Let $k$ be a perfect field and let $p$ be a prime number different from the
characteristic of $k$. Let $C$ be a smooth, projective and geometrically
integral $k$-curve and let $X$ be a Severi-Brauer $C$-scheme of relative
dimension $p-1$ . In this paper we show that $CH^{d}(X)_{{\rm{tors}}}$ contains
a subgroup isomorphic to $CH_{0}(X/C)$ for every $d$ in the range $2\leq d\leq
p$. We deduce that, if $k$ is a number field, then $CH^{d}(X)$ is finitely
generated for every $d$ in the indicated range.Comment: 6 page

Gonzalez-Aviles, Cristian D.

English

arXiv

Abstract. Let k be a perfect ﬁeld and let p be a prime number diﬀerent from the characteristic of k. Let C be a smooth, projective and geometrically integral k-curve and let X be a Severi-Brauer C-scheme of relative dimension p − 1 . In this paper we show
that CHd (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 number ﬁeld, then CHd (X) is ﬁnitely generated for every d in the indicated range

Gonzalez-Aviles, Cristián

Repositorio Institucional Académico Universidad Andrés Bello

arXiv:math/0701861v1  [math.NT]  29 Jan 2007ALGEBRAIC CYCLES ON SEVERI-BRAUERSCHEMES OF PRIME DEGREE OVER A CURVECRISTIAN D. GONZA´LEZ-AVILE´SAbstract. Let k be a perfect field and let p be a prime numberdifferent from the characteristic of k. Let C be a smooth, projectiveand geometrically integral k-curve and let X be a Severi-BrauerC-scheme of relative dimension p − 1 . In this paper we showthat CHd(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 anumber field, then CHd(X) is finitely generated for every d in theindicated range.1. Introduction.Let k be a perfect field with algebraic closure k. Very little isknown about algebraic cycles on algebraic k-varieties, especially incodimension greater than 2 or dimension greater than zero. Let pbe a prime number different from the characteristic of k and let C be asmooth, projective and geometrically integral k-curve. In this paper westudy a certain subgroup of CHd(X)tors for a Severi-Brauer C-schemeq : X → C of relative dimension p − 1 and any integer d such that2 ≤ d ≤ p. LetCH0(X/C) = Ker[CH0(X)q∗−→ CH0(C)]and let π∗ : CHd(X) → CHd(X)be induced by the extension-of-scalars map X → X, where X = X ⊗k k. Then the following holds.Main Theorem. For any d as above, there exists a canonical isomor-phismKer[CHd(X)π∗−→ CHd(X)]≃ CH0(X/C).Consequently, if k is a number field, then CHd(X) is finitely generated.2000 Mathematics Subject Classification. Primary 14C25; Secondary 14C15 .Key words and phrases. Algebraic cycles, Chow groups, curves, Severi-Brauerschemes.The author is partially supported by Fondecyt grant 1061209 and UniversidadAndre´s Bello grant DI-29-05/R.12 CRISTIAN D. GONZA´LEZ-AVILE´SAcknowledgement.I thank B.Kahn for some helpful comments.2. Preliminaries.Let k be a perfect field, fix an algebraic closure k of k and letΓ = 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,§8]. There exists a proper and flat k-morphism q : X → C all of whosefibers are Severi-Brauer varieties of dimension m− 1 (m ≥ 1) over theappropriate residue field [loc.cit.]. We will write Xη for the genericfiber X ×C Speck(C) of q and A for the central simple k(C)-algebraassociated to Xη. We defineCH0(X/C) = Ker[CH0(X)q∗−→ CH0(C)].Now let C0 be the set of closed points of C. The group of divisorialnorms of X/C (cf. [6]) is the groupk(C)∗dn = {f ∈ k(C)∗ : ∀y ∈ C0, ordy(f) ∈ (qy)∗(CH0(Xy))}where, for each y ∈ C0, qy : Xy → Speck(y) is the structural morphismof the fiber Xy. This group is closely related to CH0(X/C) (see [2,Proposition 3.1]). Indeed, there exists a canonical isomorphismCH0(X/C) ≃ k(C)∗dn/k∗NrdA∗.Now fix an integer d such that 1 ≤ d ≤ m and letCHd(X)′ = Ker[CHd(X)π∗−→ CHd(X)Γ ],where π : X → X is the canonical map. A simple transfer argumentshows that CHd(X)′ is a subgroup of CHd(X)tors. Now, since X → Chas a section, X is a projective bundle over C. Thus there exists anisomorphismCHd(X)≃ Z⊕ CH0(C).(see [3, Theorem 3.3(b), p.34]). Therefore, if JC(k) is finitely generated,where JC is the Jacobian variety of C (e.g., k is a number field orC = P1k), then CHd(X) is finitely generated if and only if CHd(X)′ isfinite.ALGEBRAIC CYCLES ON SEVERI-BRAUER SCHEMES 33. 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 flatmorphism q : X → C whose generic fiber Xη is geometrically integral.We have an exact sequence [7](1)Hd−1(Xη,Kd)δ−→⊕y∈C0CHd−1(Xy)→ CHd(X)j∗→ CHd(Xη)→ 0,where j : Xη → X is the natural map and the map which we havelabeled δ will play a role later when k = k. A similar exact sequenceexists over k, and we have two natural exact commutative diagrams:0 // Ker j∗ //CHd(X) //CHd(Xη)// 00 // (Ker  ∗)Γ // CHd(X)Γ// CHd(Xη)Γand(2) 0 //Hd−1(Xη ,Kd)j∗Hd−1(X,Kd)//⊕y∈C0CHd−1(Xy)⊕y |yπ∗y// // Ker j∗0 //(Hd−1(Xη ,Kd) ∗Hd−1(X,Kd))Γδ//⊕y |yCHd−1(Xy)Γy// (Ker  ∗)Γwhere, for each y ∈ C0, we have fixed a closed point y of C lying abovey and written Γy = Gal(k/k(y)). SetCHd(Xη)′ = Ker[CHd(Xη)→ CHd(Xη)Γ ]and, for each y ∈ C0,CHd−1(Xy)′ = Ker[CHd−1(Xy)π∗y−→ CHd−1(Xy)Γy ].Now define(3) E(X/C)= Coker Hd−1(Xη, Kd)j∗Hd−1(X, Kd)−→(Hd−1(Xη, Kd)∗Hd−1(X, Kd))Γ  .4 CRISTIAN D. GONZA´LEZ-AVILE´SThen, applying the snake lemma to the preceding diagrams, we obtain1Proposition 3.1. There exists a natural exact sequence⊕y∈C0CHd−1(Xy)′ → Ker[CHd(X) ′ → CHd(Xη)′]→ Ker[E(X/C)→⊕y∈C0CHd−1(Xy)Γyπ∗y CHd−1(Xy)]→ 0 ,where E(X/C)is the group (3).As regards the right-hand group in the exact sequence of the propo-sition, the following holds. LetHd−1(Xη, Kd)′ = Im[Hd−1(Xη, Kd)→ Hd−1(Xη, Kd)Γ ]andSald(X/C) ={f ∈Hd−1(Xη, Kd)Γ: ∀y ∈ C0, δ y (f) ∈ π∗y CHd−1(Xy)},where δ and π∗y are the maps of diagram (2).Proposition 3.2. There exists a natural exact sequence0→Sald(X/C)(∗Hd−1(X, Kd))Γ·Hd−1(Xη, Kd) ′→ Ker[E(X/C)→⊕y∈C0CHd−1(Xy)Γyπ∗y CHd−1(Xy)]→ H1(Γ, ∗Hd−1(X, Kd)).Proof. This follows by applying the snake lemma to a diagram of theform0 // A //B //B/A// 00 // AΓ //BΓ //(A/B)Γ// H1(Γ,A)with A =  ∗Hd−1(X, Kd), B = Hd−1(Xη, Kd), etc. 1Proposition 3.1 was inspired by [1, Proposition 1.1].ALGEBRAIC CYCLES ON SEVERI-BRAUER SCHEMES 54. Proof of the main theorem.Let C and A be as in Section 2, let p be a prime number differentfrom the characteristic of k and let X be a Severi-Brauer scheme overC of relative dimension p− 1.Lemma 4.1. There exists a Γ -isomorphism ∗Hd−1(X, Kd)≃ k ∗.Proof. Clearly,  ∗Hd−1(X, Kd)is the kernel of the mapδ : Hd−1(Xη, Kd)→⊕y |yCHd−1(Xy)appearing in the exact sequence (1) over k. Now Xη ≃ Pp−1η andXy ≃ Pp−1kfor every y, whence we have Γ -isomorphismsHd−1(Xη, Kd)≃ k(C)∗andCHd−1(Xy)≃ Zfor each y. Under these isomorphisms, the map δ above correspondsto the canonical mapk(C)∗ →⊕y |yZ,f 7→ (ord y(f))y |y,which yields the lemma. Theorem 4.2. For every d such that 2 ≤ d ≤ p, there exists a canon-ical isomorphismCHd(X) ′ ≃ CH0(X/C).Proof. By Lemma 4.1, Hilbert’s Theorem 90 and Proposition 3.2, thereexists a natural isomorphismKer[E(X/C)→⊕y∈C0CHd−1(Xy)Γyπ∗y CHd−1(Xy)]≃Sald(X/C)k∗Hd−1(Xη, Kd) ′.On the other hand, by [5,(8.7.2)], Hd−1(Xη, Kd)′ = NrdA∗ for every dsuch that 2 ≤ d ≤ p and, for each y ∈ C0,π∗y CHd−1(Xy) ≃ π∗y CHp−1(Xy) ≃ (qy)∗CH0(Xy) (= Z or pZ).The latter implies that Sald(X/C) = k(C)∗dn, whenceSald(X/C)/k∗Hd−1(Xη, Kd)′ ≃ k(C)∗dn/k∗NrdA∗≃ CH0(X/C).6 CRISTIAN D. GONZA´LEZ-AVILE´SFinally, [loc.cit.] shows that the groups CHd(Xη) and CHd−1(Xy)(y ∈ C0) are torsion free, whence CHd(Xη)′ and CHd−1(Xy)′ vanish.The theorem now follows from Proposition 3.1. Corollary 4.3. Let d be such that 2 ≤ d ≤ p. Then CHd(X) ′ is finiteif(1) k is a number field, or(2) k is a field of finite type over Q, C = P1k and X has a 0-cycleof degree one.In each of these cases, the group CHd(X) is finitely generated.Proof. Indeed, in these cases the group CH0(X/C) is finite [2]. References[1] Colliot-The´le`ne, J.-L. and Skorobogatov, A.: Groupe de Chow des ze´ro-cyclessur les fibre´s en quadriques. K-Theory 7, pp. 477-500 (1993).[2] Frossard, E.: Groupe de Chow de dimension ze´ro des fibrations en varie´te´s deSeveri-Brauer. Comp. Math. 110, pp. 187-213 (1998).[3] Fulton, W.: Intersection Theory. Second Ed. Springer-Verlag, 1998.[4] Grothendieck, A.: Le Groupe de Brauer I. In: Dix Expose´s sur la Cohomologiedes Sche´mas. North-Holland, Amsterdam, pp. 46-66 (1968).[5] Merkurjev, A.S. and Suslin, A.A.: K-cohomology of Severi-Brauer varietiesand the norm residue homomorphism. Math. USSR Izv. 21, No. 2, pp. 307-340(1982).[6] Salberger, P.: Galois descent and class groups of orders. Lect. Notes in Math.1142, pp. 239-255 (1985).[7] Sherman, C.: Some theorems on the K-Theory of coherent sheaves. Comm. inAlg. 7(14), pp. 1489-1508 (1979).Departamento de Matema´ticas, Universidad Andre´s Bello, ChileE-mail address : cristiangonzalez@unab.cl

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

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