Algebraic cycles on Severi-Brauer schemes of prime degree over a curve

0.12 MB·English

by  Cristian D. Gonzalez-Aviles

ALGEBRAIC CYCLES ON SEVERI-BRAUER
SCHEMES OF PRIME DEGREE OVER A CURVE
7
0
0 CRISTIAN D. GONZA´LEZ-AVILE´S
2

n
a
J

9 Abstract. Let k be a perfect field and let p be a prime number
2 differentfromthecharacteristicofk. LetC beasmooth,projective

and geometrically integral k-curve and let X be a Severi-Brauer
]
T C-scheme of relative dimension p − 1 . In this paper we show
d
N that CH (X)tors contains a subgroup isomorphic to CH0(X/C)
for every d in the range 2 ≤ d ≤ p. We deduce that, if k is a
.
h d
number field, then CH (X) is finitely generatedfor everyd in the
t
a indicated range.
m
[
1. Introduction.

1
v Let k be a perfect field with algebraic closure k. Very little is
1 known about algebraic cycles on algebraic k-varieties, especially in
6
codimension greater than 2 or dimension greater than zero. Let p
8
1 be a prime number different from the characteristic of k and let C be a
0
smooth, projective and geometrically integral k-curve. Inthis paper we
7
0 study a certain subgroup of CHd(X) for a Severi-Brauer C-scheme
tors
/ q: X → C of relative dimension p − 1 and any integer d such that
h
t 2 ≤ d ≤ p. Let
a
m
CH (X/C) = Ker CH (X) −q→∗ CH (C)
: 0 0 0
v
Xi and let π∗: CHd(X) → CHd hX be induced by thie extension-of-
r scalars map X → X, where X = X ⊗ k. Then the following holds.
a (cid:0) (cid:1) k
Main Theorem. For any d as above, there exists a canonical isomor-
phism
Ker CHd(X) −π→∗ CHd X ≃ CH (X/C).
0
Consequently, if kh is a number field, t(cid:0)hen(cid:1)CiHd(X) is finitely generated.
2000 Mathematics Subject Classification. Primary 14C25; Secondary 14C15 .
Key words and phrases. Algebraic cycles, Chow groups, curves, Severi-Brauer
schemes.
The author is partially supported by Fondecyt grant 1061209 and Universidad
Andr´es Bello grant DI-29-05/R.
1
2 CRISTIAND. GONZA´LEZ-AVILE´S
Acknowledgement.
I thank B.Kahn for some helpful comments.
2. Preliminaries.
Let k be a perfect field, fix an algebraic closure k of k and let
Γ = Gal k/k . Now let C be a smooth, projective and geometri-
cally integral k-curve and let X be a Severi-Brauer scheme over C [4,
(cid:0) (cid:1)
§8]. There exists a proper and flat k-morphism q: X → C all of whose
fibers are Severi-Brauer varieties of dimension m−1 (m ≥ 1) over the
appropriate residue field [loc.cit.]. We will write X for the generic
η
fiber X × Speck(C) of q and A for the central simple k(C)-algebra
C
associated to X . We define
η
CH (X/C) = Ker CH (X) −q→∗ CH (C) .
0 0 0
h i
Now let C be the set of closed points of C. The group of divisorial
0
norms of X/C (cf. [6]) is the group
k(C)∗ = {f ∈ k(C)∗: ∀y ∈ C ,ord (f) ∈ (q ) (CH (X ))}
dn 0 y y ∗ 0 y
where, for each y ∈ C , q : X → Speck(y) is the structural morphism
0 y y
of the fiber X . This group is closely related to CH (X/C) (see [2,
y 0
Proposition 3.1]). Indeed, there exists a canonical isomorphism
CH (X/C) ≃ k(C)∗ /k∗NrdA∗.
0 dn
Now fix an integer d such that 1 ≤ d ≤ m and let
CHd(X)′ = Ker CHd(X) −π→∗ CHd X Γ ,
h i
(cid:0) (cid:1)
where π: X → X is the canonical map. A simple transfer argument
shows that CHd(X)′ is a subgroup of CHd(X) . Now, since X → C
tors
has a section, X is a projective bundle over C. Thus there exists an
isomorphism
CHd X ≃ Z⊕CH C .
0
(see[3,Theorem3.3(b),p.3(cid:0)4]).(cid:1)Therefore, if(cid:0)J (cid:1)(k)isfinitelygenerated,
C
where J is the Jacobian variety of C (e.g., k is a number field or
C
C = P1), then CHd(X) is finitely generated if and only if CHd(X)′ is
k
finite.
ALGEBRAIC CYCLES ON SEVERI-BRAUER SCHEMES 3
3. The general method.
Let C be as above and let X be any smooth, projective and geo-
metrically integral k-variety such that there exists a proper and flat
morphism q: X → C whose generic fiber X is geometrically integral.
η
We have an exact sequence [7]
(1)
Hd−1(X ,K ) −→δ CHd−1(X ) → CHd(X) →j∗ CHd(X ) → 0,
η d y η
yM∈C0
where j: X → X is the natural map and the map which we have
η
labeled δ will play a role later when k = k. A similar exact sequence
exists over k, and we have two natural exact commutative diagrams:
0 // Kerj∗ // CHd(X) // CHd(X ) // 0
η
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // (Ker∗)Γ // CHd X Γ // CHd X Γ
η
and (cid:0) (cid:1) (cid:0) (cid:1)
(2) 0 // Hd−1(Xη,Kd) // CHd−1(Xy) // // Kerj∗
j∗Hd−1(X,Kd) yM∈C0
(cid:15)(cid:15) (cid:15)(cid:15) ⊕y|yπy∗
0 // Hd−1(Xη,Kd) Γ δ // CHd−1 Xy Γy // (Ker(cid:15)(cid:15)∗)Γ
∗Hd−1(X,Kd) y|y
(cid:18) (cid:19) M (cid:0) (cid:1)
where, for each y ∈ C , we have fixed a closed point y of C lying above
0
y and written Γ = Gal k/k(y) . Set
y
CHd(X )′ =(cid:0) Ker C(cid:1)Hd(X ) → CHd X Γ
η η η
h i
and, for each y ∈ C , (cid:0) (cid:1)
0
CHd−1(X )′ = Ker CHd−1(X ) −π→y∗ CHd−1 X Γy .
y y y
(cid:20) (cid:21)
(cid:0) (cid:1)
Now define
Γ
Hd−1(X ,K ) Hd−1 X ,K
η d η d
(3) E X/C = Coker −→ .
j∗Hd−1(X,Kd) ∗Hd−(cid:0)1 X,Kd(cid:1) ! 
(cid:0) (cid:1)
 (cid:0) (cid:1) 
4 CRISTIAND. GONZA´LEZ-AVILE´S
Then, applying the snake lemma to the preceding diagrams, we obtain1
Proposition 3.1. There exists a natural exact sequence
CHd−1(X )′ → Ker CHd(X)′ → CHd(X )′
y η
yM∈C0 (cid:2) (cid:3)
CHd−1 X Γy
y
→ Ker E X/C → → 0,
π∗CHd−1(X )
" (cid:0) (cid:1) yM∈C0 y (cid:0) (cid:1)y #
where E X/C is the group (3).
As reg(cid:0)ards th(cid:1)e right-hand group in the exact sequence of the propo-
sition, the following holds. Let
Hd−1(X ,K )′ = Im Hd−1(X ,K ) → Hd−1 X ,K Γ
η d η d η d
h i
(cid:0) (cid:1)
and
Sal (X/C) = f∈Hd−1 X ,K Γ :∀y ∈ C , δ (f) ∈ π∗CHd−1(X ) ,
d η d 0 y y y
n o
(cid:0) (cid:1)
where δ and π∗ are the maps of diagram (2).
y
Proposition 3.2. There exists a natural exact sequence
Sal (X/C)
d
0 →
Γ
∗Hd−1 X,K ·Hd−1(X ,K )′
d η d
(cid:0) (cid:0) (cid:1)(cid:1) CHd−1 X Γy
y
→ Ker E X/C →
π∗CHd−1(X )
" (cid:0) (cid:1) yM∈C0 y (cid:0) (cid:1)y #
→ H1 Γ,∗Hd−1 X,K .
d
Proof. This follows b(cid:0)y applying(cid:0)the sna(cid:1)k(cid:1)e lemma to a diagram of the
form
0 // A // B // B/A // 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // AΓ // BΓ // A/B Γ // H1 Γ,A
(cid:0) (cid:1) (cid:0) (cid:1)
with A = ∗Hd−1 X,K , B = Hd−1 X ,K , etc. (cid:3)
d η d
(cid:0) (cid:1) (cid:0) (cid:1)
1Proposition 3.1 was inspired by [1, Proposition 1.1].
ALGEBRAIC CYCLES ON SEVERI-BRAUER SCHEMES 5
4. Proof of the main theorem.
Let C and A be as in Section 2, let p be a prime number different
from the characteristic of k and let X be a Severi-Brauer scheme over
C of relative dimension p−1.
Lemma 4.1. There exists a Γ-isomorphism
∗Hd−1 X,K ≃ k∗.
d
Proof. Clearly, ∗Hd−1 X,K (cid:0)is the k(cid:1)ernel of the map
d
δ: Hd−(cid:0)1 X ,K(cid:1) → CHd−1 X
η d y
y|y
(cid:0) (cid:1) M (cid:0) (cid:1)
appearing in the exact sequence (1) over k. Now X ≃ Pp−1 and
η η
X ≃ Pp−1 for every y, whence we have Γ-isomorphisms
y
k
Hd−1 X ,K ≃ k(C)∗
η d
and
(cid:0) (cid:1)
CHd−1 X ≃ Z
y
for each y. Under these isomorphisms, the map δ above corresponds
(cid:0) (cid:1)
to the canonical map
k(C)∗ → Z,
y|y
M
f 7→ (ord (f)) ,
y y|y
(cid:3)
which yields the lemma.
Theorem 4.2. For every d such that 2 ≤ d ≤ p, there exists a canon-
ical isomorphism
CHd(X)′ ≃ CH (X/C).
0
Proof. By Lemma 4.1, Hilbert’s Theorem 90 and Proposition 3.2, there
exists a natural isomorphism
CHd−1 X Γy Sal (X/C)
y d
Ker E X/C → ≃ .
π∗CHd−1(X ) k∗Hd−1(X ,K )′
" (cid:0) (cid:1) yM∈C0 y (cid:0) (cid:1)y # η d
On the other hand, by [5,(8.7.2)], Hd−1(X ,K )′ = NrdA∗ for every d
η d
such that 2 ≤ d ≤ p and, for each y ∈ C ,
0
π∗CHd−1(X ) ≃ π∗CHp−1(X ) ≃ (q ) CH (X ) (= Z or pZ).
y y y y y ∗ 0 y
The latter implies that Sal (X/C) = k(C)∗ , whence
d dn
Sal (X/C)/k∗Hd−1(X ,K )′ ≃ k(C)∗ /k∗NrdA∗
d η d dn
≃ CH (X/C).
0
6 CRISTIAND. GONZA´LEZ-AVILE´S
Finally, [loc.cit.] shows that the groups CHd(X ) and CHd−1(X )
η y
(y ∈ C ) are torsion free, whence CHd(X )′ and CHd−1(X )′ vanish.
0 η y
(cid:3)
The theorem now follows from Proposition 3.1.
Corollary 4.3. Let d be such that 2 ≤ d ≤ p. Then CHd(X)′ is finite
if
(1) k is a number field, or
(2) k is a field of finite type over Q, C = P1 and X has a 0-cycle
k
of degree one.
In each of these cases, the group CHd(X) is finitely generated.
Proof. Indeed, in these cases the group CH (X/C) is finite [2]. (cid:3)
0
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Departamento de Matema´ticas, Universidad Andr´es Bello, Chile
E-mail address: [email protected]