E. Friedlander and H. B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533--565.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....) Mor(X; C s Gamma1 Gamma P s Gamma1 Delta ) 13) where the latter is a weak product of topological spaces. Therefore, we obtain k (C X (C 1 ) Phi s0 L 2 H 2s Gammak (X) according to Definition 3.7. This proves assertion (a) When X is smooth, it follows from [FHBL97] that L s H 2s (X) A s (X) and that the multiplicative structure on Phi s L s H 2s (X) 0 (ZX (C 1 ) induced by the join conicides with the intersection product on A (X) cf. 6) A simple colimit argument completes the proof of assertion (b) The previous result shows ....

....) and g 2 CX (W ) one verifies that this assignment satisfies the following properties. OE V (f f 0 ) OE V (f) OE V (f 0 ) and OE V PhiW ( f; g) OE V (f) Delta OE W (g) 15) We now invoke a particular model of the group completion functor, studied in detail in [LF93a] FG93] and [FHBL97] If M belongs to a suitable class of abelian topological monoids, then one can take M to be its Grothendieck group (naive group completion) with an appropriate topology. It follows, using the functoriality of this model that the assignment V 7 ZX (V ) CX (V ) is also an I monoid ....

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E. M. Friedlander and Jr. H. B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), no. 2, 533--565.

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E. Friedlander and H. B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533--565.

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Eric M. Friedlander and H. Blaine Lawson. Duality relating spaces of algebraic cocycles and cycles. Topology, 36(2):533--565, 1997.

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Friedlander, E. and H.B. Lawson, Jr., Duality relating spaces of algebraic cocycles and cycles, Topology 36 no.2 (1997), 533-565.

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E. Friedlander and H. B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533--565.

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E. Friedlander and H.B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533-565.

....a smooth variety relating topological cycle cohomology with supports to topological cycle homology. Construction of the cap product on the level of presheaves of chain complexes is relatively straight forward, whereas the duality theorem is a direct consequence of duality proved in [F2] following [FL 2]. As a consequence, we show that topological cycle cohomology of a smooth variety equals morphic cohomology. Proposition 5.1. There is a natural pairing of presheaves in SX M( Gamma; s) 2s] Omega L( Gamma; r) L( Gamma Theta A s ; r) 5:1:1) where L( Gamma Theta A s ; r) sends U 2 ZarX ....

....a continuous algebraic map f : U C 0 (P s ) to its graph Gamma f 2 C d (U Theta P s ) for any 2 ZarX . Hence, whenever X satisfies the condition that topological cycle cohomology equals morphic cohomology (e.g. X smooth by Corollary 5. 6 below) the induced map is the duality map of [FL 2], F2] D : H j (X; s) H 2d Gammaj (X; d Gamma s) 5:3:0) BLOCH OGUS PROPERTIES FOR TOPOLOGICAL CYCLE THEORY 15 Proposition 5.3. Consider the following Cartesian square of varieties Y 0 Gamma Gamma Gamma Gamma X 0 fi y y ff Y Gamma Gamma Gamma Gamma X whose horizontal ....

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E. Friedlander and H.B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533-565.

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E. Friedlander and H.B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533-565.

....[13] Friedlander and H.B. Lawson prove a moving lemma for families of cycles on a smooth scheme which enables one to make all effective cycles of degree bounded by some constant to intersect properly all effective cycles of similarly bounded degree. This was used to establish duality isomorphisms [12], 9] between Lawson homology (cf. 18] and morphic cohomology (cf. 11] topological analogues of motivic homology with locally compact supports and motivic cohomology. Theorem 5.3 presents the result of adapting the moving lemma of [13] to our present context of DM k . As consequences of this ....

E. Friedlander and H.B. Lawson. Duality relating spaces of algebraic cocycles and cycles. Topology, 36 (1997), 533--565.

.... Nevertheless, just such a construction applied to the set of morphisms from X to certain Chow varieties of cycles in projective space leads to the morphic cohomology of X as introduced in [FL 1] In this paper, we show that, in general, the topology of bounded convergence (introduced in [FL 2]) on Mor(X; Y ) has a natural algebraic description arising from the enriched structure on Mor(X; Y ) as a contravariant functor on the category of smooth curves. This functorial interpretation leads to a convenient formulation of the technique of demonstrating uniqueness of specialization ....

....(X Theta Y ) an called the duality map . This extends to the map of presheaves on X Zar D : M( Gamma; s) Gamma2s] Gamma L( Gamma; d Gamma s) 2d Gamma 2s] or, equivalently, to the map D : M( Gamma; s) Gamma L( Gamma; d Gamma s) 2d] If X;Y are both smooth, then the main results of [FL 2], F2] assert that the map D : Mor(X; C r Y ) C r d (X Theta Y ) is a quasi isomorphism. In particular, for X smooth of pure dimension d, this duality isomorphism has the form D : M( Gamma; s) Gamma L( Gamma; d Gamma s) 2d] 3:4:0) where we have implicitly used the homotopy invariance ....

E. Friedlander and H.B. Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533-565.

....been developed (cf. 13] 3] in which the role of rational equivalence is replaced by algebraic equivalence, and a bivariant extension L H (Y; X) 7] has been introduced. This more topological approach suggested a duality relating covariant and contravariant theories as established in [8]. Our bivariant cycle cohomology groups A r;i (Y; X) de ned for schemes Y; X of nite type over a eld k, satisfy A r;i (Spec(k) X) CH n r (X; i) whenever X is an ane scheme over k of pure dimension n. This bivariant theory is a somewhat more sophisticated version of a theory brie y ....

....that C (G) U) C (F ) U) is a quasiisomorphism. 7 Duality. In this section, we prove duality theorems relating z equi (U; X; r) to z equi (X U; r dim(U) for a smooth scheme U . The proofs of these theorems use techniques which were originally developed for the duality theorems of [8]. In the next section, we shall apply duality to conclude the basic properties of bivariant cycle cohomology groups A r;i (Y; X) for all schemes of nite type Y; X over a eld k which admit resolution of singularities. We begin with the following duality theorem for projective, smooth varieties ....

Eric M. Friedlander and H. Blaine Lawson. Duality relating spaces of algebraic cocycles and cycles. Topology, 36(2):533-565, 1997.

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Friedlander, E. and H.B. Lawson, Jr., Duality relating spaces of algebraic cocycles and cycles, Topology 36 no.2 (1997), 533-565.

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E.M. Friedlander and H. B. Lawson, Jr., Duality relating spaces of algebraic cocycles and cycles, Topology, 36 (1997), 533 - 565.

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