On cohomology of Witt vectors of algebraic integers and a conjecture of Hesselholt
aa r X i v : . [ m a t h . N T ] N ov ON COHOMOLOGY OF WITT VECTORS OF ALGEBRAICINTEGERS AND A CONJECTURE OF HESSELHOLT
AMIT HOGADI AND SUPRIYA PISOLKAR
Abstract.
Let K be a complete discrete valued field of characteristic zerowith residue field k K of characteristic p >
0. Let
L/K be a finite Galois exten-sion with the Galois group G and suppose that the induced extension of residuefields k L /k K is separable. In [3], Hesselholt conjectured that H ( G, W ( O L ))is zero, where O L is the ring of integers of L and W ( O L ) is the Witt ring of O L w.r.t. the prime p . He partially proved this conjecture for a large classof extensions. In this paper, we prove Hesselholt’s conjecture for all Galoisextensions. Introduction
Let p be a prime number and K be a complete discrete valued field of charac-teristic zero with residue field k K of characteristic p . L/K be a finite Galoisextension of complete discrete valued fields as above with Galois group G . Sup-pose that k L /k K is separable. In [3], Hesselholt conjectured that the proabeliangroup lim ←− H ( G, W n ( O L )) vanishes, where W n ( O L ) is the ring of Witt vectors oflength n in O L . As explained in [3], this can be viewed as an analogue of Hilberttheorem 90 for the Witt ring W ( O L ).In order to prove this conjecture one easily reduces to the case where L/K is atotally ramified Galois extension of degree p (see Lemma 4.1). For such extension,let s = s (L / K) be the ramification break (see [1], III, (1.4)) in the ramificationfiltration of Gal(L / K). Hesselholt proved the following theorem, which proves hisconjecture for a large class of extensions.
Theorem 1.1 ([3]) . Let
L/K be a totally ramified cyclic extension of order p with s > e K / ( p − . Then lim ←− H ( G, W n ( O L )) is zero. The proabelian group lim ←− H ( G, W n ( O L )) can be identified with H ( G, W ( O L ))(see Corollary 2.2). The main result of this paper is, Theorem 1.2.
Let
L/K be a finite Galois extension of complete discrete valuedfields with Galois group G . Then H ( G, W ( O L )) is zero. Although the proof of Theorem 1.1 does not generalize (for instance due to useof ([3], 2 . Lemma 1.3. ([3], 1.1)
Let
L/K be as in Theorem 1.2. Let m ≥ be an integerand suppose that the induced map H ( G, W m + n ( O L )) → H ( G, W n ( O L )) is zero for n = 1 . Then the same is true, for all n ≥ . In particular lim ←− H ( G, W n ( O L )) = 0Thus, because of Lemmas 4.1, 1.3 and Corollary 2.2, to prove Theorem 1.2, it isenough to prove the following. Theorem 1.4.
Let K be as above and L / K be degree- p totally ramified cyclicextension with Galois group G . Then there exists a positive integer m ∈ N suchthat the homomorphism H ( G, W m ( O L )) → H ( G, O L ) is equal to zero. Acknowledgement : Our sincere thanks to Prof. C. S. Rajan for his much helpand useful discussions. The second author would also like to thank Joel Riou forhis interest and useful comments.2.
Preliminaries
In this section we show that lim ←− H ( G, W n ( O L )) coincides with H ( G, W ( O L ))(see Corollary 2.2). Note that in general group cohomology does not commutewith inverse limits. Proposition 2.1.
Let G be a finite group and { A i } i ∈ N be an inverse system of G modules indexed by N . For j > i , let φ ji : A j → A i denote the given maps. Thenthe following two statements hold. (i) If φ ji is surjective for all j > i then H ( G, lim ←− A i ) → lim ←− H ( G, A i ) is surjective. (ii) If the induced maps φ Gji : A Gj → A Gi are surjective for all j > i , then H ( G, lim ←− A i ) → lim ←− H ( G, A i ) is injective. Corollary 2.2.
Let
L/K be a finite Galois extension of complete discrete valuedfields. Then the natural map
Φ : H ( G, W ( O L )) → lim ←− H ( G, W n ( O L )) is an isomorphism.Proof. By construction of Witt vectors, the projection maps W n +1 ( O L ) → W n ( O L ) N HESSELHOLT’S CONJECTURE 3 are surjective. Thus by the above proposition, Φ is surjective. In order to proveinjectivity of Φ we need to prove surjectivity of W n +1 ( O L ) G → W n ( O L ) G This follows from the fact that W i ( O L ) G = W i ( O K ) for all i and from the surjec-tivity of the projection maps W n +1 ( O K ) → W n ( O K ). (cid:3) Proof of Proposition 2.1. ( i ) Suppose we are given an element α ∈ lim ←− H ( G, A i ).This is equivalent to given α i ∈ H ( G, A i ) for all i such that α i +1 α i . We nowinductively construct cocycles a ig representing the class α i as follows. For i = 1,choose a g arbitrarily. Now, suppose a ng has been constructed. Then construct a n +1 g as follows. First start with any cocycle b n +1 g which represents α n +1 . For anelement b ∈ A n +1 , let b denote its image in A n . Thus b n +1 g is a cocycle in A n which represents the same class as that represented by a ng . Thus, there exists c ∈ A n such that b n +1 g − a ng = gc − c Since by assumption, A n +1 → A n is surjective, there exists an element d ∈ A n +1 such that d = c . Now define a n +1 g = b n +1 g − ( gd − d )This completes the inductive construction of the cocyles a ig . The cocyles havethe property that for all i and g , a i +1 g a ig and thus they define a cocyles with values in lim ←− A i whose class obviously mapsto the element α we started with.( ii ) Suppose α is a class in H ( G, lim ←− A i ) which maps to zero in lim ←− H ( G, A i ), orequivalently maps to zero in H ( G, A i ) for each i . Under the given assumption wewill show that α = 0. Choose a cocyle a g representing α . By abuse of notation,we will denote the image of a g in A n by a g . The n will be clear from context.For each n , we will now inductively construct an element b n ∈ A n such that a g = gb n − b n ∀ g ∈ G and for all n , b n +1 maps to b n . For n = 1, we know that the image of a g in A isa coboundary. Thus there exists an element b ∈ A such that a g = gb − b ∀ g ∈ G Now suppose we have defined b n . To define b n +1 we first choose an element c n +1 ∈ A n +1 such that a g = gc n +1 − c n +1 ∀ g ∈ G A. HOGADI AND S. PISOLKAR
However the image of c n +1 in A n , denoted by c n +1 satisfies gc n +1 − c n +1 = gb n − b n which means, there exists a d ∈ A Gn such that b n = c n +1 + d Since the map A Gn +1 → A Gn is assumed to be surjective, we can lift d to an element e d ∈ A Gn +1 . Now define b n +1 = c n +1 + e d The b ′ n defined above are compatible elements and hence define an element b oflim ←− A i . Also, from the construction it is clear that a g = gb − b ∀ g ∈ G holds, since it holds after taking image in A i for all i . Thus the cocycle a g isactually a coboundary and hence the class α we started with is trivial. (cid:3) Remarks on addition of Witt vectors
The main observation of this section is Lemma 3.2, which lies at the heart of theproof of Theorem 1.2. We first recall from ([4],II) how addition of Witt vectorsis defined. For every positive integer n , define polynomials w n ∈ Z [ X , ..., X n ] by w n ( X , ..., X n ) = X p n − + pX p n − + p X p n − + · · · + p n − X n One now defines addition of Witt vectors (thanks to Theorem 3.1) in such a waythat if(1) ( X , ..., X n ) + ( Y , ..., Y n ) = ( Z , ..., Z n )then w i ( X , ..., X i ) + w i ( Y , ..., Y i ) = w i ( Z , ..., Z i ) ∀ i ≤ n Theorem 3.1. ([4], II. § For every positive integer n , there exists a unique φ n ∈ Z [ X , .., X n , Y , .., Y n ] such that w n ( X , ..., X n ) + w n ( Y , ..., Y n ) = w n ( φ , ..., φ n ) ∀ n ∈ N In other words, in equation (1) above Z i = φ i ( X , ..., X i , Y , .., Y i )Note that since φ ′ i s are polynomials with integral coefficients, the expressionmakes sense in all characteristics.We now consider addition of p Witt vectors. Let( x , ..., x n ) + ... + ( x p , ..., x pn ) = ( z , ..., z n )By above discussion, for every i ≤ n , there exist polynomials in ni variables, g i ∈ Z [ X , .., X i , ...X p , ..., X pi ] such that z i = g i ( x , ..., x i , ..., x p , ..., x pi ) N HESSELHOLT’S CONJECTURE 5
The following observation is about the nature of these polynomials.
Lemma 3.2.
Let R be a ring, p be a prime, n ∈ N and W n ( R ) be the ring ofWitt vectors of length n . Let x i = ( x i , x i , · · · , x in ) ∈ W n ( R ) for ≤ i ≤ p . Let ( z , z , · · · , z n ) := p X i =1 ( x i , x i , · · · , x in )(1) For all ≤ ℓ ≤ n there exists a polynomial expression f ℓ ∈ Z [ { x ij } ] where ≤ i ≤ p, ≤ j ≤ ℓ − such that z ℓ = p X i =1 x iℓ + f ℓ where each monomial of f l has degree ≥ p . (2) There exists a polynomial expression h ℓ − ∈ Z [ { x ij } ] where ≤ i ≤ p, ≤ j ≤ ℓ − such that f ℓ = P pi =1 x pi,ℓ − − ( P pi =1 x i,ℓ − ) p p + 1 p p − X j =1 (cid:18) pj (cid:19) p X i − x p − ji,ℓ − f jℓ − + h ℓ − and each monomial appearing in h ℓ − has degree ≥ p .Proof. (1) By definition of addition of Witt vectors in Witt ring we have p X i =1 w ℓ ( x i , ..., x pℓ ) = w ℓ ( z , ..., z ℓ )Using the expression for the polynomials w ℓ and rearranging, we get z ℓ = p X i =1 x iℓ + f ℓ where f ℓ = 1 p ℓ − p X i =1 x p ℓ − i − z p ℓ − ! + ... + 1 p p X i =1 x pi ( ℓ − − z pℓ − ! The claim that f ℓ has integral coefficients follows from Theorem 3.1. Note thateach z t , t ≤ ℓ in the above expression is again a polynomial expression in thevariables x ′ ij s, j ≤ t . It is straightforward to observe from the expression of f ℓ that every monomial appearing in the expression has degree ≥ p .(2) Substitute z ℓ − = P pj =1 x j ( ℓ − + f ℓ − in the expression of f ℓ and rewrite f ℓ as f ℓ = (cid:16)P pi =1 x pi ( ℓ − (cid:17) − (cid:0)P pi =1 x i ( ℓ − (cid:1) p p − p p − X j =2 (cid:18) pj (cid:19) p X i =1 x i ( ℓ − ! p − j · f jℓ − + h ℓ − where A. HOGADI AND S. PISOLKAR h ℓ − = − p f pℓ − + p (cid:16) x p ℓ − + x p ℓ − + · · · + x p p ( ℓ − − z p ℓ − (cid:17) + · · · + p ℓ − (cid:16) x p ℓ − + x p ℓ − + · · · + x p ℓ − p − z p ℓ − (cid:17) Note that since p is a prime number, every binomial coefficient (cid:0) pj (cid:1) with 1 ≤ j < p ,is divisible by p . Thus the first two terms in the above expressions of f ℓ haveintegral coefficients. Since we know that f ℓ has integral coefficients, it followsthat h ℓ − has integral coefficients too. Moreover, since all monomials appearingin f ℓ − have degree ≥ p , all monomials appearing in f pℓ − have degree ≥ p . It isalso clear that for 1 ≤ i ≤ ℓ −
2, all monomials in1 p ℓ − i (cid:16) x p ℓ − i i + x p ℓ − i i + · · · + x p ℓ − i pi − z p ℓ − i i (cid:17) have degree ≥ p . This shows that all monomials appearing in the expression of h ℓ − have degree ≥ p . (cid:3) Proof of the main theorem
We will prove Theorem 1.2 in this section.
Lemma 4.1.
Let p be a prime number and L/K be a finite Galois extension ofcomplete discrete fields with G = Gal( L/K ) . Suppose that k L /k K is separable.Then the following two statements are equivalent. (i) H ( G, W ( O L )) = 0 for all extensions L/K as above. (ii) H ( G, W ( O L )) = 0 for all L/K as above which are ramified and of degree p .Proof. ( i ) = ⇒ ( ii ) is obvious. Now we prove ( i ) assuming ( ii ).Let L/K be any Galois extension of complete discrete valued fields. Let L t bethe maximal subfield of L which is tamely ramified over K . The extension L t /K isGalois and let H = Gal( L/L t ). Since L t /K is tame, O L t is a projective O K [ G/H ]module (see [2], I. Theorem(3)) which can be used to show the vanishing of H ( G/H, W ( O L t )). Moreover, because of the following inflation-restriction exactsequence0 → H ( G/H, W ( O L t )) inf −→ H ( G, W ( O L )) res −→ H ( H, W ( O L ))vanishing of H ( G, W ( O L )) is implied by that of H ( H, W ( O L )). Thus withoutloss of generality, we may replace K by L t and assume that our extension L/K istotally wildly ramified Galois extension. Thus G is a p -group. Since any p -grouphas a normal subgroup of index p , again by induction and inflation-restrictionexact sequence, we reduce ourselves to the case when L/K is of degree p . But inthis case the vanishing of H ( G, W ( O L )) is guaranteed by ( ii ). This proves thelemma. (cid:3) N HESSELHOLT’S CONJECTURE 7
Let G be any finite cyclic group with a generator σ . Let M a G -module. Thenthe cohomology group H i ( G, M ) is isomorphic to the i th cohomology group ofthe complex M − σ −→ M tr −→ M − σ −→ M tr −→ M → · · · where for a ∈ M , tr ( a ) = P g ∈ G ga . Thus in the case at hand, where L/K is acyclic Galois extension, we have a canonical isomorphism H ( G, W m ( O L )) ∼ = W m ( O L ) tr =0 / ( σ − W m ( O L )Henceforth, for K as before, we assume L/K is a totally ramified cyclic extensionof degree p . For such an extension we will denote by s the ramification break.To prove the theorem 1.4 we need following lemmas and results from [3]. Lemma 4.2. ([3], 2.4)
Let L / K be as above. Suppose that x ∈ O tr =0 L representsa non-zero class in H (G , O L ) . Then v L ( x ) ≤ s − . Lemma 4.3. ([3], 2.1)
Let L / K be as above. For all a ∈ O L , v K ( tr ( a )) ≥ ( v L ( a ) + s ( p − /p Lemma 4.4. ([3], 2.2.)
Let L / K be as above. For all a ∈ O L , v K ( tr ( a p ) − tr ( a ) p ) = e K + v L ( a ) Lemma 4.5.
Let x = ( x , x , · · · , x n ) ∈ W n ( O L ) tr =0 then for all ≤ ℓ ≤ n − tr ( x ℓ ) = tr ( x pℓ − ) − tr ( x ℓ − ) p p − C.tr ( x ℓ − ) p + h ℓ − where C is the integer defined by C = 1 p p − X j =1 ( − j (cid:18) pj (cid:19) and h ℓ − is a polynomial expression in ( x , · · · , x ℓ − ) and it’s all p − conjugates.Further each monomial of h ℓ − is of degree ≥ p .Proof. Since x ∈ W n ( O L ) tr =0 we have p X i =1 ( σ i − x , ..., σ i − x n ) = (0 , ... x ij = σ i − x j ≤ i ≤ p, ≤ j ≤ n, and z i = 0 ∀ ≤ i ≤ n (cid:3) A. HOGADI AND S. PISOLKAR
Lemma 4.6.
For ℓ ≥ , h ℓ − ∈ O K . Further v K ( h ℓ − ) ≥ p · min { v L ( x i ) | ≤ i ≤ ℓ − } Proof.
The polynomial expression for h ℓ − in x , ..., x n and its conjugates can beseen to be invariant under the Galois action. Hence it belongs to O K . Furthersince h ℓ − is a sum of monomials in x ′ i s, i ≤ ℓ − ≥ p , we have v L ( h ℓ − ) ≥ p · min { v L ( x i ) | ≤ i ≤ ℓ − } The lemma now follows from the fact that v L ( h ℓ − ) = p · v K ( h ℓ − ). (cid:3) Proof of Theorem 1.4.
By the Lemma 4.2, to prove the Theorem 1.4 it is suffi-cient to find M ∈ N such that, for all x = ( x , ..., x M ) ∈ W M ( O L ) tr =0 , v L ( x ) ≥ s .Step(1): Let n be a positive integer and ( x , ..., x n ) ∈ W n ( O L ) tr =0 . We will proveby induction on ℓ that v L ( x ℓ ) ≥ s ( p − p for 1 ≤ ℓ ≤ n − h = 0, tr ( x ) = 0 we have − tr ( x ) = 1 p ( tr ( x p ) − tr ( x ) p )But by Lemma 4.3, v K ( tr ( x )) ≥ s ( p − p . Thus v K ( tr ( x p ) − tr ( x ) p ) − e K = v K ( tr ( x )) ≥ s ( p − p By Lemma 4.4 v K ( tr ( x p ) − tr ( x ) p ) = v L ( x ) + e K . Therefore v L ( x ) ≥ s ( p − p .This proves the claim for ℓ = 1.Now assume that for all i ≤ ℓ − v L ( x i ) ≥ s ( p − p . We will prove v L ( x ℓ ) ≥ s ( p − p .By Lemma 4.5, we have − tr ( x ℓ +1 ) = tr ( x pℓ ) − tr ( x ℓ ) p p − C · tr ( x ℓ ) p + h ℓ − Thus, using Lemma 4.4 we get v L ( x ℓ ) = v K ( tr ( x pℓ ) − tr ( x ℓ ) p p ) ≥ inf { v K ( tr ( x ℓ +1 )) , v K ( C · tr ( x ℓ ) p ) , v K ( h ℓ − ) } Using Lemma 4.3, we have v K ( tr ( x ℓ +1 )) ≥ s ( p − /p and v K ( C · tr ( x ℓ ) p ) ≥ s ( p − N HESSELHOLT’S CONJECTURE 9
By Lemma 4.6, and by induction hypothesis v K ( h ℓ − ) ≥ s ( p − v L ( x ℓ ) ≥ s ( p − p Step(2): Now we will show existance of M ∈ N such that for all x ∈ W M ( O L ), v L ( x ) ≥ s . For any positive integer n and ( x , ..., x n ) ∈ W n ( O L ) tr =0 , by Step(1)we have v L ( x i ) ≥ s ( p − p , ∀ ≤ i ≤ n − . For a fixed n , and 1 ≤ i ≤ n −
1, we claim that v L ( x n − i ) ≥ s ( p − p (1 + 1 p + · · · + 1 p i − )We prove this by induction on i . For i = 1, this is the claim that v L ( x n − ) ≥ s ( p − p which follows from Step(1). Now assuming the claim for a general 1 ≤ i ≤ n − i + 1. By induction hypothesis v L ( x n − i ) ≥ s ( p − p (1 + 1 p + · · · + 1 p i − )Therefore by using Lemma 4.3 we get v K ( tr ( x n − i )) ≥ v L ( x n − i ) + s ( p − p ≥ s ( p − p (1 + 1 p + · · · + 1 p i )By Lemma 4.5 − tr ( x n − i ) = tr ( x pn − ( i +1) ) − tr ( x n − ( i +1) ) p p − C · tr ( x n − ( i +1) ) p + h n − ( i +2) By Lemma 4.3, v K ( C · tr ( x n − ( i +1) ) p ) ≥ s ( p − v K ( h n − ( i +2) ) ≥ s ( p − v L ( x n − ( i +1) ) = v K (cid:16) tr ( x pn − ( i +1) ) − tr ( x n − ( i +1) ) p p (cid:17) ≥ min { v K ( tr ( x n − i )) , v K ( C · tr ( x n − ( i +1) ) p ) , v K ( h n − ( i +2) ) }≥ min { s ( p − p (1 + p + · · · + p i ) , s ( p − , s ( p − } = s ( p − p (1 + p + · · · + p i )This proves the claim. Hence v L ( x ) ≥ s ( p − p (1 + 1 p + · · · + 1 p n − ) We know that as n → ∞ , s ( p − p (1 + p + · · · + p n − ) → s . There exists an integer M , such that s ( p − p (1 + 1 p + · · · + 1 p M − ) > s − v L is a discrete valuation, for such M and for any ( x , ..., x M ) ∈ W M ( O L ) tr =0 ,we have shown that v L ( x ) ≥ s (cid:3) References [1] Fesenko, I. B., Vostokov, S. V.;
Local fields and their extensions . Second edition. Trans-lations of Mathematical Monographs, . American Mathematical Society, Providence,RI, 2002.[2] Fr¨ohlich, Albrecht;
Galois module structure of algebraic integers . . Springer-Verlag,Berlin, 1983.[3] Lars Hesselholt; Galois cohomology of Witt vectors of algebraic integers. Math. Proc.Cambridge Philos. Soc. (2004), no. , 551557.[4] Jean-Pierre Serre; Local fields . Translated from the French by Marvin Jay Greenberg. GTM . Springer-Verlag, New York-Berlin, 1979. School of Mathematics, Tata Institute of Fundamental Research, Homi BhabhaRoad, Bombay 400005, India
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