# zbMATH — the first resource for mathematics

On the cycle map for torsion algebraic cycles of codimension two. (English) Zbl 0764.14004
For complete varieties $$X$$ over a field $$k$$ and positive integers $$n$$, conditions that entail the injectivity of the map $$\rho_{n,tor}:CH^ 2(X)_{tor}\to H^ 4(X,\mu_ n^{\otimes 2})$$ obtained by composing the inclusion $$CH^ 2(X)_{tor}\hookrightarrow CH^ 2(X)$$, the quotient map $$CH^ 2(X)\to CH^ 2(X)/n$$ and the cycle map $${\rho_ n:CH^ 2(X)/n\to H^ 4(X,\mu_ n^{\otimes 2})}$$ are discussed $$(\mu_ n$$ is the sheaf of $$n$$-th roots of unity on $$X$$ and $$CH^ 2(X)$$ is the group of rational classes of codimension 2 cycles on $$X)$$.
We cannot reproduce here the several sets of subtle conditions that are proved to suffice for the injectivity, but the flavour of them can be appreciated by the following sample case: if $$X$$ is projective and smooth over a number field $$k$$, if the Picard variety of $$X$$ has potentially good reduction, and if $$H^ 2(X_{zar},{\mathcal O}_ X)=0$$, then there exists an integer $$N>0$$ such that $$\rho_{n,tor}$$ is injective for any $$n$$ divisible by $$N$$. Let us also state one of the theorems that are established with the methods introduced to prove the injectivity results: If $$X$$ is smooth over a field $$k$$ which is a finitely generated extension of $$\mathbb{Q}$$, if $$H^ 2(X_{zar}{\mathcal O}_ X)=H^ 1(X_{zar},{\mathcal O}_ X)=0$$, and if $$X$$ has a $$k$$-rational point, then $$CH^ 2(X)_{tor}$$ is finite.

##### MSC:
 14C25 Algebraic cycles 14C05 Parametrization (Chow and Hilbert schemes)
Full Text:
##### References:
  Bloch, S.: On the Chow group of certain rational surfaces. Ann. Sci. Éc. Norm. Supér, IV. Sér.14, 41-59 (1981) · Zbl 0524.14006  Bloch, S.: Lectures on Algebraic Cycles. (Duke Univ. Math. Ser.) Durham: Duke University Press 1980 · Zbl 0436.14003  Bloch, S.: Algebraic K-theory and class field theory for arithmetic surfaces. Ann. Math.114, 229-265 (1981) · Zbl 0512.14009  Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Ec. Norm. Supér., IV. Sér.7, 181-202 (1974) · Zbl 0307.14008  Coombes, K.: The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero. (Preprint) · Zbl 0787.14002  Colliot-Thélène, J.-L.: Hilbert’s theorem 90 for K2, with application to the Chow groups of rational surfaces. Invent. Math.71, 1-20 (1983) · Zbl 0527.14011  Colliot-Thélène, J.-L., Raskind, W.: K2-cohomology and the second Chow group. Math. Ann.270, 165-199 (1985) · Zbl 0548.14001  Colliot-Thélène, J.-L., Raskind, W.: Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: un théorème de finitude pour la torsion. (Preprint) · Zbl 0752.14004  Colliot-Thélène, J.-L., Raskind, W.: On the reciprocity law for surfaces over finite fields. J. Fac. Sci., Univ. Tokyo, Sect. I A33, 283-294 (1986) · Zbl 0595.14015  Colliot-Thélène, J.-L., Sansuc, J.-J.: On the Chow group of certain rational surfaces: A sequel to a paper of S. Bloch. Duke Math. J.48, 421-447 (1981) · Zbl 0479.14006  Colliot-Thélène, J.-L., Sansuc, J.-J., Soule, C.: Torsion dans le groupe de Chow de codimension deux. Duke math. J.51 (1984)  Deligne, P.: La conjecture de Weil II. Publ. Math., Inst. Hautes Étud. Sci.52 (1981) · Zbl 0456.14014  Fulton, W.: Intersection Theory (Ergeb. Math. Grenzgeb., 3. Folge) Berlin Heidelberg New York: Springer 1984  Grothendieck, A.: Le groupe de Brauer II. In: Dix Exposés sur la cohomologie des schémas, pp. 67-87. Amsterdam: North-Holland 1968  Grayson, D.: Universal exactness in algebraic K-theory. J. Pure Appl. Algebra36, 139-141 (1985) · Zbl 0558.18007  Gros, M.: 0-cycles de degré zéro sur les surfaces fibrées en coniques. J. Reine Angew. Math.373, 166-184 (1987) · Zbl 0593.14005  Jannsen, U.: Continuous Etale Cohomology. Math. Ann.280, 207-245 (1988) · Zbl 0649.14011  Jannsen, U.: On the ?-adic cohomology of varieties over number fields and its Galois cohomology. In: Ihara, Y., Ribet, K.A., Serre, J.-P. (eds.) Galois Group over Q Berlin Heidelberg New York: Springer 1989 · Zbl 0703.14010  Milne, J.S.: Étale Cohomology. Princeton, New Jersey: Princeton University Press 1980 · Zbl 0433.14012  Milne, J.S.: Arithmetic Duality Theorems. (Perspect. Math.) Boston: Academic Press 1986 · Zbl 0613.14019  Mercurjev, A.S., Suslin, A.A.: K-cohomology of Severi-Brauer Varieties and the norm residue homomorphism. Math. USSR Izv.21, 307-341 (1983) · Zbl 0525.18008  Okochi, T.: On the Chow group of a conic bundle surface. Master’s thesis. University of Tokyo: 1984  Quillen, D.: Higher algebraic K-theory I. (Lect. Notes Math. vol.) Berlin Heidelberg New York: Springer 1973 · Zbl 0292.18004  Raskind, W.: Torsion algebraic cycles on varieties over local fields. In: Jardine, J.F., Snaith, V.P. (eds.) Algebraic K-theory: Connection with Geometry and Topology (Lake Louise 1987). Dordrecht: Kluwer Academic Publishers (1989) · Zbl 0709.14005  Salberger, P.: Lectures in Italy. June 1989  Salberger, P.: Torsion cycles of codimension two andl-adic realization of motivic cohomology (preprint) · Zbl 0833.14004  Serre, J.-P.: Cohomologie Galoisienne. (Lect. Notes Math. vol. 5) Berlin Heidelberg New York: Springer 1965  Suslin, A.A.: Torsion in K2 of fields. Prepr. LOMI E-2-82; K-Theory1, no. 1, 5-29 (1987)  Tate, J.: Relations between K2 and Galois cohomology. Invent. Math.36, 257-274 (1976) · Zbl 0359.12011  Deligne, P.: Cohomologie Etale. (Lect. Notes Math. vol. 569) Berlin Heidelberg New York: Springer 1977  Artin, M., Grothendieck, A., Verdier, J.L.: Théorie des Topos et Cohomologie Etale des Schémas (Tome 2). (Lect. Notes Math. vol. 270; ibid (Tome 3), ibid vol. 305) Berlin Heidelberg New York: Springer 1972; 1973 · Zbl 0234.00007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.