Hilbert 90 for Galois cohomology

0.17 MB·English

by  Nicole Lemire

HILBERT 90 FOR GALOIS COHOMOLOGY
6
0 NICOLE LEMIRE†, JA´N MINA´Cˇ‡, ANDREW SCHULTZ, AND JOHN
0 SWALLOW
2

n
Abstract. Assuming the Bloch-Kato Conjecture, we determine
a
J precise conditions under which Hilbert 90 is valid for Milnor k-

7 theory and Galois cohomology. In particular, Hilbert 90 holds for
2 degreen whenthe cohomologicaldimensionofthe Galoisgroupof
the maximal p-extension of F is at most n.
]
T
N
.
h
The key to the Bloch-Kato Conjecture is Hilbert 90 for Milnor K-
t
a
theory for cyclic extensions E/F of degree p. It is desirable to know
m
when Hilbert 90 holds for Galois cohomology Hn(E,F ) as well. In
[ p
this paper we develop precise conditions under which Hilbert 90 holds
1
for Galois cohomology.
v
3
6
Let p be a prime number, E/F a cyclic extension of degree p with
6
1 group G, and assume that F contains a primitive pth root of unity ξ .
p
0
Let σ be a generator of G. We say that Galois cohomology satisfies
6
0 Hilbert 90 for n N if the following sequence is exact:
∈
/
h
at Hn(E,F ) σ−1 Hn(E,F ) corE/F Hn(F,F ). (1)
p p p
m −−→ −−−−→
:
v We say then that (h90) is valid.
n
i
X
r In section 1 we formulate our results. We recall some basic results
a
relatedtothe Bloch-KatoConjecture insection2, andinsections 3and
4 we prove our results. In section 5 we compare (h90) with vanishing
n
ofthefirst cohomologyofF [G]-moduleHn(E,F ). Werefer thereader
p p
to [Lo] and the references therein for interesting comments on classical
Hilbert 90 and its extensions by Speiser and Noether.
Date: January 25, 2006.
†Research supported in part by NSERC grant R3276A01.
‡ResearchsupportedinpartbyNSERCgrantR0370A01,andbyaDistinguished
Research Professorshipat the University of Western Ontario.
1
2 LEMIRE, MINA´Cˇ,SCHULTZ, ANDSWALLOW
1. Conditions for Hilbert 90 for Galois cohomology
Choose a F such that E = F(√p a). To simplify notation we ab-
breviate Hn(∈E,F ) and Hn(F,F ) by Hn(E) and Hn(F), respectively,
p p
and express cup products α β as α.β.
∪
Theorem 1. Suppose that p = 2 and n N. Then (h90) is valid if
n
∈
and only if
Hn(F) = corHn(E)+(a).Hn−1(F).
Let E× := E 0 and suppose still that p = 2. If a is a sum of two
\{ }
squares in F then a N E×, and by the projection formula (see
E/F
∈
[NSW, Prop. 1.5.3iv]) we obtain (a).Hn−1(F) corHn(E). Therefore
⊂
we have thefollowing corollary. (Observe thataisa sum oftwo squares
in F if and only if 1 N E×.)
E/F
− ∈
Corollary 1.1. Suppose that p = 2, n N, and a F2 +F2. Then
∈ ∈
(h90) is valid if and only if cor : Hn(E) Hn(F) is surjective.
n
→
Theorem 2. Suppose that p > 2 and n N. Then (h90) is valid if
n
∈
and only if
Hn(F) = corHn(E)+(ξ ).Hn−1(F).
p
Nowifξ N E×,thenbytheprojectionformula,(ξ ).Hn−1(F)
p E/F p
∈ ⊂
corHn(E). Therefore we have
Corollary 2.1. Suppose that p > 2 and ξ N E×. Then (h90)
p E/F n
∈
is valid if and only if cor : Hn(E) Hn(F) is surjective.
→
Corollaries 1.1 and 2.1 together imply
Corollary 3. Suppose that n N and ξ N E×. Then (h90) is
p E/F n
∈ ∈
valid if and only if cor : Hn(E) Hn(F) is surjective.
→
Now let F(p) be the compositum of all finite Galois p-power exten-
sions of F in a fixed separable closure of F, and consider the cohomo-
logical dimension of G (p) := Gal(F(p)/F). If cdG (p) n, then by
F F
≤
[NSW, Prop. 3.3.8] we have the surjectivity of cor : Hn(E) Hn(F).
→
In fact, since ξ F, the statement cdG (p) n is equivalent to the
p F
∈ ≤
surjectivity of cor on Hn(E) for all cyclic extensions E of F of degree
p [LLMS, Thm. 2]. Moreover, if a primitive p2th root of unity is in F,
then for a suitable choice of the p2th root of unity ξp2 F we obtain
∈
NE/F(ξp2) = ξp for every cyclic extension E/F of degree p. Hence from
Theorems 1 and 2 and Corollaries 1.1 and 2.1 we have
HILBERT 90 FOR GALOIS COHOMOLOGY 3
Corollary 4. Suppose that n N. If cdG (p) n, then (h90) is
F n
∈ ≤
valid for all cyclic extensions E/F of degree p. Moreover, if ξp2 F
∈
and if (h90) is valid for all cyclic extensions E/F of degree p, then
n
cdG (p) n.
F
≤
We present two further results: an interpretation of (h90) in terms
n
ofdirectsummandsofHn(E)definedoverF,andahereditaryproperty
of (h90) .
n
Theorem 5. Suppose that n N. Then (h90) is valid if and only if
n
∈
for each pair Q Hn(F), P Hn(E) satisfying
⊂ ⊂
Hn(E) = resQ P,
⊕
we have resQ = 0 .
{ }
Theorem 6. Suppose that n N. Then if (h90) is valid, (h90) is
n m
∈
valid for all m n.
≥
At the end of the paper we compare (h90) with H1(G,Hn(E)) =
n
0 .
{ }
2. Bloch-Kato and Milnor K-theory
For i 0, let K F denote the ith Milnor K-group of the field F,
i
≥
with standard generators denoted by f f , f ,...,f F×.
1 i 1 i
{ }···{ } ∈
(See [Mi] and [FV, Chap. IX].) We use the usual abbreviation k F for
n
K F/pK F. For an extension of fields E/F, we use i for the natural
n n E
inclusions of K-theory and k-theory, and we also use N for both
E/F
the norm map K E K F and the induced map k E k F.
n n n n
→ →
We prove our results first for Milnor k-theory, using Hilbert 90 for
Milnor K-theory, and then we use the Bloch-Kato Conjecture to iden-
tify k F and Hn(F). (See [V1, Lemma 6.11 and 7] and [V2, 6 and
n
§ §
Theorem 7.1]. For further expositions of the work of Rost and Voevod-
sky on Bloch-Kato Conjecture, see [Ro], [MVW], and [Su].)
We say that Milnor k-theory satisfies Hilbert 90 at n N for a
∈
cyclic extension E/F of degree p with Gal(E/F) = σ if the following
h i
sequence is exact:
k E σ−1 k E NE/F k F. (2)
n n n
−−−→ −−−→
4 LEMIRE, MINA´Cˇ,SCHULTZ, ANDSWALLOW
We say then that (h90)M is valid. By the Bloch-Kato Conjecture,
n
there exists a G-equivariant isomorphism k E Hn(E), and there-
n
→
fore(h90)M isequivalent to(h90) . Todetermineconditionsfor(1)itis
n n
then sufficient to determine conditions for (2). Since the G-equivariant
isomorphism sends products to cup products, conditions for k E ex-
n
pressed in terms of products will carry over to the analogous conditions
for Hn(E) expressed in terms of cup products.
We use the following two results in Voevodsky’s work on the Bloch-
Kato Conjecture. Because we apply Voevodsky’s results in the case
when the base field contains a primitive pth root of unity we give for-
mulations restricted to this case. The first result is Hilbert 90 for
Milnor K-theory.
Theorem 7 ([V1, Lemma 6.11 and 7] and [V2, 6 and Thm. 7.1]).
Let F be a field containing a primitiv§e pth root of§unity and m N.
∈
For any cyclic extension E/F of degree p with Gal(E/F) = σ , the
h i
sequence
K E σ−1 K E NE/F K F
m m m
−−−→ −−−→
is exact.
As is standard, we then have the so-called “Small Hilbert 90 for k ”:
n
Corollary 7.1. Assume the hypotheses of Theorem 7. Then the se-
quence
N
(σ 1)k E +i k F k E E/F k F
m E m m m
− −→ −−−→
is exact. In particular,
(h90)M i k F (σ 1)k E.
m ⇐⇒ E m ⊂ − m
Proof. Since N (σ 1) = 0 on k E and N is multiplication by p
E/F m E/F
−
on i k F, our sequence is a complex.
E n
Now let α K E and write α¯ for the class of α in k E. Suppose
m m
∈
that N α¯ = 0 k F. Then N α = pβ for some β K F.
E/F m E/F m
∈ ∈
Consider α′ = α i (β). Then N α′ = 0 and by Theorem 7 there
E E/F
−
exists γ K F such that (σ 1)γ = α′ = α i (β). Then modulo p
m E
we have∈(σ 1)γ¯ = α¯ i (β¯)−and so α¯ = (σ −1)γ¯ +i (β¯). (cid:3)
E E
− − −
The following theorem is a strengthening of [V1, Prop. 5.2]. Again
a F× is chosen so that E = F(√p a).
∈
HILBERT 90 FOR GALOIS COHOMOLOGY 5
Theorem 8 ([LMS2, Thm. 6]). Let F be a field containing a primitive
pth root of unity. Then for any cyclic extension E/F of degree p and
m N the sequence
∈
k E NE/F k F {a}·− k F iE k E
m−1 m−1 m m
−−−→ −−−−→ −→
is exact.
3. Proofs of Theorems 1 and 2
Proof of Theorem 1. By Corollary 7.1, (h90)M is valid if and only if
n
i k F (σ 1)k E = (σ +1)k E = i N k E.
E n n n E E/F n
⊂ −
Hence (h90)M is valid if and only if k F = N k E + keri . By
n n E/F n E
Theorem 8, keri = a k F. Therefore
E n−1
{ }·
(h90)M k F = N k E + a k F.
n ⇐⇒ n E/F n { }· n−1
(cid:3)
Using the Bloch-Kato Conjecture, the theorem follows.
Remark 1. The theorem also follows from Arason’s long exact se-
quence (see [A]). Therefore when p = 2 we can deduce (h90) from
n
basic Galois cohomology, without reference to the Bloch-Kato Conjec-
ture.
Proposition 1. Let F be a field containing a primitive pth root ξ
p
of unity and n N. For any cyclic extension E/F of degree p with
∈
Gal(E/F) = σ ,
h i
(σ 1)k E (k E)G = i ( ξ k F)+i N k E.
n n E p n−1 E E/F n
− ∩ { }·
Beforeproving Proposition1we introducesome further notationand
establish Lemma 1 below.
For y k E 0 , set the length l(y) of y to be
n
∈ \{ }
l(y) := max t N : (σ 1)t−1y = 0 .
{ ∈ − 6 }
Observe that since 1+σ + +σp−1 = (σ 1)p−1 in F G, i N =
p E E/F
··· −
(σ 1)p−1 on k E and therefore l(y) p.
n
− ≤
Since ξ k F = ξc k F for (p,c) = 1, we assume without
{ p}· n−1 { p}· n−1
loss of generality for the proofs of the proposition and the following
σ−1
lemma that √p a = ξ .
p
6 LEMIRE, MINA´Cˇ,SCHULTZ, ANDSWALLOW
Lemma 1. Suppose y k E with l = l(y) 2. Then if l 3,
n
∈ ≥ ≥
(σ 1)l−1(y) i N k E,
E E/F n
− ∈
and if l = 2
(σ 1)y i ( ξ k F +N k E).
E p n−1 E/F n
− ∈ { }·
Proof. If l = p, then (σ 1)p−1k E = i N k E shows the result in
n E E/F n
−
this case. Hence we may assume that p > 2.
Suppose l < p. Then y ker(σ 1)p−1 and so i N (y) = 0. By
E E/F
∈ −
Theorem 8, there exists b k F such that N y = a b. By the
n−1 E/F
∈ { }·
projection formula [FW, p. 81],
N (cid:0)y √p a i (b)(cid:1) = 0.
E/F E
−{ }·
Then by Corollary 7.1, there exist ω k E and f k F such that
n n
∈ ∈
y = (σ 1)ω + √p a i (b)+i f
E E
− { }·
and hence
(σ 1)l−1y = (σ 1)lω +(σ 1)l−2i ( ξ b).
E p
− − − { }·
If l(y) 3 we deduce
≥
(σ 1)l−1y = (σ 1)lω
− −
where l(ω) = l+1. Set y = ω and repeat the argument. We obtain
l+1
y k E of lengths l k p with
k n
∈ ≤ ≤
(σ 1)l−1y = (σ 1)k−1y .
k
− −
Take α = y to obtain
p
(σ 1)l−1y i N k E,
E E/F n
− ∈
as required.
If l(y) = 2 we have that
2
(σ 1)y = (σ 1) ω +i ( ξ b)
E p
− − { }·
for some ω k E and some b k F. We see that l(ω) 3. If
n n−1
∈ ∈ ≤
l(ω) < 3 then
(σ 1)y i ( ξ k F),
E p n−1
− ∈ { }·
while if l(ω) = 3 then by the previous case we see that (σ 1)2ω
i N k E, so the result holds in either case. − (cid:3)∈
E E/F n
HILBERT 90 FOR GALOIS COHOMOLOGY 7
Proof of Proposition 1. The right-hand side is contained in i k F and
E n
so is fixed by G. Furthermore,
i N k E = (σ 1)p−1k E (σ 1)k E.
E E/F n n n
− ⊂ −
Let f k F. Then because
n−1
∈
(σ 1)( √p a i (f)) = i ( ξ f),
E E p
− { }· { }·
we have i ( ξ k F) (σ 1)k E, and so
E p n−1 n
{ }· ⊂ −
i ( ξ k F)+i N k E (σ 1)k E (k E)G.
E p n−1 E E/F n n n
{ }· ⊂ − ∩
Now let 0 = x (k E)G (σ 1)k E. Then x = (σ 1)γ for some
n n
γ k E wi6 th l(∈γ) = 2. Th∩e res−ult follows from Lemma−1. (cid:3)
n
∈
Proof of Theorem 2. By Corollary 7.1, (h90)M is valid if and only if
n
i k F (σ 1)k E, or, equivalently,
E n n
⊂ −
(h90)M (σ 1)k E i k F = i k F.
n ⇐⇒ − n ∩ E n E n
By Proposition 1,
(σ 1)k E i k F = i ( ξ k F)+i N k E.
n E n E p n−1 E E/F n
− ∩ { }·
Hence (h90)M is valid if and only if
n
i k F = i ( ξ k F +N k E).
E n E p n−1 E/F n
{ }·
Since p > 2, a a = 0. Then by Theorem 8, keri N k E.
E E/F n
{ } · { } ⊂
Hence
(h90)M k F = ξ k F +N k E.
n ⇐⇒ n { p}· n−1 E/F n
(cid:3)
Using the Bloch-Kato Conjecture, the theorem follows.
4. Proofs of Theorems 5 and 6
Werecall someresults onF [G]-modulesfor Ga cyclic groupoforder
p
p. The indecomposable F [G]-modules are precisely the cyclic F [G]-
p p
modules V := F [G]/ (σ 1)i of dimensions i = 1,...,p; hence V
i p i
is annihilated by (σ h1)i−butinot by (σ 1)i−1. The trivial F [G]-
p
module V F is the−unique simple F [G]-−module up to isomorphism.
1 p p
≃
Recall that a semisimple module is any direct sum, possibly infinite,
of simple modules. For each i, the fixed submodule VG of V is (σ
i i −
1)i−1V, and each V has finite composition length and therefore (see
i
[AF, Lemma 12.8]) its endomorphism ring is local.
Proposition 2. Let M be an F [G]-module. Then M = S T where
p
⊕
S is a maximal semisimple direct summand and TG (σ 1)T.
⊂ −
8 LEMIRE, MINA´Cˇ,SCHULTZ, ANDSWALLOW
Proof. Since F [G] is an Artinian principal ideal ring, every F [G]-
p p
module W is a direct sum of cyclic F [G]-modules [SV, Thm. 6.7].
p
By the Krull-Schmidt-Azumaya Theorem (see [AF, Thm. 12.6]), all
decompositions of W into direct sums of indecomposable modules are
equivalent. Now decompose M = M where for each i = 1,...,p, M
i i
is a direct sum of cyclic F [G]-mod⊕ules V of dimension i. Set S = M ,
p i 1
T = M M . Then S is a semisimple direct summand and
2 p
⊕ ··· ⊕
T contains no nonzero semisimple direct summand. Hence S is maxi-
mal. Since for each i 2, T is a direct sum of modules V satisfying
i i
VG = (σ 1)i−1V , we≥have that T satisfies TG (σ 1)T. (cid:3)
i − i ⊂ −
Proof of Theorem 5. Assume first that (h90)M is valid, and suppose
n
that k E = i Q P for Q k F, P k E. Then N i Q = pQ =
n E n n E/F E
⊕ ⊂ ⊂
0 . By (h90)M,
{ } n
i Q (σ 1)k E = (σ 1)i Q (σ 1)P = (σ 1)P P.
E n E
⊂ − − ⊕ − − ⊂
Hence i Q i Q P = 0 , as desired.
E E
⊂ ∩ { }
For the other direction, assume that if k E = i Q P for Q k F,
n E n
⊕ ⊂
P k E, then i Q = 0 . By Proposition 2 we may write k E =
n E n
⊂ { }
S T where S is a maximal semisimple direct summand and TG
⊕ ⊂
(σ 1)T. We claim that U = i k F S = 0 , as follows. Because
E n
− ∩ { }
S is semisimple there exists V such that U V = S. Let Q k F be
n
⊕ ⊂
the inverse image of U under i . Then k E = i Q (V T) so that
E n E
⊕ ⊕
U = 0 , as desired.
{ }
Nowlet f k F bearbitrary, andwritex := i f = s+talongS T.
n E
∈ ⊕
Since (σ 1)x = 0, we see that (σ 1)t = 0 and t TG (σ 1)T.
− − ∈ ⊂ −
By Proposition 1, t i k F and therefore s = x t i k F. But
E n E n
∈ − ∈
since U = 0 , s = 0. Hence i k F (σ 1)k E. By Corollary 7.1,
E n n
{ } ⊂ −
we deduce that (h90)M is valid.
n
(cid:3)
Using the Bloch-Kato Conjecture, the theorem follows.
Remark 2. The same argument as in the first paragraph of the proof
shows directly that if (h90) is valid and Hn(E) = resQ P for Q
n
⊕ ⊂
Hn(F) and P Hn(E), then resQ = 0 .
⊂ { }
Proof of Theorem 6. Let n N and assume first that p > 2. We prove
∈
the result by induction on m. The base case m = n is given. Assume
then that (h90)M holds. By Theorem 2,
m
k F = N k E + ξ k F.
m E/F m p m−1
{ }·
HILBERT 90 FOR GALOIS COHOMOLOGY 9
Then each element in k F is a sum of the form f r+ g ξ s
m+1 p
{ }· { }·{ }·
where f,g F×, r N k E, and s k F.
E/F m m−1
∈ ∈ ∈
Bytheprojectionformula(see[FV,Thm.3.8]), f r N k E.
E/F m+1
{ }· ∈
Moreover, g ξ s = ξ g−1 s ξ k F. Therefore
p p p m
{ }·{ }· { }·{ }· ∈ { }·
k F = N k E + ξ k F.
m+1 E/F m+1 p m
{ }·
By the proof of Theorem 2, we have (h90)M . Using the Bloch-Kato
m+1
Conjecture, the theorem for p > 2 follows.
The case p = 2 follows by replacing ξ with a and Theorem 2 with
p
(cid:3)
Theorem 1 in the argument above.
5. Hilbert versus Noether and Speiser
Itisinterestingtocompare(h90) withtheconditionH1(G,Hn(E)) =
n
0 . Write ann (a) for the annihilator of the cup-product with (a) in
n
{ }
Hn(F).
Theorem 9. Suppose n N. Then
∈
H1(G,Hn(E)) = 0 = (h90) .
n
{ } ⇒
If p > 2 and n N, the following are equivalent:
∈
(1) H1(G,Hn(E)) = 0
(2) Hn(E) is a free F{[G}]-module
p
(3) cor : Hn−1(E) Hn−1(F) is surjective.
→
If p = 2 and n N, the following are equivalent:
∈
(1) H1(G,Hn(E)) = 0
(2) Hn(E) is a free F{[G}]-module
p
(3) Hn(E) = ann (a) (a).Hn−1(F).
n
⊕
Proof. If H1(G,Hn(E)) = 0 then (σ 1)Hn(E) = kerN, where the
{ } −
endomorphism N : Hn(E) Hn(E) is defined by N = rescor. Since
→
kerN kercor, we obtain (h90) .
n
⊇
The equivalences (1) (2) follow from [L, III.1, Prop. 1.4]. The
⇔ § (cid:3)
equivalences (2) (3) follow from [LMS, Thm. 1].
⇔
10 LEMIRE, MINA´Cˇ,SCHULTZ, ANDSWALLOW
6. Acknowledgements
AndrewSchultzwouldliketothankRaviVakilforhisencouragement
and direction in this and all other projects. John Swallow would like
to thank Universit´e Bordeaux I for its hospitality during 2005–2006.
References
[A] J. Kr. Arason. Cohomologische invarianten quadratischer Formen. J. Al-
gebra 36 (1975), 448–491.
[AF] F. Anderson and K. Fuller. Rings and categories of modules. Graduate
Texts in Mathematics 13. New York: Springer-Verlag, 1973.
[FV] I.FesenkoandS.Vostokov.Localfieldsandtheirextensions,2nded.Trans-
lationsofMathematicalMonographs121.Providence,RI:AmericanMath-
ematical Society, 2002.
[FW] E. M. Friedlander and C. W. Weibel. An overview of algebraic K-theory.
Algebraic K-theoryand its applications (Trieste, 1997),editedbyH.Bass,
A.O.Kuku,andC.Pedrini,1–119.RiverEdge,NJ:WorldSci.Publishing,
1999.
[L] S. Lang. Topics in cohomology of groups. Lecture Notes in Math., 1625.
New York: Springer-Verlag,1996.
[LLMS] J.Labute,N.Lemire,J.Min´aˇc,andJ.Swallow.Cohomologicaldimension
andSchreier’sformulainGaloiscohomology.Can. Math. Bull., toappear.
[LMS] N. Lemire, J. Min´aˇc, and J. Swallow. When is Galois cohomology free or
trivial? New York J. of Math. 11 (2005), 291–302.
[LMS2] N. Lemire, J. Min´aˇc, and J. Swallow. Galois module struc-
ture of Galois cohomology and partial Euler-Poincar´e characteristics.
arXiv:Math.NT/0409484v3 (2006).
[Lo] F. Lorenz. Ein Scholion zum Satz 90 von Hilbert. Anh. Math. Sem. Univ.
Hamburg 68 (1998), 347–362.
[MVW] C. Mazza, V. Voevodsky, and Ch. Weibel. Lecture notes on motivic coho-
mology. math.rutgers.edu/˜weibel/MVWnotes/xprova.pdf(2005).
[Mi] J. Milnor. Algebraic K-theory and quadratic forms. Invent. Math. 9
(1970), 318–344.
[NSW] J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields.
Berlin: Springer-Verlag,2000.
[Ro] M. Rost. Chain lemma for symbols. www.math.uni-bielefeld.de/˜rost/
chain-lemma.html.
[SV] D. Sharpe and P. Va´mos. Injective modules. Cambridge Tracts in Math-
ematics and Mathematical Physics 62. London: Cambridge University
Press, 1972.
[Su] A. Suslin. Norm varieties. Notes by Seva Joukhovitski.K-theory preprint
archive, no. 742 (2005). www.math.uiuc.edu/K-theory/0742/.
[V1] V. Voevodsky. Motivic cohomology with Z/2-coefficients. Publ. Inst.
Hautes E´tudes Sci. 98 (2003), 59–104.
[V2] V. Voevodsky. On motivic cohomology with Z/l-coefficients. K-theory
preprint archive 639. www.math.uiuc.edu/K-theory/0639/(2003).