On the K(π,1)-property for rings of integers in the mixed case
aa r X i v : . [ m a t h . N T ] J a n On the K ( π, by Alexander SchmidtNovember 7, 2018 Abstract
We investigate the Galois group G S ( p ) of the maximal p -extension un-ramified outside a finite set S of primes of a number field in the (mixed)case, when there are primes dividing p inside and outside S . We show thatthe cohomology of G S ( p ) is ‘often’ isomorphic to the ´etale cohomology ofthe scheme Spec (Ø k r S ), in particular, G S ( p ) is of cohomological dimen-sion 2 then. We deduce this from the results in our previous paper [Sch2],which mainly dealt with the tame case. Let Y be a connected locally noetherian scheme and let p be a prime number.We denote the ´etale fundamental group of Y by π ( Y ) and its maximal pro- p factor group by π ( Y )( p ). The Hochschild-Serre spectral sequence inducesnatural homomorphisms φ i : H i ( π et ( Y )( p ) , Z /p Z ) −→ H i et ( Y, Z /p Z ) , i ≥ , and we call Y a ‘ K ( π,
1) for p ’ if all φ i are isomorphisms; see [Sch2] Proposi-tion 2.1 for equivalent conditions. See [Wi2] for a purely Galois cohomologicalapproach to the K ( π, Theorem 1.1.
Let k be a number field and let p be a prime number. Assumethat k does not contain a primitive p -th root of unity and that the class numberof k is prime to p . Then the following holds:Let S be a finite set of primes of k and let T be a set of primes of k ofDirichlet density δ ( T ) = 1 . Then there exists a finite subset T ⊂ T such that Spec (Ø k ) r ( S ∪ T ) is a K ( π, for p . Remarks.
1. If S contains the set S p of primes dividing p , then Theorem 1.1holds with T = ∅ and even without the condition ζ p / ∈ k and Cl ( k )( p ) = 0,see [Sch2], Proposition 2.3. In the tame case S ∩ S p = ∅ , the statement ofTheorem 1.1 is the main result of [Sch2]. Here we provide the extension to the‘mixed’ case ∅ S ∩ S p S p . 1. For a given number field k , all but finitely many prime numbers p satisfy thecondition of Theorem 1.1. We conjecture that Theorem 1.1 holds without therestricting assumption on p .Let S be a finite set of places of a number field k . Let k S ( p ) be the maximal p -extension of k unramified outside S and put G S ( p ) = Gal ( k S ( p ) | k ). If S R denotes the set of real places of k , then G S ∪ S R ( p ) ∼ = π ( Spec (Ø k ) r S )( p ) (wehave G S ( p ) = G S ∪ S R ( p ) if p is odd or k is totally imaginary). The followingTheorem 1.2 sharpens Theorem 1.1. Theorem 1.2.
The set T ⊂ T in Theorem 1.1 may be chosen such that (i) T consists of primes p of degree with N ( p ) ≡ p , (ii) ( k S ∪ T ( p )) p = k p ( p ) for all primes p ∈ S ∪ T . Note that Theorem 1.2 provides nontrivial information even in the case S ⊃ S p , where assertion (ii) was only known when k contains a primitive p -th rootof unity (Kuz’min’s theorem, see [Kuz] or [NSW], 10.6.4 or [NSW ], 10.8.4,respectively) and for certain CM fields (by a result of Mukhamedov, see [Muk]or [NSW], X § ], X § G S ( p ) beinga duality group. If ζ p / ∈ k , this is interesting even in the case that S ⊃ S p , whereexamples of G S ( p ) being a duality group were previously known only for realabelian fields and for certain CM-fields (see [NSW], 10.7.15 and [NSW ], 10.9.15,respectively, and the remark following there).Previous results in the mixed case had been achieved by K. Wingberg [Wi1],Ch. Maire [Mai] and D. Vogel [Vog]. Though not explicitly visible in this paper,the present progress in the subject was only possible due to the results on mildpro- p groups obtained by J. Labute in [Lab].I would like to thank K. Wingberg for pointing out that the proof of Propo-sition 8.1 in my paper [Sch2] did not use the assumption that the sets S and S ′ are disjoint from S p . This was the key observation for the present paper.The main part of this text was written while I was a guest at the Departmentof Mathematical Sciences of Tokyo University and of the Research Institute forMathematical Sciences in Kyoto. I want to thank these institutions for theirkind hospitality. We start with the observation that the proofs of Proposition 8.2 and Corol-lary 8.3 in [Sch2] did not use the assumption that the sets S and S ′ are disjointfrom S p . Therefore, with the same proof (which we repeat for the convenienceof the reader) as in loc. cit., we obtain 2 roposition 2.1. Let k be a number field and let p be a prime number. Assume k to be totally imaginary if p = 2 . Put X = Spec (Ø k ) and let S ⊂ S ′ be finitesets of primes of k . Assume that X r S is a K ( π, for p and that G S ( p ) = 1 .Further assume that each p ∈ S ′ r S does not split completely in k S ( p ) . Thenthe following hold. (i) X r S ′ is a K ( π, for p . (ii) k S ′ ( p ) p = k p ( p ) for all p ∈ S ′ r S .Furthermore, the arithmetic form of Riemann’s existence theorem holds, i.e.,setting K = k S ( p ) , the natural homomorphism ∗ p ∈ S ′ \ S ( K ) T ( K p ( p ) | K p ) −→ Gal ( k S ′ ( p ) | K ) is an isomorphism. Here T ( K p ( p ) | K p ) is the inertia group and ∗ denotes the freepro- p -product of a bundle of pro- p -groups, cf. [NSW], Ch. IV, §
3. In particular,
Gal ( k S ′ ( p ) | k S ( p )) is a free pro- p -group. Proof.
The K ( π, H i ( G S ( p ) , Z /p Z ) ∼ = H i et ( X r S, Z /p Z ) = 0 for i ≥ , hence cd G S ( p ) ≤
3. Let p ∈ S ′ r S . Since p does not split completely in k S ( p ) and since cd G S ( p ) < ∞ , the decomposition group of p in k S ( p ) | k is anon-trivial and torsion-free quotient of Z p ∼ = Gal ( k nr p ( p ) | k p ). Therefore k S ( p ) p is the maximal unramified p -extension of k p . We denote the normalization ofan integral normal scheme Y in an algebraic extension L of its function field by Y L . Then ( X r S ) k S ( p ) is the universal pro- p covering of X r S . We consider the´etale excision sequence for the pair (( X r S ) k S ( p ) , ( X r S ′ ) k S ( p ) ). By assumption, X r S is a K ( π,
1) for p , hence H i et (( X r S ) k S , Z /p Z ) = 0 for i ≥ Z /p Z from the notation, this impliesisomorphisms H i et (cid:0) ( X r S ′ ) k S ( p ) (cid:1) ∼ → M ′ p ∈ S ′ r S ( k S ( p )) H i +1 p (cid:0) (( X r S ) k S ) p (cid:1) for i ≥
1. Here (and in variants also below) we use the notational convention M ′ p ∈ S ′ r S ( k S ( p )) H i +1 p (cid:0) (( X r S ) k S ( p ) ) p (cid:1) := lim −→ K ⊂ k S ( p ) M p ∈ S ′ r S ( K ) H i +1 p (cid:0) (( X r S ) K ) p (cid:1) , where K runs through the finite extensions of k inside k S ( p ). As k S ( p ) real-izes the maximal unramified p -extension of k p for all p ∈ S ′ r S , the schemes(( X r S ) k S ( p ) ) p , p ∈ S ′ r S ( k S ( p )), have trivial cohomology with values in Z /p Z and we obtain isomorphisms H i (( k S ( p )) p ) ∼ → H i +1 p (cid:0) (( X r S ) k S ( p ) ) p (cid:1) for i ≥
1. These groups vanish for i ≥
2. This implies H i et (( X r S ′ ) k S ( p ) ) = 03or i ≥
2. Since the scheme ( X r S ′ ) k S ′ ( p ) is the universal pro- p covering of( X r S ′ ) k S ( p ) , the Hochschild-Serre spectral sequence yields an inclusion H ( Gal ( k S ′ ( p ) | k S ( p ))) ֒ → H et (( X r S ′ ) k S ( p ) ) = 0 . Hence
Gal ( k S ′ ( p ) | k S ( p )) is a free pro- p -group and H ( Gal ( k S ′ ( p ) | k S ( p ))) ∼ → H et (( X r S ′ ) k S ( p ) ) ∼ = M ′ p ∈ S ′ r S ( k S ( p )) H ( k S ( p ) p ) . We set K = k S ( p ) and consider the natural homomorphism φ : ∗ p ∈ S ′ \ S ( K ) T ( K p ( p ) | K p ) −→ Gal ( k S ′ ( p ) | K ) . By the calculation of the cohomology of a free product ([NSW], 4.3.10 and 4.1.4), φ is a homomorphism between free pro- p -groups which induces an isomorphismon mod p cohomology. Therefore φ is an isomorphism. In particular, k S ′ ( p ) p = k p ( p ) for all p ∈ S ′ r S . Using that Gal ( k S ′ ( p ) | k S ( p )) is free, the Hochschild-Serre spectral sequence E ij = H i (cid:0) Gal ( k S ′ ( p ) | k S ( p )) , H j et (( X r S ′ ) k S ′ ( p ) ) (cid:1) ⇒ H i + j et (( X r S ′ ) k S ( p ) )induces an isomorphism0 = H et (( X r S ′ ) k S ( p ) ) ∼ −→ H et (( X r S ′ ) k S ′ ( p ) ) Gal ( k S ′ | k S ) . Hence H et (( X r S ′ ) k S ′ ( p ) ) = 0, since Gal ( k S ′ ( p ) | k S ( p )) is a pro- p -group. Now[Sch2], Proposition 2.1 implies that X r S ′ is a K ( π,
1) for p .In order to prove Theorem 1.1, we first provide the following lemma. Foran extension field K | k and a set of primes T of k , we write T ( K ) for the set ofprolongations of primes in T to K and δ K ( T ) for the Dirichlet density of theset of primes T ( K ) of K . Lemma 2.2.
Let k be a number field, p a prime number and S a finite set ofnonarchimedean primes of k . Let T be a set of primes of k with δ k ( µ p ) ( T ) = 1 .Then there exists a finite subset T ⊂ T such that all primes p ∈ S do not splitcompletely in the extension k T ( p ) | k . Proof.
By [NSW], 9.2.2 (ii) or [NSW ], 9.2.3 (ii), respectively, the restrictionmap H ( G T ∪ S ∪ S p ∪ S R ( p ) , Z /p Z ) −→ Y p ∈ S ∪ S p ∪ S R H ( k p , Z /p Z )is surjective. A class in α ∈ H ( G T ∪ S ∪ S p ∪ S R ( p ) , Z /p Z ) which restricts to anunramified class α p ∈ H nr ( k p , Z /p Z ) for all p ∈ S ∪ S p ∪ S R is contained in H ( G T ( p ) , Z /p Z ). Therefore the image of the composite map H ( G T ( p ) , Z /p Z ) ֒ → H ( G T ∪ S ∪ S p ∪ S R ( p ) , Z /p Z ) → Y p ∈ S H ( k p , Z /p Z )4ontains the subgroup Q p ∈ S H nr ( k p , Z /p Z ). As this group is finite, it is alreadycontained in the image of H ( G T ( p ) , Z /p Z ) for some finite subset T ⊂ T . Weconclude that no prime in S splits completely in the maximal elementary abelian p -extension of k unramified outside T . Proof of Theorems 1.1 and 1.2. As p = 2, we may ignore archimedean primes.Furthermore, we may remove the primes in S ∪ S p and all primes of degree greaterthan 1 from T . In addition, we remove all primes p with N ( p ) p from T . After these changes, we still have δ k ( µ p ) ( T ) = 1.By Lemma 2.2, we find a finite subset T ⊂ T such that no prime in S splitscompletely in k T ( p ) | k . Put X = Spec (Ø k ). By [Sch2], Theorem 6.2, appliedto T and T r T , we find a finite subset T ⊂ T r T such that X r ( T ∪ T ) isa K ( π,
1) for p . Then Proposition 2.1 applied to T ∪ T ⊂ S ∪ T ∪ T , showsthat also X r ( S ∪ T ∪ T ) is a K ( π,
1) for p . Now put T = T ∪ T ⊂ T .It remains to show Theorem 1.2. Assertion (i) holds by construction of T .By [Sch2], Lemma 4.1, also X r ( S ′ ∪ T ) is a K ( π,
1) for p . By [Sch2], Theorem 3,the field k T ( p ) realizes k p ( p ) for p ∈ T , showing (ii) for these primes. Finally,assertion (ii) for p ∈ S follows from Proposition 2.1. We start by investigating the relation between the K ( π, S : If p ∤ p is a prime with ζ p / ∈ k p ,then every p -extension of the local field k p is unramified (see [NSW], 7.5.1 or[NSW ], 7.5.9, respectively). Therefore primes p / ∈ S p with N ( p ) p cannot ramify in a p -extension. Removing all these redundant primes from S ,we obtain a subset S min ⊂ S , which has the property that G S ( p ) = G S min ( p ).Furthermore, by [Sch2], Lemma 4.1, X r S is a K ( π,
1) for p if and only if X r S min is a K ( π,
1) for p . Theorem 3.1.
Let k be a number field and let p be a prime number. Assumethat k is totally imaginary if p = 2 . Let S be a finite set of nonarchimedeanprimes of k . Then any two of the following conditions (a) – (c) imply the third. (a) Spec ( O k ) r S is a K ( π, for p . (b) lim ←− K ⊂ k S ( p ) Ø × K ⊗ Z p = 0 . (c) ( k S ( p )) p = k p ( p ) for all primes p ∈ S min .The limit in (b) runs through all finite extensions K of k inside k S ( p ) . If (a)–(c) hold, then also lim ←− K ⊂ k S ( p ) Ø × K,S min ⊗ Z p = 0 . Remarks:
1. Assume that ζ p ∈ k and S ⊃ S p . Then (a) holds and condition(b) holds for p > S > r + 2 (see [NSW ], Remark 2 after 10.9.3). In the5ase k = Q ( ζ p ), S = S p , condition (b) holds if and only if p is an irregular primenumber.2. Assume that S ∩ S p = ∅ and S min = ∅ . If condition (a) holds, then either G S ( p ) = 1 (which only happens in very special situations, see [Sch2], Proposition7.4) or (b) holds by [Sch2], Theorem 3 (or by Proposition 3.2 below). Proof of Theorem 3.1.
We may assume S = S min in the proof. Let K runthrough the finite extensions of k in k S ( p ) and put X K = Spec (Ø K ). Applyingthe topological Nakayama-Lemma ([NSW], 5.2.18) to the compact Z p -modulelim ←− Ø × K ⊗ Z p , we see that condition (b) is equivalent to(b)’ lim ←− K ⊂ k S ( p ) Ø × K /p = 0.Furthermore, by [Sch2], Proposition 2.1, condition (a) is equivalent to(a)’ lim −→ K ⊂ k S ( p ) H i et (( X r S ) K , Z /p Z ) = 0 for i ≥ i = 1, i ≥
4, and it holds for i = 3 provided that G S ( p ) is infinite or S is nonempty or ζ p / ∈ k (see [Sch2], Lemma 3.7). The flatKummer sequence 0 → µ p → G m · p → G m → −→ Ø × K /p −→ H fl ( X K , µ p ) −→ p Pic ( X ) → K . As the field k S ( p ) does not have nontrivial unramified p -extensions,class field theory implieslim ←− K ⊂ k S ( p ) p Pic ( X K ) ⊂ lim ←− K ⊂ k S ( p ) Pic ( X K ) ⊗ Z p = 0 . As we assumed k to be totally imaginary if p = 2, the flat duality theorem ofArtin-Mazur ([Mil], III Corollary 3.2) induces natural isomorphisms H et ( X K , Z /p Z ) = H fl ( X K , Z /p Z ) ∼ = H fl ( X K , µ p ) ∨ . We conclude that( ∗ ) lim −→ K ⊂ k S ( p ) H et ( X K , Z /p Z ) ∼ = (cid:0) lim ←− K ⊂ k S ( p ) Ø × K /p (cid:1) ∨ . We first show the equivalence of (a) and (b) in the case S = ∅ . If (a)’ holds,then ( ∗ ) shows (b)’. If (b) holds, then ζ p / ∈ k or G S ( p ) is infinite. Hence weobtain (a)’ for i = 3. Furthemore, (b)’ implies (a)’ for i = 2 by ( ∗ ). This finishesthe proof of the case S = ∅ .Now we assume that S = ∅ . For p ∈ S ( K ), a standard calculation of localcohomology shows that H i p ( X K , Z /p Z ) ∼ = i ≤ ,H ( K p , Z /p Z ) /H nr ( K p , Z /p Z ) for i = 2 ,H ( K p , Z /p Z ) for i = 3 . i ≥ . p ∈ S = S min , every proper Galois subextension of k p ( p ) | k p admits ramified p -extensions. Hence condition (c) is equivalent to(c)’ lim −→ K ⊂ k S ( p ) L p ∈ S ( K ) H i p ( X K , Z /p Z ) = 0 for all i ,and to(c)” lim −→ K ⊂ k S ( p ) L p ∈ S ( K ) H p ( X K , Z /p Z ) = 0.Consider the direct limit over all K of the excision sequences · · · → M p ∈ S ( K ) H i p ( X K , Z /p Z ) → H i et ( X K , Z /p Z ) → H i et (( X r S ) K , Z /p Z ) → · · · . Assume that (a)’ holds, i.e. the right hand terms vanish in the limit for i ≥ ∗ ) shows that (b)’ is equivalent to (c)”.Now assume that (b) and (c) hold. As above, (b) implies the vanishing ofthe middle term for i = 2 , ], 10.3.12, respectively)0 → Ø × K → Ø × K,S → M p ∈ S ( K ) ( K × p /U p ) → Pic ( X K ) → Pic (( X r S ) K ) → Z -algebra) Z p , we obtain exact sequences of finitely generated, hencecompact, Z p -modules. Passing to the projective limit over the finite extensions K of k inside k S ( p ) and using lim ←− Pic ( X K ) ⊗ Z p = 0, we obtain the exactsequence0 → lim ←− K ⊂ k S ( p ) Ø × K ⊗ Z p → lim ←− K ⊂ k S ( p ) Ø × K,S ⊗ Z p → lim ←− K ⊂ k S ( p ) M p ∈ S ( K ) ( K × p /U p ) ⊗ Z p → . Condition (c) and local class field theory imply the vanishing of the right handlimit. Therefore (b) implies the vanishing of the projective limit in the middle.If G S ( p ) = 1 and condition (a) of Theorem 1.1 holds, then the failure incondition (c) can only come from primes dividing p . This follows from the next Proposition 3.2.
Let k be a number field and let p be a prime number. Assumethat k is totally imaginary if p = 2 . Let S be a finite set of nonarchimedeanprimes of k . If Spec (Ø k ) r S is a K ( π, for p and G S ( p ) = 1 , then every prime p ∈ S with ζ p ∈ k p has an infinite inertia group in G S ( p ) . Moreover, we have k S ( p ) p = k p ( p ) for all p ∈ S min r S p . roof. We may assume S = S min . Suppose p ∈ S with ζ p ∈ k p does not ramifyin k S ( p ) | k . Setting S ′ = S r { p } , we have k S ′ ( p ) = k S ( p ), in particular, H et ( X r S ′ , Z /p Z ) ∼ −→ H et ( X r S, Z /p Z ) . In the following, we omit the coefficients Z /p Z from the notation. Using thevanishing of H et ( X r S ), the ´etale excision sequence yields a commutative exactdiagram H ( G S ′ ( p )) ∼ / / (cid:127) _ (cid:15) (cid:15) H ( G S ( p )) ≀ (cid:15) (cid:15) H p ( X ) (cid:31) (cid:127) / / H et ( X r S ′ ) α / / H et ( X r S ) / / H p ( X ) / / / / H et ( X r S ′ ) . Hence α is split-surjective and Z /p Z ∼ = H p ( X ) ∼ → H et ( X r S ′ ). This implies S ′ = ∅ , hence S = { p } , and ζ p ∈ k . The same applies to every finite extensionof k in k S ( p ), hence p is inert in k S ( p ) = k ∅ ( p ). This implies that the naturalhomomorphism Gal ( k nr p ( p ) | k p ) −→ G ∅ ( k )( p )is surjective. Therefore G S ( p ) = G ∅ ( p ) is abelian, hence finite by class fieldtheory. Since this group has finite cohomological dimension by the K ( π, p ∈ S with ζ p ∈ k p ramify in k S ( p ). As this applies toevery finite extension of k inside k S ( p ), the inertia groups must be infinite. For p ∈ S min r S p this implies k S ( p ) p = k p ( p ). Theorem 3.3.
Let k be a number field and let p be a prime number. As-sume that k is totally imaginary if p = 2 . Let S be a finite nonempty set ofnonarchimedean primes of k . Assume that conditions (a)–(c) of Theorem 3.1hold and that ζ p ∈ k p for all p ∈ S . Then G S ( p ) is a pro- p duality group ofdimension . Proof.
Condition (a) implies H ( G S ( p ) , Z /p Z ) ∼ → H et ( X r S, Z /p Z ) = 0. Hence cd G S ( p ) ≤
2. On the other hand, by (c), the group G S ( p ) contains Gal ( k p ( p ) | k p )as a subgroup for all p ∈ S . As ζ p ∈ k p for p ∈ S , these local groups have coho-mological dimension 2, hence so does G S ( p ).In order to show that G S ( p ) is a duality group, we have to show that D i ( G S ( p ) , Z /p Z ) := lim −→ U ⊂ G S ( p ) cor ∨ H i ( U, Z /p Z ) ∨ vanish for i = 0 ,
1, where U runs through the open subgroups of G S ( p ) and thetransition maps are the duals of the corestriction homomorphisms; see [NSW],3.4.6. The vanishing of D is obvious, as G S ( p ) is infinite. Using (a), wetherefore have to show thatlim −→ K ⊂ k S ( p ) H (( X r S ) K , Z /p Z ) ∨ = 0 .
8e put X = Spec ( O k ) and denote the embedding by j : ( X r S ) K → X K . Bythe flat duality theorem of Artin-Mazur, we have natural isomorphisms H (( X r S ) K , Z /p Z ) ∨ ∼ = H fl ,c (( X r S ) K , µ p ) = H fl ( X K , j ! µ p ) . The excision sequence together with a straightforward calculation of local co-homology groups shows an exact sequence( ∗ ) M p ∈ S ( K ) K × p /K × p p → H fl ( X K , j ! µ p ) → H fl (( X r S ) K , µ p ) . As ζ p ∈ k p and k S ( p ) p = k p ( p ) for p ∈ S by assumption, the left hand term of( ∗ ) vanishes when passing to the limit over all K . We use the Kummer sequenceto obtain an exact sequence( ∗∗ ) Pic (( X r S ) K ) /p −→ H fl (( X r S ) K , µ p ) −→ p Br (( X r S ) K ) . The left hand term of ( ∗∗ ) vanishes in the limit by the principal ideal theorem.The Hasse principle for the Brauer group induces an injection p Br (( X r S ) K ) ֒ → M p ∈ S ( K ) p Br ( K p ) . As k S ( p ) realizes the maximal unramified p -extension of k p for p ∈ S , the limitof the middle term in ( ∗∗ ), and hence also the limit of then middle term in ( ∗ )vanishes. This shows that G S ( p ) is a duality group of dimension 2. Remark:
The dualizing module can be calculated to D ∼ = tor p (cid:0) C S ( k S ( p ) (cid:1) , i.e. D is isomorphic to the p -torsion subgroup in the S -id`ele class group of k S ( p ).The proof is the same as in ([Sch1], Proof of Thm. 5.2), where we dealt withthe tame case. References [Kuz] L. V. Kuz’min
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