Chow group of 0-cycles on surface over a p-adic field with infinite torsion subgroup

0.2 MB·English

by  Masanori Asakura

Chow group of 0-cycles on surface over a p-adic field
with infinite torsion subgroup
Masanori Asakura and Shuji Saito
7
0 Contents
0
2
1 Introduction 1
n
a
J 2 Review on cycle class map and ℓ-adic regulator 4

0
3
3 Counterexample to Bloch-Kato conjecture over p-adic field 7

]
G
4 Proof of Theorem 1.1 10
A
h. 5 Appendix: SK1 of curve over p-adic field 12
t
a
m
Abstract
[

2 We give an example of a projective smooth surface X over a p-adic field K such
v
that for any prime ℓ different from p, the ℓ-primary torsion subgroup of CH (X),
1 0
6 the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving
1 a variant of the Block-Kato conjecture which characterizes the image of an ℓ-adic
1
regulator map from a higher Chow group to a continuous ´etale cohomology of X by
1
6 using p-adic Hodge theory. By aid of theory of mixed Hodge modules, we reduce
0 the problem to showing the exactness of de Rham complex associated to a variation
/
h of Hodge structure, which follows from Nori’s connectivity theorem. Another key
at ingredient is the injectivity result on ´etale cycle class map for Chow group of 1-
m cycles on a proper smooth model of X over the ring of integers in K of due to K.
: Sato and the second author.
v
i
X
r
a
1 Introduction
Let X be a smooth projective variety over a base field K and let CHm(X) be the Chow
group of algebraic cycles of codimension m on X modulo rational equivalence. In case K
is a number field, there is a folklore conjecture that CHm(X) is finitely generated, which
in particular implies that its torsion part CHm(X) is finite. The finiteness question
tor
has been intensively studied by many authors, particularly for the case m = 2 and m =
dim(X) (see nice surveys [23] and [7]).
1
When K is a p-adic field (namely the completion of a number field at a finite place),
RosenschonandSrinivas[25]haveconstructedthefirstexamplethatCHm(X) isinfinite.
tor
They proves that there exists a smooth projective fourfold X over a p-adic field such that
the ℓ-torsion subgroup CH (X)[ℓ] (see Notation) of CH (X), the Chow group of 1-cycles
1 1
on X, is infinite for each ℓ 5,7,11,13,17 .
∈ { }
A purpose of this paper is to give an example of a projective smooth surface X over
a p-adic field such that for any prime ℓ different from p, the ℓ-primary torsion subgroup
CH (X) ℓ (see Notation) of CH (X), the Chow group of 0-cycles on X, is infinite.
0 0
{ }
Here we note that for X as above, CH (X) ℓ is known to be always cofinite type over
0
{ }
Z (namely the direct sum of a finite group and a finite number of copies of Q /Z .
ℓ ℓ ℓ
The fact follows from Bloch’s exact sequence (2.3)). Thus our example presents infinite
phenomena of different nature from the example in [25]. Another noteworthy point is
that the phenomena discovered by our example happens rather generically.
To make it more precise, we prepare a notion of ‘generic surfaces’ in P3. Let
M P(H0(P3, (d))) = P(d+3)(d+2)(d+1)/6−1
⊂ Q OP ∼ Q
bethemodulispaceoverQofthenonsingularsurfacesinP3 (thesubscription‘Q’indicates
Q
the base field), and let
f : M
X −→
be the universal family over M. For X P3 , a nonsingular surface of degree d defined
⊂ K
over a field K of characteristic zero, there is a morphism t : SpecK M such that
→
X = SpecK. We call X generic if t is dominant (i.e. t factors through the generic
∼ M
X ×
point of M). In other words, X is generic if it is defined by an equation
F = a zI (a K)
I I
∈
I
X
([z : z : z : z ] is the homogeneous coordinate of P3, I = (i , ,i ) are multi-indices
0 1 2 3 0 3
···
and zI = zi0 zi3) satisfying the following condition:
0 ··· 3
( ) a = 0 for I and a /a are algebraically independent over Q where I =
∗ I 6 ∀ { I I0}I6=I0 0
(1,0,0,0).
The main theorem is:
Theorem 1.1 Let K be a finite extension of Q and X P3 a nonsingular surface of
p ⊂ K
degree d 5. Suppose that X is generic and has a projective smooth model X P3
≥ OK ⊂ OK
over the ring of integers in K. Let r be the rank of the N´eron-Severi group of the
K
O
smooth special fiber. Then we have
CH (X) ℓ = (Q /Z )⊕r−1 (finite group)
0 ∼ ℓ ℓ
{ } ⊕
for ℓ = p.
6
For example let a K∗ be elements such that a are algebraically independent
I I I
∈ { }
over Q and ord (a ) > 0. Then the surface X P3 defined by an equation
K I ⊂ K
zd zd +zd zd + a zI with d 5
0 − 1 2 − 3 I ≥
I
X
2
satisfies the assumption in Theorem 1.1 and r 2 and hence CH (X) has an infinite
0
≥
torsion subgroup. Theorem 1.1 may be compared with the finiteness results [8] and [24]
onCH (X) forasurfaceX overap-adicfieldundertheassumptionthatH2(X, ) = 0
0 tor X
O
or that the rank of the N´eron-Severi group does not change by reduction.
A distinguished role is played in the proof of Theorem 1.1 by the ℓ-adic regulator map
ρ : CH2(X,1) Q H1 (Spec(K),H2(X ,Q (2))) (X = X K)
X ⊗ ℓ −→ cont K ℓ K ×K
from higher Chow group to continuous ´etale cohomology ([18]), where K is an algebraic
closure of K and ℓ is a prime different from ch(K). It is known that the image of ρ is
X
contained in the subspace
H1(Spec(K),V) H1 (Spec(K),V) (V = H2(X ,Q (2)))
g ⊂ cont K ℓ
introduced by Bloch and Kato [5]. In case ℓ = p this is obvious since H1 = H1 by
6 g
definition. For ℓ = p this is a consequence of a fundamental result in p-adic Hodge theory,
which affirms that every representation of G = Gal(K/K) arising from cohomology of
K
a variety over K is a de Rham representation (see a discussion after [5], (3.7.4)).
When K is a number field or a p-adic field, it is proved in [27] that CH2(X) ℓ is finite
{ }
in case the image of ρ coincides with H1(Spec(K),V). Bloch and Kato conjecture that
X g
it should be always the case if K is a number field.
The first key step in the proof of Theorem 1.1 is to disprove the variant of the Block-
Kato conjecture for a generic surface X P3 over a p-adic field K (see Theorem 3.6). In
⊂ K
terms of Galois representations of G = Gal(K/K), our result implies the existence of a
K
1-extension of Q -vector spaces with continuous G -action:
ℓ K
( ) 0 H2(X ,Q (2)) E Q 0,
∗ → K ℓ → → l →
such that E is a de Rham representation of G but that there is no 1-extension of motives
K
over K:
0 h2(X)(2) M h(Spec(K)) 0
→ → → →
which gives rise to ( ) under the realization functor. The main ingredient in the proof of
∗
the first key result is Nori’s connectivity theorem (cf. [16]). This is done in 3 after in 2
§ §
we review some basic facts on cycle class map for higher Chow groups.
Another key ingredient is the injectivity result on´etalecycle class map forChow group
of 1-cycles on a proper smooth model of X over the ring O of integers in K due to Sato
K
and the second author [26]. It plays an essential role in deducing the main theorem 1.1
from the first key result, which is done in 4.
§
Finally, in 5 Appendix, we will apply our method to produce an example of a curve
§
C over a p-adic field such that SK (C) is infinite.
1 tor
Acknowledgment. The authors are grateful to Dr. Kanetomo Sato for the stimulating
discussions and helpful comments.
Notations. For an abelian group M, we denote by M[n] (resp. M/n) the kernel (resp.
cokernel) of multiplication by n. For a prime number ℓ we put
M ℓ := M[ℓn], M := M ℓ .
tor
{ } { }
n ℓ
[ M
3
For a nonsingular variety X over a field CHj(X,i) denotes Bloch’s higher Chow groups.
We write CHj(X) := CHj(X,0) the (usual) Chow groups.
2 Review on cycle class map and ℓ-adic regulator
In this section X denotes a smooth variety over a field K and n denotes a positive integer
prime to ch(K).
2.1: By [15] we have the cycle class map
ci,j : CHi(X,j,Z/nZ) H2i−j(X,Z/nZ(i)),
´et → ´et
where the right hand side is the ´etale cohomology of X with coefficient µ⊗i, Tate twist
n
of the sheaf of n-th roots of unity. The left hand side is Bloch’s higher Chow group with
finite coefficient which fits into the exact sequence
0 CHi(X,j)/n CHi(X,j,Z/nZ) CHi(X,j 1)[n] 0. (2.1)
→ → → − →
In this paper we are only concerned with the map
c = c2,1 : CH2(X,1,Z/nZ) H3(X,Z/nZ(2)). (2.2)
´et ´et → ´et
By [6] it is injective and its image is equal to
NH3(X,Z/nZ(2)) = Ker H3(X,Z/nZ(2)) H3(Spec(K(X)),Z/nZ(2)),
´et ´et → ´et
(cid:0)
where K(X) is the function field of X. In view of (2.1) it implies an exact sequence
0 CH2(X,1)/n c´et NH3(X,Z/nZ(2)) CH2(X)[n] 0. (2.3)
−→ −→ ´et −→ −→
2.2: We also need the cycle map to continuous ´etale cohomology group (cf. [18]):
c : CH2(X,1) H3 (X,Z (2)),
cont −→ cont ℓ
where ℓ is a prime different from ch(K). Note that in case K is a p-adic field we have
H3 (X,Z (2)) = lim H3(X,Z/ℓnZ(2))
cont ℓ ´et
←n−
and c is induced by c by passing to the limit. We have the Hochschild-Serre spectral
cont ´et
sequence
Ei,j = Hi (Spec(K),Hj(X ,Z (2))) Hi+j(X,Z (2)). (2.4)
2 cont K ℓ ⇒ cont ℓ
If K is finitely generated over the prime subfield and X is proper smooth over K, the Weil
conjecture proved by Deligne implies that H0(Spec(K),H3(X ,Q (2))) = 0. The same
K ℓ
conclusion holds if K is a p-adic field and X is proper smooth having good reduction over
K. Thus we get under these assumptions the following map
ρ : CH2(X,1) H1 (Spec(K),H2(X ,Q (2))) (2.5)
X −→ cont K ℓ
4
as the composite of c Q and an edge homomorphism
cont ℓ
⊗
H3 (X,Q (2)) H1 (Spec(K),H2(X ,Q (2))).
cont ℓ → cont K ℓ
2.3: For later use, we need an alternative definition of cycle class maps. For an integer
i 1 we denote by the sheaf on X , the Zariski site on X, associated to a presheaf
i Zar
≥ K
U K (U). By [20], 2.5, we have canonical isomorphisms
i
7−→
CH2(X,1) H1 (X, ), CH2(X,1,Z/nZ) H1 (X, /n). (2.6)
≃ Zar K2 ≃ Zar K2
Let ǫ´et : X X be the natural map of sites and put
´et Zar
→
i (Z/nZ(r)) = Riǫ´etµ⊗r.
H´et ∗ n
The universal Chern classes in the cohomology groups of the simplicial classifying space
for GL (n 1) give rise to higher Chern class maps on algebraic K-theory (cf. [14],
n
≥
[31]). It gives rise to a map of sheaves:
/n i (Z/nZ(i)). (2.7)
Ki −→ H´et
By [22] it is an isomorphism for i = 2 and induces an isomorphism
H1 (X, /n) ∼= H1 (X, 2 (Z/nZ(2))). (2.8)
Zar K2 −→ Zar H´et
By the spectral sequence
Epq = Hp (X, q (Z/nZ(2))) = Hp+q(X,Z/nZ(2)).
2 Zar H´et ⇒ ´et
together with the fact Hp (X, q (Z/nZ(2))) = 0 for p > q shown by Bloch-Ogus [6], we
Zar H´et
get an injective map
H1 (X, 2 (Z/nZ(2))) H3(X,Z/nZ(2))
Zar H´et −→ ´et
Again by the Bloch-Ogus theory the image of the above map coincides with the coniveau
filtration NH3(X,Z/nZ(2)). Combined with (2.6) and (2.8) we thus get the map
´et
c : CH2(X,1,Z/nZ) ∼= H1 (X, /n) ∼= NH3(X,Z/nZ(2)) ⊂ H3(X,Z/nZ(2)).
´et −→ Zar K2 −→ ´et −→ ´et
One can check the map agrees with the map (2.2).
2.4: Now we work over the base field K = C. Let X be the site on the underlying
an
analytic space X(C) endowed with the ordinary topology. Let ǫan : X X be the
an Zar
→
natural map of sites and put
Hi (Z(r)) = RiǫanZ(r) (Z(r) = (2π√ 1)rZ).
an ∗ −
Higher Chern class map then gives a map of sheaves
i (Z(i)).
Ki −→ Han
By the same argument as before, it induces a map
c : CH2(X,1) ∼= H1 (X, ) H3 (X(C),Z(2)) (2.9)
an −→ Zar K2 −→ an
5
Lemma 2.1 The image of can is contained in F2Ha3n(X(C),C), the Hodge filtration de-
fined in [11].
Proof. Let r (Z(i)) be the sheaf on X associated to a presheaf
HD Zar
U Hr(U,Z(i))
7→ D
where H• denotes Deligne-Beilinson cohomology (cf. [12], 2.9). Higher Chern class maps
D
to Deligne-Beilinson cohomology give rise to the map K 2 (Z(2)) and c factors as
2 → HD an
in the following commutative diagram
H1 (X, ) H1 (X, 2 (Z(2))) H1 (X, 2 (Z(2)))
Zar K2 −−−→ Zar HD −−−→ Zar Han
a
H3(X,Z(2)) b H3 (X(C),Z(2)).
D y −−−→ an y
Here the map a is induced from the spectral sequence
Epq = Hp (X, q (Z(2))) = Hp+q(X,Z(2))
2 Zar HD ⇒ D
inview ofthe fact Hp (X, 1 (2)) = 0for p > 0 since 1 (2) = C/Z(2) (constant sheaf).
Zar HD ∀ HD ∼
Since the image of b is contained in F2H3 (X(C),C) (see [12], 2.10), so is the image of
an
c . Q.E.D.
an
Lemma 2.2 We have the following commutativity diagram
CH2(X,1) can H3 (X(C),Z(2))
an
−−−→
(2.10)
CH2(X,1,Z/nZ) c´et H´e3t(X,Z/nZ(2))
y −−−→ y
Here the right vertical map is the composite
H3 (X(C),Z(2)) H3 (X(C),Z(2) Z/nZ) ∼= H3(X,Z/nZ(2))
an an ´et
→ ⊗ −→
and the isomorphism comes from the comparison isomorphism between ´etale cohomology
and ordinary cohomology (SGA41, Arcata, 3.5) together with the isomorphism
2
Z(1) Z/nZ (ǫan)∗µ
n
⊗ ≃
given by the exponential map.
Proof. This follows from the compatibility of the universal Chern classes ([14] and [31]).
Q.E.D.
6
3 Counterexample to Bloch-Kato conjecture over p-
adic field
In this section K denotes a p-adic field and let X be a proper smooth surface over K. We
fix a prime ℓ (possibly ℓ = p) and consider the map (2.5)
ρ : CH2(X,1) H1 (Spec(K),V) (V = H2(X ,Q (2)))). (3.1)
X −→ cont ´et K ℓ
Define the primitive part V of V by:
V :=eH´e2t(XK,Qℓ(2))/V0, V0 = [HX]⊗Qℓ(1), (3.2)
where [HX] ∈ Hc2ont(XeK,Qℓ(1)) is the cohomology class of a hyperplane section. Let
ρ : CH2(X,1) H1 (Spec(K),V)
−→ cont
be the induced map. e e
Theorem 3.1 Let X P3 be a generic smooth surface of degree d 5. Then ρ is the
⊂ K ≥
zero map for arbitrary ℓ.
e
Remark 3.2 (1) The key point of the proof is Nori’s connectivity theorem. This is an
analogue of [34] 1.6 (where she worked on Deligne-Beilinson cohomology).
(2) Bloch-Kato [5] considers regulator maps such as (3.1) for a smooth projective variety
over a number field and conjectures that its image coincides with H1. We will see
g
later (see Theorem 3.6) that the variant of the conjecture over a p-adic field is false
in general.
(3) The construction of a counterexample mentioned in (2) hinges on the assumption
that the surface X P3 is generic. One may still ask whether the image of l-adic
⊂ K
regulator map coincides H1 for a proper smooth variety X over a p-adic field when
g
X is defined over a number field.
Proof. Let f : M be as in the introduction and let t : Spec(K) M be a dominant
X → →
morphism such that X Spec(K). For a morphism S M of smooth schemes
M
≃ X × →
over Q let f : X = S S be the base change of f. The same construction of
S S M
X × →
(2.5) give rise to the regulator map
ρ : CH2(X ,1) H1 (S,V ),
S S → cont S
where V = R2(f ) Q (2) is a smooth Q -sheaf on S. Define the primitive part of V :
S S ∗ l l S
V = R2(f ) Q (2)/[H] Q (1),
S S ∗ l l
⊗
where [H] H0(S,R2(fS)∗Qel(1)) is the class of a hyperplane section. Let
∈
ρ : CH2(X ,1) H1 (S,V ),
S S → cont S
e
e
7
be the induced map. Note
CH2(X,1) = lim CH2(X ,1),
S
−→S
where S M ranges over the smooth morphisms which factor t : Spec(K) M. Note
→ →
also that we have the commutative diagram for such S:
CH2(X ,1) ρeS H1 (S,V )
S −−−→ cont S
e
CH2(X,1) ρe H1 (Spec(K),V).
y −−−→ cont y
Thus it suffices to show
e
H1 (S,V ) = 0.
cont S
Without loss of generality we suppose S is a affine smooth variety over a finite extension
e
L of Q.
Claim 3.3 Assume d 4. The natural map
≥
H1 (S,V ) H1(S ,V ) (S := S Spec(Q))
cont S −→ ´et Q S Q ×L
is injective.
e e
Indeed, by the Hochschild-Serre spectral sequence, it is enough to see H0(S ,V ) = 0,
´et Q S
which follows from [3], Th.5.3(2).
e
By SGA41, Arcata, Cor.(3.3) and (3.5.1) we have
2
H1(S ,V ) = H1(S ,V ) H1 (S(C),Van) Q , (S := S Spec(C))
´et Q S ∼ ´et C S ≃ an S ⊗ l C ×L
where VSan is thee primitive paert of VSan = R2(efSan)∗Q(2) with fSan : (XSC)an → (SC)an,
the natural map of sites. By definition Van is a local system on S(C) whose fiber over
S
s S(Ce) is the primitive part of H2 (X (C),Q(2)) for X , the fiber of X S over s.
∈ an s s S →
Due to Lemma 2.2, it suffices to show thaet the triviality of the image of the map
ρan : CH2(X ,1) H1 (S(C),Van)
S SC −→ an S
which is induced from
e e
c : CH2(X ,1) H3 (X (C),Q(2))
an SC −→ an S
by using the natural map
H3 (X (C),Q(2)) H1 (S(C),Van)
an S → an S
arising from the Leray spectral sequence for fan : (X ) (S ) and the vanishing
S SC an → C an
R3(fan) Q(2) = 0.
S ∗
Claim 3.4 The image of ρan is contained in the Hodge filtration F2H1 (S(C),Van C)
S an S ⊗
defined by theory of Hodge modules [29].
e e
This follows from the functoriality of Hodge filtrations and Lemma 2.1.
8
Claim 3.5 For integers m,p 0 there is a natural injective map
≥
FpHm(S(C),Van C) Hm(S ,GpDR(Van))
an S ⊗ → C S
where GpDR(Van) is the complex of sheaves on S :
S e C e
FpH2 (X /Se) Fp−1H2 (X /S) Ω1
dR S prim ⊗OSC → dR S prim⊗ SC/C → ···
Fp−rH2 (X /S) Ωr Fp−rH2 (X /S) Ωr+1
··· → dR S prim ⊗ SC/C → dR S prim ⊗ SC/C → ···
Here H• (X /S) denotes the de Rham cohomology of X /S, and H• (X /S) is its
dR S S dR S prim
primitive part defined by the same way as before, and the maps are induced from the
Gauss-Manin connection.
This follows from [1] Lemma 4.2. We note that its proof hinges on theory of mixed
Hodge modules. A key point is degeneration of Hodge spectral sequence for cohomology
with coefficient.
By the above claims we are reduced to show the exactness at the middle term of the
following complex:
F2H2 (X /S) F1H2 (X /S) Ω1 H2 (X /S) Ω2 . (3.3)
dR S prim⊗OSC → dR S prim⊗ SC/C → dR S prim⊗ SC/C
For this it suffices to show that
f Ω2 (R1f Ω1 ) Ω1 R2f Ω2
∗ XS/S ⊗OSC → ∗ XS/S prim ⊗ SC/C → ∗OXS ⊗ SC/C
is exact at the middle term and
f Ω2 Ω1 (R1f Ω1 ) Ω2
∗ XS/S ⊗ SC/C → ∗ XS/S prim ⊗ SC/C
is injective. The former holds when d 5 and the latter holds when d 3 by Nori’s
≥ ≥
connectivity theorem (cf. [16]). This completes the proof of 3.1. Q.E.D.
Let K be the ring of integers and k be the residue field. In order to construct
K
O ⊂
an example where the image of the regulator map
ρ : CH2(X,1) ρX H1 (Spec(K),V) (V = H2(X ,Q (2))))
X −→ cont ´et K ℓ
is not equal to H1(Spec(K),V), we now take a proper smooth surface X having good
g
reduction over K so that X has a proper smooth model X over Spec( ). We denote
OK OK
thespecialfiberbyY. By[21](seethediagrambelow5.7onp.341),thereisacommutative
diagram
CH2(X,1) ρe H1(Spec(K),V)
−−−→ g
(3.4)
∂
CH1(Y) α H1 (Spec(K),V)/H1(Spec(K),V)
y −−−→ cont y f
where H1 H1 H1 are the subspaces introduced by Bloch-Kato [5] and ∂ is a
f ⊂ g ⊂ cont
boundary map in localization sequence for higher Chow groups.
9
Theorem 3.6 Let X P3 be a generic smooth surface of degree d 5. Assume that X
⊂ K ≥
has a projective smooth model X P3 over and let Y P3 be its special fiber.
OK ⊂ OK OK ⊂ k
(1) The image of ∂ Q is contained in the subspace of CH1(Y) Q generated by the
⊗ ⊗
class [H ] of a hyperplane section of Y.
Y
(2) Let r be the Picard number of Y. Then
dim H1(Spec(K),V)/Image(ρ ) r 1.
Qℓ g X ≥ −
(cid:0) (cid:1)
Proof. Letting V V be as in (3.2), we have a decomposition V = V V as G -
0 0 K
⊂ ⊕
modules. Let W CH2(X,1) be the image of Z [H ] K× under the product map
X
⊂ · ⊗
CH1(X) K× CH2(X,1). Then it is easy to see ρ induces an isomorpehism
X
⊗ →
W Q H1(Spec(K),V ) = H1 (Spec(K),V )
⊗ ℓ ≃ g 0 cont 0
and that ∂(W) = Z [H ] CH1(Y). Hence (1) follows from Theorem 3.1 together with
Y
· ⊂
injectivity of α in (3.4) proved by [21], Lemma 5-7.
As for (2) we first note that dim H1 (Spec(K),V )/H1(Spec(K),V ) = 1 (see [5],
Qℓ cont 0 f 0
3.9). Moreover the same argument (except using the Tate conjecture) in the last part of
(cid:0) (cid:1)
5 of [21] shows
§
dim (CH1(Y) Q ) dim H1(Spec(K),V)/H1(Spec(K),V) .
Qℓ ⊗ ℓ ≤ Qℓ g f
(cid:0) (cid:1)
Hence (2) follows from (1). Q.E.D.
Remark 3.7 Let the assumption be as in Corollary 3.6. Then we have
r 1 ℓ = p
dim H1(Spec(K),V)/Image(ρ ) − 6
Qℓ g X ≥ (r 1+(h0,2 +h1,1 1)[K : Qp] ℓ = p
− −
(cid:0) (cid:1)
where hp,q := dim Hq(X,Ωp ) denotes the Hodge number. Moreover the equality holds
K X/K
if and only if the Tate conjecture for divisors on Y holds. It follows from Theorem 3.1 and
computation of dim H1(Spec(K),V) using [5] 3.8 and 3.8.4. The details are omitted.
Qℓ g
4 Proof of Theorem 1.1
Let K be a p-adic field and K the ring of integers and k the residue field. Let us
K
O ⊂
consider schemes
j
X X Y
−−−→ OK ←−i−−
(4.1)
  
Spec(K) Spec( ) Spec(k)
  K 
y −−−→ yO ←−−− y
10