morphism X −→T whose domainX is connected has necessarily a connected targetT.

The following fact will be repeatedly used :

Two morphisms of opposite direction, T → U and U →T, between connected sets are necessarily iso- morphisms.

2. Pointing

The pointing of a connected setT is the choice of an elementt∈T. Such a choice allows us to introduce the stabilizerGt={g∈G, gt=t} oft, and the isomorphism of G-sets

(2.1) G/Gt−→gT.

This association between a connected set and a subgroup may sometimes help in offering intuition of these things. But,if, instead of sets, we consider sheaves, or objects in a topos, the pointing process may be impossible, at least globally ; moreover, the choice of an element, when possible, is mostly arbitrary, and different choices lead to different isomorphisms ; for example, the stabilizer of an other element of T, written as gt, is the conjugate subgroup : G_gt =gG_tg⁻¹. For these reasons, we try, in this note, to avoid the pointing process at least in defining/constructing objects ; but we are less reluctant to use it when it seems that some proofs are easier for a quotient G/H; it can then be looked upon as a kind of localization.

1. Bourbakiuses the adjectivehomogeneous [A, I 5.5.]

3. Degree

Let f : X −→ Y be a morphism. The degree of f, denoted by deg(f), is the map Y → N defined by y7−→Card(f⁻¹(y)).

A morphism is surjective if its degree map is nowhere zero.

IfY is connected, the degree map is constant since f⁻¹(gy) =gf⁻¹(y).

3.1. Lemma Let f :X −→Y be a morphism. Then f is of constant degreedif and only if there exists a surjective morphism Y⁰−→Y such thatX×Y Y⁰ be the disjoint union ofdcopies deY⁰.

This strong analogy with the ´etale coverings from the algebraic geometry explains why a morphismf whose degree is constant, say degf =d, will often be called acovering, or even a covering of degree d The proof uses an induction on the degree d, starting from the obvious case whered= 1. Suppose now thatX be broke as the disjoint unionX=Y^tmtZ, i.e. thatf ownsm G-sections with disjoint images²; ifZ is non empty, its degree overY isd−m6= 0, thus the restriction off toZ is a surjective morphism f⁰ :Z →Y; moreover, one has X×f,Y,f⁰Z =Z^tmtZ×Y Z, and the morphismZ×Y Z −→Z admits a section disjoint from the previous ones, the diagonal ; then fZ :X×Y Z −→ Z admits at least d+ 1 sections with disjoint images.

4. Automorphisms

4.1 LetT be a connectedG-set. Then the groupW(T) of theG-automorphisms ofT acts freely onT; in other words, the map

W(T)×T −→T×T

is injective ; in fact, given two automorphisms w and w⁰, if one element t ∈ T has the same image w(t) =w⁰(t), then for all g∈G, one has w(gt) =gw(t) = gw⁰(t) =w⁰(gt), and thusw=w⁰, since the action ofGis transitive.

For a morphismf :X →Y, we denote byW(X/Y), orW(f), the group of theG-automorphismsw ofX which are compatible withf, i.ef w=f.

The morphism f is said to be galoisian, or a Galois covering, if it is surjective and if there exists a subgroup Γ⊂W(f)such that the map

Γ×X−→X×_Y X, (w, x)7→(w(x), x) is bijective, or equivalently, if the mapX/Γ −→ Y is bijective.

As usual, the group Γ is not uniquely determined byf ifX is not connected ; for a morphism between connected sets (hence f is surjective), things are easer :

4.2 Lemma A morphismf :T →U between connected sets is galoisian if and only if Card(W(T /U)) = deg(f).

Since f is surjective, the degree offT =f×1T :T×UT −→T is equal to deg(f) ; from this we deduce the equality

deg(f).Card(T) = Card(T×UT).

Hence the claim.

4.3. Lemma Let Γ be a subgroup ofW(T); then the quotient setT /Γ is aG-set, and the morphism f :T −→ T /Γ

is a Galois covering and one has Γ =W(f).

Since T is acted on freely by Γ, the orbits have all the same cardinality, namely the order d of Γ ; these orbits are the fibers of the morphismf :T −→T /Γ =S; thusf is a covering of degreed. The two morphisms

Γ× T −→W(T /S)× T −→T×_ST

2. IfY were connected, the images of distinct sections would necessarily be disjoint.

are injective. SinceT →S is of degreed, the cardinality ofT×_ST is equal tod.Card(T) = Card(Γ×T) ; therefore one has Γ =W(T /S), andf is a Galois covering.

4.4. Lemma LetSbe connected, and letf :X−→Sbe a Galois covering with groupW. Any connected component T ⊂X is galoisian overS with group the stabilizer ofT inW.

In fact, consider two elements t, t⁰ ∈T in the same fiber. By definition of a Galois covering, there exists a uniquew∈W such thatw(t) =t⁰. This automorphismwstabilizesT since every element inT may be written asgt, withg∈G, and one hasw(gt) =gw(t) =gt⁰∈T.

4.5. Remark. The result is no more true for a G-subset T of X which would not be connected.

Consider for example a finite setI endowed with a cyclic permutationσ; and letX =S×I the disjoint union of copies of S indexed by I; let W be the group generated by the automorphism w given by w(s, i) = (s, σ(i)) ; the projectionX =S×I−→S is clearly a Galois covering with groupW, sinceI is a torsor under the group< σ >. It is easy to find a specific set Icontaining a subset J whose stabilizer in < σ >is reduced to the identity. Then the componentS×J ⊂S×I can’t be galoisian over S for a subgroup ofW.

5. The case of pointed G-sets

LetH ⊂Kbe two subgroups ofG, and letf :G/H−→G/K be the corresponding morphism. Its degree is the indice :

deg(f) = (K:H).

The group of the G-automorphisms of theG-set G/H is isomorphic with (Norm_G(H)/H)^o ([A] I 5.5, prop. 5) ; precisely, the (right) action of an n∈Gwhich normalizesH on the classgH is given by

(gH)? n = gnH = gHn.

Thus the relative automorphisms, i.e. the elements ofW(f), are given by thenwhich, moreover, fix the classesgK; therefore

W(f) = (NormK(H)/H)^o. We deduce the following remark :

5.1. Lemma The morphism f : G/H −→ G/K is galoisian if and only if H is normal in K; the Galois group is thenW(f) = (K/H)^o.

6. Existence of maximal p-galoisian coverings, and of maximal solvable coverings

I shall use the following definitions.

A covering is p-galoisian, also said a p-Galois covering, if it is a Galois covering whose Galois group is ap-group.

A solvable covering is a Galois covering whose Galois group is solvable.

6.1. Lemma LetT, U, V be connectedG-sets, and letU −→^f T ←−^g V be Galois coverings (resp.p-Galois coverings, resp. solvable coverings). Let S ⊂ U ×T V be a connected component of the fibered product.

Then the composite morphism h:S−→T is galoisian (resp. etc.)

S

h

//U

f

V _g //T

To prove the lemma we choose a point s ∈ S, we denote by u, v, t its images, and we consider the stabilizers G_s, G_u, G_v, G_t; sinceS is a subset of U×V, one hasG_s=G_u∩G_v. According to (5.1), the

hypothesis onf andg means thatG_u andG_v are normal inG_t; thusG_u∩G_v is also normal inG_t, i.e.

his galoisian. The group inclusion

Gt/Gu∩Gv ⊂ Gt/Gu ×Gt/Gv

implies the more restrictive statements.

6.2. Proposition Let T be a connectedG-set, and letpbe a prime.

i) Let ϕ:T^(p)−→ T be a p-Galois covering for which Card(T^(p)) is maximal. Thenϕ is also maximal in the “arrow“ sense : namely, for any galoisian p-covering ψ : U −→ T, there exists a (non unique) morphism κ:T^(p)−→U such that ϕ=ψ◦κ. In particular, this covering is unique up to a non unique isomorphism.

ii) Similarly, a maximal solvable coveringσ:T^slv −→T is also maximal in the above sense, and thus it is unique up to an isomorphism.

This is a direct consequence of the lemma above, since a subgroup of a solvable group is solvable.

6.3. Corollary Any morphism U −→ V between connected sets may be extended as a morphism U^(p)−→ V^(p) of their maximal p-Galois coverings, and also as a morphismU^slv−→V^slv.

If the morphism U −→ V is a p-Galois covering, then the composite U^(p) −→ U −→ V is still a p-Galois covering, and thusU^(p)−→V^(p)is an isomorphism ; moreover this construction is idempotent : (U^(p))^(p)=U^(p).

The same is true for solvability : If the morphismU −→V is a solvable covering, then the composite U^slv −→U −→V is still a solvable covering, and thus U^slv −→V^slv is an isomorphism ; moreover this construction is idempotent : (U^slv)^slv=U^slv.

LetP be the Galois group ofV^(p)−→V ; by the base change we get ap-Galois coveringU×VV^(p)−→U with the same group ; let T be a connected component of U×V V^(p); by (4.4), ϕ:T −→U is a Galois covering with group the stabilizer ofT inP, that is ap-Galois covering.

T

ϕ

&&

$$

U×_V V^(p)

//U

V^(p) //V

From the proposition above the maximal p-galoisian coveringU^(p)−→U factors throughϕ:T −→U, and that gives the required morphism U^(p)−→V^(p).

Now suppose that the morphism f : U −→ V be p-galoisian. To conclude that f^(p) : U^(p) −→V^(p) is an isomorphism, we have to show there exists a morphism in the other direction, and for doing so it is enough to show that the compositeU^(p)−→U −→V is a Galois covering. Choose a elementt∈U^(p)and denote by uand v its images. Considering the normalizers, we get the sequence of inclusions of normal subgroups

Gt C Gu C Gv

and we must check that Gtis normal in Gv. SinceU^(p)→U is a maximalp-Galois covering, Gtis the minimal normal subgroup ofGuwhose quotient is ap-group ; its unicity implies thatGtis stable by any automorphism ofGu, and in particular, by any conjugation by an element ofGv.

The proof in the solvable case is formally the same.

6.4. Corollary Let T be a connectedG-set, and letpbe a prime. Suppose that there does not exist non trivial p-Galois covering T⁰ →T. Let P ⊂W(T) be ap-Sylow subgroup of the automorphisms group of T. Then the morphism (T /P)^(p)−→T is an isomorphism.

Since the morphism T −→ T /P is a p-Galois covering (4.3), the maximal one (T /P)^(p) −→ T /P factors troughT :

(T /P)^(p)−→^ϕ T −→^ψ T /P,

and ϕ is a p-Galois covering since the composite ψ◦ϕ is. The assumption on T implies that ϕ is an isomorphism.

7. Prime ideals in the Burnside ring.

The class of a (finite)G-set X in the Burnside ring Burn(G) will be notedcl(X).

LetT be a connectedG-set ; then the set Hom_G(T, X) is finite, and its cardinality is denoted bym_T(X), in homage toBurnsidewho called this number themark ofT in X ([B], p.236)³.

m_T(X) = Card(Hom_G(T, X)).

The marks extend to ring homomorphisms

mT : Burn(G) −→ Z.

We choose a set T of connectedG-sets containing exactly one element of each isomorphism classe. The marks together define a homomorphism

m: Burn(G) −→ M(T,Z) x7−→(T 7→mT(x)) whose target denotes the ring of all maps from the setT to Z.

7.1. Lemma The set (cl(T))_T∈T is a basis of Burn(G) and the morphism m is injective. In conse- quence,the ringBurn(G)is reduced.

Since any finite G-set is the disjoint union of its orbits, the family (cl(T))_T∈T generates the additive group Burn(G). An elementx∈Burn(G) may thus be written asx=Pa_Tcl(T) with coefficientsa_T in Z. Suppose that some of these coefficients are non zero (it is the case ifx6= 0). LetS∈ T be a connected set with minimal cardinality among those for which aT 6= 0 ; since any morphism S →T is surjective, implying that Card(S)≥Card(T), we have mS(T) = 0 for the other T, that is if aT 6= 0 and T 6=S; therefore, one hasmS(x) =aSmS(S)6= 0. That implies the two assertions.

7.2. With standard algebra arguments one deduces from this lemma a first characterization of minimal and maximal prime ideals of Burn(G), via their traces on the subringZ⊂Burn(G) ; namely

7.2.1.A prime idealp is minimal if and only ifZ∩p= 0.

7.2.2.A prime idealm is maximal if and only ifZ∩m=pZ, for a prime number p.

Consider a commutative algebraK→Bof finite dimension over a fieldK; then each prime idealpofB is maximal, and thus also minimal, since for xnon zero in B/p, theK-linear mapB/p→B/p, y7→xy is injective, hence bijective.

This remark shows that the above conditions are sufficient. Let now p be a minimal prime ideal in B = Burn(G). Since B is reduced, the local ringB_p is a field, and the image of an elementa∈Z∩pin B_p is zero. But, B being free over Z, the composite map Z→B →B_p is flat ; thus, it sends a regular (i.e non zero) element of Zonto a regular element in Bp; henceZ∩p= 0.

Now let m be a maximal ideal in B, we have to check that Z/m∩Z is a field. A non zero element a in that ring has a non zero, hence invertible, image in the field B/m; its inverse, b = a⁻¹ is the root of a monic polynomial p(X), of degree say n, with coefficients in Z/m∩Z. We thus have the relation 0 =aⁿp(b) = 1 +ac, with c∈Z/m∩Z; it shows that ais invertibleZ/m∩Z.

The next theorem summarizes the description of Spec(Burn(G)), due toA. Dress.

The number of automorphisms of a connected setT is denoted byw(T) = mT(T) = Card(W(T)).

3. In fact Burnside identifiesT with a quotientG/H, and speaks of the mark ofHin the representationX.

7.3. Theorem Let G be a finite group, and, as above, choose a set T of connected G-sets containing exactly one element from each isomorphism class.

i) The assignment cl(T)7→pT = Ker(Burn(G) −→^mT Z) defines a bijection between the set T and the set of minimal prime ideals of Burn(G).

ii) For T ∈ T and for a prime p, the kernelmT ,p of the homomorphism

Burn(G)−−−−−−→^mTmod.p Fp

is a maximal ideal. Moreover, the assignment (cl(T), p) 7→ mT ,p defines a bijection between the set of couples (cl(T), p)withT ∈ T andpa prime number not dividingw(T), and the set of maximal ideals of Burn(G).

iii) LetS andT inT, and letpbe a prime. Then the inclusionpS ⊂mT ,pis equivalent to the congruence mS≡mTmod.p.

If, moreover, the prime pdoes not divide w(T), then the following conditions are equivalent a)mS ≡mTmod.p.

b) There is a sequence ofp-Galois coveringsS→S₁→ · · · →S_n =T.

c) The maximalp-Galois coverings (6.2) ofS andT have isomorphic domains : S^(p)−→gT^(p). d) There exists ap-Galois coveringT^(p)−→S.

We begin with a key lemma.

7.4. Lemma Let k be an integral domain, and let f : Burn(G)−→k be a ring homomorphism. Let T be a connected set which is maximal (i.e. whose cardinality is maximal) among those such thatf(T)6= 0.

Thenf factors throughmT :

Burn(G)

f

''

m_T //Z _κ //k ,

where κis the unique ring homomorphismZ→k.Two such connected sets are isomorphic.

Moreover, if κis injective, then all the connected setsS such thatf =ϕ_S are isomorphic.

LetX be aG-set. We first compute f(T ×X). It is the sum of thef(U) for the connected components U ofT×X. By projection, such a component comes equipped with two maps

T ←−^τ U −→^u X

If τ is not an isomorphism, then the choice of T implies that f(U) = 0. If τ is an isomorphism, we associate with U the map uτ⁻¹:T −→X; conversely, to any mapv:T −→X is associated its graph, which is a component of T ×X isomorphic with T. These two constructions are clearly inverse each others, showing that the number of connected components of T ×X, which are isomorphic with T, is equal to Card(Hom_G(T, X)) =m_T(X). We have thus shown that

f(T×X) =mT(X).f(T).

But,f being a ring homomorphism, one hasf(T×X) =f(T).f(X). Sincef(T) is non zero in the domain k, we conclude that

f = κ◦mT.

Now, ifT⁰is another connected set withf(T⁰)6= 0, then a fortiori the integermT(T⁰) = Card(HomG(T, T⁰)) is not zero, i.e. there is a morphism T →T⁰; by reversing the roles ofT and T⁰, we see that T and T⁰ are isomorphic.

Suppose now that κ is injective, and thusf may be seen as a map to Z, i.e. f =mT. Let S be a connected set such that mS =mT; the integermS(S) = Card(W(S)) is not zero, showing that there is a morphism T →S; similarly, there is a morphismS→T; thusS andT are isomorphic.

7.5. Proof of the partsi) andii) of the theorem.

Let p be a prime ideal in Burn(G), and let f : Burn(G) −→ Burn(G)/p be the quotient morphism.

Consider a maximal connected set T such that cl(T) ∈/ p; the above lemma implies that f factors throughmT, i.e. thatf =κ◦mT; note thatκis surjective sincef is. We thus have, for aG-setX,

(7.5.1) cl(X) ≡ m (X) mod.p.

For any T, the compositeZ−→^can. Burn(G)−→^mT Zis bijective. From that and (7.2.1), we deduce that the minimality of pi.e.Z∩p= 0, is equivalent to the morphismκ:Z−→Burn(G)/pbeing injective (hence an isomorphism). The part i) follows easily.

Consider now the case where κ is not injective. Since the morphismκ:Z−→ Burn(G)/p is surjective one has Burn(G)/p=F_pfor a prime number p. The choice ofT and the congruence (7.5.1) imply that

mT(T) 6≡ 0 mod.p

In other words, the prime p does not divide the cardinality w(T) of the group W(T). Together with (7.2.2), it is thus proven that each maximal ideal is of the form mT ,p withp-w(T). It remains to check the injectivity of the asignment : ifmT ≡mT⁰mod.p, then in particular, one has

m_T(T⁰)≡m_T⁰(T⁰)6≡0 mod.p;

this implies the existence of a morphism T →T⁰, and by symmetry, also a morphismT⁰ →T; henceT andT⁰ are isomorphic.

The following result is classical.

7.6. Lemma Let E be a finite set acted on by a finite p-group P. Then, denoting byE^P the subset of invariants, one has

Card(E) ≡ Card(E^P) mod.p.

In particular, iff :T −→T /P is ap-Galois covering of G-sets, thenm_T ≡m_{T /P} mod.p.

The congruence between the marks comes from the bijection

HomG(T /P, X) −→ HomG(T, X)^P, u7−→u◦f.

7.7.Proof of the partiii) of the theorem.

pS ⊂mT ,p⇐⇒m_S ≡m_Tmod.p

The inclusion pS ⊂ mT ,p induces the canonical homomorphism Burn(G)/pS −→ Burn(G)/mT ,p which makes the following diagram commutative

Burn(G) ^mS //Z ^∼ //

Burn(G)/pS

Burn(G)_m

T

//Z //Fp

∼ //Burn(G)/mT ,p

We thus have the congruencem_S≡m_Tmod.p. The converse is clear.

a)⇒b).

Since mT(T) = w(T) is not a multiple of p, the congruence mS ≡ mTmod.p implies that mS(T) 6≡

0 mod.p, and, in particular, that there exists a morphismS→T. If it is an isomorphism, there is nothing more to prove. If not, then m_T(S) = 0 ; but, by the property a), one hasm_T(S)≡m_S(S)mod.p;thusp divides the order ofW(S), and there is a non trivialp-Sylow subgroupP ofW(S) ; it acts on the right on Hom_G(S, T). The lemma (7.6) above and the hypothesisp-w(T) imply the congruences

Card(Hom_G(S, T)^P) ^(7.6)≡ Card(Hom_G(S, T)) ^(a)≡ Card(Hom_G(T, T))6≡0 modp

Therefore, there exists a morphism f :S −→T such thatf w=f for allw∈P; lettingS1=S/P, the morphismf may now be factorized as

S −→^f1 S1 −→ T,

where f1 is ap-Galois covering, which is not trivial since P 6= 1. As mS ≡mS₁mod.p, the conclusion follows by induction on the degree of f.

b)⇒c)

This implication is proved in (6.3).

c)⇒d)

By construction, the morphismS^(p)−→S is ap-Galois covering.

d)⇒a)

The lemma (7.6) applied to thep-Galois coveringsT^(p)−→T andT^(p)−→S gives the congruences m_T(p) ≡ mT ≡ mS mod.p

8. Connected components of Spec(Burn(G)).

In this paragraph, the adjective connected will have two meanings, the second one being relative to Spec(Burn(G)) endowed with the Zariski topology, and its subsets. That may not cause any confusion.

As usual, for a prime idealp of Burn(G), we denote byV(p) the set of the primes which containp; it is a closed connected set. In any topological space, ifEandF are two connected sets such thatE∩F6=∅, then E∪F is connected. Thus a connected componentC of Spec(Burn(G)) is a union ofV(p) withpa minimal prime ; in fact, it is a finite union since the ring Burn(G) has only a finite number of minimal prime ideals. Conversely, the setsE which may be written in the following manner are connected : (?) E=V(p₁)∪V(p₂)∪ · · · ∪V(p_n)

where the p_i are minimal prime ideals, such that V(p_i)∩V(p_i+1) 6= ∅, for i = 1,2,· · · , n−1. In this description we must allow somep_i being equal : in fact, ifpis a minimal prime such that V(p)∩E6=∅, but V(p)∩V(pj) =∅forj6=i, then a sequence attached to the connected setV(p)∪E is

(p1,· · ·,pi,p,pi,· · · ,pn).

Now letEbe a maximal set of the type described by (?). Let us show thatEis a connected component of the spectrum. SinceEis closed, it is enough to prove thatEis open ; but the remark above shows that the complement ofE is the union of theV(p) which are disjoint fromE. Since, again, the ring Burn(G) has only a finite number of minimal prime ideals, this union is a closed subset.

8.1. Theorem a) The minimal prime ideals p_S and p_T are in the same Zariski connected component of Spec(Burn(G))if and only if there exits a connectedG-setU and two solvable coverings

S ←−^σ U −→^τ T,

or, equivalently, if the sourcesS^slv andT^slv of the maximal solvable coverings ofS andT are isomorphic (cf. (6.3)).

b) For any non empty G-set U let C(U) ⊂ Spec(Burn(G)) be the union of the V(pT) for the various solvable coverings U →T. ThenC(U)is connected (for the Zariski topology), and C(U) is a connected component if and only ifU has no non trivial abelian coverings, that is if the morphismU^slv−→U is an isomorphism.

c) In particular, Spec(Burn(G))is connected if and only if the groupGis solvable.

a) For proving that the condition is sufficient, it is enough to show that p_U and p_T are in the same connected component. By choosing a Jordan-H¨older sequence of its (solvable) Galois groupW(U/T), the solvable coveringτ:U −→T may be written as the composite of a sequence

U −→^τ1 U1 τ₂

−→ U2→ · · · →Un=T,

where eachU_i−1 −→^τi Ui is a cyclic Galois covering of orderpi for some prime (depending ofi). Let pi = Ker(Burn(G) −→^mUi Z)

be the minimal prime associated withU_i. The lemma (7.6) implies the following congruence between the marks

mU_i−1 ≡ mU_imod.pi

From the theorem (7.3 ;iii)) we deduce that the setV(p_i−1)∩V(pi) contains the maximal idealmU_i,p_i. Hence pU andpT are in the same connected component of Spec(Burn(G)).

Conversely, suppose that the minimal primespS andpT are in the same connected component, and that a sequence like (?) above has been chosen with p1 =pS, and pn =pT; let Si be a G-set corresponding withpi. By assumption, a maximal ideal contains bothpiandpi+1; by (7.3iii)), its is of the formmU_i,p, for a suitable prime p, and there are two p-Galois coverings Si ←− Ui −→ Si+1; but a p-group being solvable, we see from (6.3) that the maximal solvable coveringsS^slv_i andS_i+1^slv are isomorphic. Hence the conclusion.

b) The first part shows that a minimal primep_S is in the connected component containingp_U if and only ifS^slv andU^slv are isomorphic, that is ifp_S ∈C(U^slv). The conclusion follows.

c) Consider theG-set reduced to one point ; it is canonically pointed with stabilizer equal toG, and thus it is isomorphic toG/G; denoting byD^∞(G) the intersection of the derived series, the maximal solvable covering of the point is G/D^∞(G)−→ •. Let p_• be the prime ideal associated to the point ; if it is in the same component as the prime pG = Ker(Card), then, froma) we deduce that the G-sets G/D^∞(G) and Gare isomorphic, that isD^∞(G) = 1, i.e.Gis solvable. Conversely, if Gis solvable, then for each subgroup H, the morphismG−→G/H is a Galois covering with group the solvable group H, showing that the spectrum is connected.

References

[A]N. Bourbaki, Algebra I, (English edition), Springer-Verlag (1989)

[B]W. Burnside, Theory of groups of finite order, second edition (1911), reprinted by Dover (1955) [D 69],A. Dress,A Characterisation of Solvable Groups, Math. Z., vol 110, p.213-217, (1969) [D 71],A. Dress, Notes on the theory of representations of finite groups, Bielefeld Notes (1971)