E. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361--428.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... It is immediate from the definitions that all the assertions in Proposition 2.2 hold for the new collection (M n , Q n , q q,n , i n,q ) above. These assertions guarantee that the spaces and maps involved, along with their restrictions to fixed point sets, satisfy the hypothesis of [FL92, Prop. 2.13] The corollary then follows. In order to fully understand the equivariant homotopy type of Z 0 we are reduced to understanding Z 0 . However, P = T is the Thom space of the Real bundle O P(H) It follows from Propositions 1.11 and 1.12 ....

Eric Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math 136 (1992), no. 2, 361--428.

.... and hence the induced maps from algebraic to topological Lawson homology with finite coefficients L r H alg 2r i (Z; Z=n) Gamma L r H 2r i (Z; Z=n) The simplicial abelian groups used to define L r H alg 2r i (Z) may be given a slightly different description. E.Friedlander and H. Lawson [4] proved that for any normal connected scheme S the abelian monoid Hom(S;C r (Z) coincides with the monoid of effective cycles on S Theta Z every component of which is equidimensional of relative dimension r over S. Thus Hom( Delta q ; C r (Z) coincides with the group of cycles in Delta ....

E. Friedlander and H. Lawson. A theory of algebraic cocycles. Ann. of Math., 136:361--428, 1992.

....moduli spaces, and are functorial on X. The coefficients for the corresponding cohomology theories reflect algebraic geometric and topological invariants for the variety X. In the morphic case, the coefficients are given in terms of FriedlanderLawson s morphic cohomology for varieties [FL92] The theory carries total Chern class maps and total cycle maps, extending to the stable category classical constructions in algebraic geometry. Contents 1. Introduction 2 2. Moduli spaces of algebraic cycles 3 2.1. Algebraic cycles and Chow varieties 4 2.2. Holomorphic Mapping spaces 6 3. ....

....FAMILIES OF E1 SPECTRA 3 1. In Theorems 3.2 and 3.8 we show that ZX is an E1 ring spectrum, which satisfies k (Z X ) Y s L s H 2s Gammak (X) where L s H 2s Gammak (X) denotes the morphic cohomology of the variety X. These are invariants for X, introduced and studied in [FL92] which have a hybrid algebraic geometric and topological nature. 2. If X is a smooth variety, then 0 (Z X ) A (X) as a ring, where A (X) is the graded Chow ring of cycles modulo algebraic equivalence; cf. Ful84] 3. The spectrum MX is essentially obtained via the action of an E1 ....

[Article contains additional citation context not shown here]

E. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361--428.

....approach is natural and has several original features. An important fact is that all spaces and maps involved are colimits of directed systems in the category Var C of projective algebraic varieties and algebraic maps. This is quite relevant for the study of the morphic cohomology introduced in [FL92] and of holomorhic K theory performed in [CLF] We expect that the constructions made here, as well as in [CLF] can be generalized to arbitrary varieties once the appropriate category is chosen, such as in [Fri97] The constructions of classifying spaces made, together with their ....

....(BU; Q) with H (BU=S1 ; Q) via ae . Then we denote by i 2j the image in rational cohomology of the integral class represented by the composition SP1 (P(C 1 ) Gamma sp 1 Y j=1 K(Z; 2j) Gamma Gamma pr j K(Z; 2j) where sp is the canonical splitting equivalence presented in [FL92] and pr j is the projection. The first main result is the following. Theorem 3.2. Let f : SP1 (P(C 1 ) BU=S1 be the homomorphism of Proposition 2.12, and identify H (BU=S1 ; Q) with H (BU ; Q) via the projection ae : BU BU=S1 . Then, for j 1 one has f ( j ch j ) i 2j ; ....

[Article contains additional citation context not shown here]

E. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361--428.

.... case, following Kapranov, Sturmfels and Zelevinski [KSZ91] we introduce the Chow quotient X= C of X by C , which comes equipped once again with a trace map t p Gamma1 : C p Gamma1 Gamma X= C Delta C p Gamma X Delta , defined using techniques from Friedlander and Lawson [FL92]. In a similar fashion to the previous theorem we introduce a monoid morphism Psi p : Pi p Gamma1 Gamma X= C Delta Theta Pi p Gamma X 1 Delta Theta Pi p Gamma X 2 Delta Pi p Gamma X Delta which produces the following result. Theorem 5.5. Let X be an smooth ....

....study the projective closure of line bundles and exhibit various explicit examples. In Section 5 we introduce the Chow quotient X= C ) n of a projective variety X under an algebraic action of (C ) n , following [KSZ91] We combine the notions of Chow quotients and the trace maps of [FL92] to prove Theorem 5.5. Various examples of Chow quotients, resulting trace maps, and Euler Chow series are exhibited. In the Appendix A we present an algebraic framework which places the Euler Chow series E p Gamma X Delta in a broader context, as an invariant of the Pontrjagin ring of the ....

[Article contains additional citation context not shown here]

E. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361--428.

No context found.

E. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361--428.

No context found.

E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361--428.

No context found.

Eric M. Friedlander and H. Blaine Lawson, Jr. A theory of algebraic cocycles. Ann. of Math. (2), 136(2):361--428, 1992.

....#, C ) and = Hom(# 0# # 0 , C ) The linear projection H induces a bundle mapping # : # covering # which is an isomorphism on fibres. The map # in (6.4) is given by #(h) h (# 1 # 1 ) for h #, C ) Our main assertion here is the following. Proposition 6.2. The flat pull back of cycles gives a Z 2 homotopy equivalence 2n. Interesting Note 6.3 A quick proof of Proposition 6.2 can be given for q 2n as follows. By [LLM 1 ] and [La] the flat pull back of cycles gives equivariant homotopy 11 for all q and n. Equivariant excision ....

....h (# 1 # 1 ) for h #, C ) Our main assertion here is the following. Proposition 6.2. The flat pull back of cycles gives a Z 2 homotopy equivalence 2n. Interesting Note 6.3 A quick proof of Proposition 6.2 can be given for q 2n as follows. By [LLM 1 ] and [La] the flat pull back of cycles gives equivariant homotopy 11 for all q and n. Equivariant excision arguments (cf. Li 1 ] Li 2 ] LLM 1 ] then show is also a Z 2 homotopy equivalence, and the Proposition follows from the commutativity of (6.4) Unfortunately this will not ....

[Article contains additional citation context not shown here]

Friedlander, E. and H.B. Lawson, Jr., A theory of algebraic cocycles, Annals of Math., 136 (1992), 361-428.

....Z 2 action on induced by conjugation, extends linearly to a Z 2 action on SP n ) in such a way that the map r q,n becomes Z 2 equivariant. This map, in turn, induces an equivariant homomorphism (8.4) r q,n : n ) One has an evident trace map (see [Proposition 7. 1, FL 1 ] 8.5) tr : n ) defined as the extension to the free abelian group n ) of the natural inclusion SP n ) This is easily seen to be Z 2 equivariant and can be used to define an equivariant homomorphism (8.6) # q,n : as the composition r q,n n ....

...., the result follows. # Corollary 8.4. If M q , Q n , q q,n and i q,n are as above, then the group homomorphism Q 0 Q given by the product q q,0 q q,q is a homotopy equivalence. Proof. The proposition above guarantees that all three collections considered satisfy the hypothesis of [FL 1 , Proposition 2.13] from which the corollary follows. Recall that, given a CW pair (X, Y ) with X connected and a basepoint x 0 Y , then the quotient group 0 (X) Z 0 (Y ) is naturally identified with 0 (X Y ) o , the connected component of 0 in 0 (X Y ) Furthermore, this ....

Friedlander, E. and H.B. Lawson, Jr., A theory of algebraic cocycles, Annals of Math., 136 (1992), 361-428.

No context found.

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....by the N.S.F. Contents 0. Conventions and terminology 1. Cocycles on projective varieties 2. Duality map 3. Duality theorems 4. Compatibility with Poincar e duality 5. Applications C. Cocycles and the compact open topology M. The Moving Lemma for Families T. Tractable monoids In [FL 1 ] the authors introduced the notion of an e ective algebraic cocycle on an algebraic variety X with values in a variety Y , and developed a bivariant morphic cohomology theory based on such objects. The theory was shown to have a number of intriguing properties, including Chern classes for ....

....and for smooth varieties it intertwines certain Gysin maps. It is also compatible with the s operations of [FM] which act on both theories. This shows that for smooth varieties Poincar e duality preserves the ltrations induced by these operations on singular theory (with Z coecients) FM] FL 1 ] The basic results have a wide range of applications. For example it is shown that for generalized ag manifolds X;Y (smooth varieties with cell decompositions) there is an isomorphism = H 2(m r) X Y ; Z) where m = dim(X) Furthermore, for any smooth m dimensional variety X there are ....

[Article contains additional citation context not shown here]

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....the spaces of complex points of real varieties. We suggest that these latter unstable spaces are worthy of study in their own right. 8 Real morphic cohomology and characteristic classes We introduce real morphic cohomology, the analogue in the real context of morphic cohomology considered in [FL1]. This is a cohomology theory presumably dual (for smooth varieties) to real Lawson homology as constructed by P. dos Santos in [Sa] We show that there are natural Chern classes from K alg # (# . top R X) to this real morphic cohomology for any real quasi projective variety X . These Chern ....

E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361--428.

.... We next formulate two lemmas which will enable us to identify the space r0 Z r Gamma1 (X; P 1 ) 1 with its H space structure given by the I(2) an pairing I(2) an Theta 0 a r0 Z r Gamma1 (X; P 1 ) 1 1 A Theta2 Gamma a r0 Z r Gamma1 (X; P 1 ) 1 : We recall from [10] the notation Z j (X) j Z 0 (X; P j ) Z 0 (X; P j Gamma1 ) j 1; for the quotient topological abelian group associated to the map Z 0 (X; P j Gamma1 ) Z 0 (X; P j ) induced by inclusion P j Gamma1 , P j into the first j homogeneous coordinates. For j = 0, we may identify ....

....Consequently, there is a natural isomorphism j 0 Omega 1 Sigma 1 ( a r0 Z r Gamma1 (X; P 1 ) 1 ) 1 A = M q0 L q H 2q Gammaj (X) 38 ERIC M. FRIEDLANDER AND MARK E. WALKER for any j 0, where L q H 2q Gammaj (X) denotes the morphic cohomology of X as introduced in [10] and formulated in [9] Proof. The description of 0 i r0 Z r Gamma1 (X; P 1 ) 1 j given in Lemma 6.4 shows that to group complete this monoid we merely need to invert the element (1; X] 2 N Theta AD (X) where D = dimX. Let ffl 2 Z 0 (X; P 1 ) correspond to the effective cycle X ....

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....completions. Observe now that scalar multiplication t : Hom C (p 0 ; H ) Hom C (p 0 ; H ) by real numbers t 0 gives bundle maps commuting with the Z 2 action, and pulling to the normal cone gives a Z 2 deformation retraction T 2 e n Z 0 (Q n ) o (cf. LLM 1 ] FL] L] Therefore, it remains only to show that the inclusion (6.10) is a Z 2 homotopy equivalence. 13 We shall proceed in analogy with the arguments in [LLM 1 ] We consider the direct sum Hom C (p 0 ; H ) Hom C (p 0 ; H ) Q n and choose two distinct projections 0 ; 1 : ....

.... and that D = D jD (and all scalar multiples tD for 0 t 1) do not meet the vertices 0 ; 1 of our projections To each D 2 U(d) we associate the pull back cycle e D = p D and de ne a transformation D : C 2 (Hom C (p 0 ; H ) C 2 (Hom C (p 0 ; H ) as in [LLM 1 ] FL] L] by setting D (c) 1 ) n 0 c e D o : Note that 1 and 0 are proper on e D . Note also that if deg(D) d, then lim t 0 tD = d 2 Id. The arguments given in [LLM 1 ] may now be repeated in this context. The important point is to show that there is a function N(d) ....

[Article contains additional citation context not shown here]

Friedlander, E. and H.B. Lawson, Jr., A theory of algebraic cocycles, Annals of Math., 136 (1992), 361-428. 26

....the condition that the ring of analytic functions of X wn consists of those continuous functions on X whose composition with p are analytic on X . The following proposition gives some insight into why continuous algebraic maps occurred in the formulation of morphic cohomology theory (cf. [FL 1]) BLOCH OGUS PROPERTIES FOR TOPOLOGICAL CYCLE THEORY 5 Proposition 1.4. Let Y be a projective variety and let Mor( Gamma; Y ) denote the qfhsheaf on varieties with values in topological spaces whose value on a normal variety X is the space of morphisms from X to Y as topologized in Proposition ....

....on Chow monoids; let tr : C r (C t (X Theta P ) C r t (X Theta P ) denote the trace map of [FL 1;7. 1] Then Gamma f jZ j tr ffi F (Z) BLOCH OGUS PROPERTIES FOR TOPOLOGICAL CYCLE THEORY 13 This defines a pairing C t (P ) X) Theta C r (X) C r t (X Theta P ) As argued for example in [FL 1], one verifies that this pairing is continuous algebraic (and thus, in particular, continuous) The generic algebraicity is shown by applying the above construction to a generic f 2 C t (P ) X) and a generic Z 2 C r (X) and concluding that the resulting cycle on X Theta P is rational over the ....

[Article contains additional citation context not shown here]

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

No context found.

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....the relation established by A. Grothendieck between K 0 (X) and A (X) More recently, Lawson homology theory for complex algebraic varieties has been developed (cf. 13] 4] in which the role of rational equivalence is replaced by algebraic equivalence, and a bivariant extension L H (Y; X) [8] has been introduced. This more topological approach suggested a duality relating covariant and contravariant theories as established in [7] Our bivariant cycle cohomology groups A r;i (Y; X) defined for schemes Y; X of finite type over a field k, satisfy A r;i (Spec(k) X) CH n Gammar (X; ....

....cosuspension isomorphisms, and a Gysin exact triangle. We also provide three pairings for our theory: the first is the direct analogue of the pairing that gives operations in Lawson homology [11] the second is a multiplicative pairing inspired by the multiplicative structure in morphic cohomology [8], and the final one is a composition product viewed in the context of Lawson homology as composition of correspondences. Because of its good properties and its relationship to higher algebraic K theory, one frequently views the higher Chow groups CH n Gammar (X; i) as at least one version of ....

[Article contains additional citation context not shown here]

E. Friedlander and H. B. Lawson. A theory of algebraic cocycles. Annals of Math., 136:361--428, 1992.

....T ae T . Proof. The condition that an r cycle on Y t lies in S j Z r (Y t ) is equivalent to the condition that there exists a j cycle ffi t on C r Gammaj (Y t ) homologically equivalent to 0 with the property that t = tr(ffi t ) where tr is the trace map Z j (C r Gammaj (Y t ) Z r (Y t ) of [FL 1]. Let [C j (C r Gammaj (Y=T ) T ) Theta2 T ] hom denote the kernel of the map C j (C r Gammaj (Y=T ) T ) Theta T C j (C r Gammaj (Y=T ) T ) M t2T H 2r (Y t ) sending (ffi; ffi 0 ) to [tr(ffi) Gamma tr(ffi 0 ) Let i = i Gamma i Gamma be a minimal representation of i 2 Z ....

E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Annals of math 136 (1992), 361-428.

.... of effective r cycles on X (cf. F1] A Chow correspondence f has an associated cycle Z f in Y Theta X; if Y is normal, then every (effective) algebraic cycle in Y Theta X equi dimensional of fiber dimension r over Y is the cycle associated to a unique Chow correspondence f : Y C r (X) cf. [F L]) In an earlier paper, we obtained a correspondence homomorphism OE f : H (Y ; Q) H 2r (X; Q) associated to a Chow correspondence f : Y C r (X) whose domain of definition is the subspace H (Y ; Q) ae H (X; Q) of classes of lowest weight for the Mixed Hodge Structure on the rational ....

E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

.... AND MARK E. WALKER We next formulate two lemmas which will enable us to identify the space r 0 Z r 1 (X; P 1 ) 1 with its H space structure given by the I(2) an pairing I(2) an 0 a r 0 Z r 1 (X; P 1 ) 1 1 A 2 a r 0 Z r 1 (X; P 1 ) 1 : We recall from [10] the notation Z j (X) Z 0 (X; P j ) Z 0 (X; P j 1 ) j 1; for the quotient topological abelian group associated to the map Z 0 (X; P j 1 ) Z 0 (X; P j ) induced by inclusion P j 1 , P j into the rst j homogeneous coordinates. For j = 0, we may identify d Mor(X;C 0;d ....

.... r 1 (and which restricts to the evident constant map when r = 0) Consequently, there is a natural isomorphism j 0 1 1 ( a r 0 Z r 1 (X; P 1 ) 1 ) 1 A = M q 0 L q H 2q j (X) for any j 0, where L q H 2q j (X) denotes the morphic cohomology of X as introduced in [10] and formulated in [9] Proof. The description of 0 r 0 Z r 1 (X; P 1 ) 1 given in Lemma 6.4 shows that to group complete this monoid we merely need to invert the element (1; X] 2 N AD (X) where D = dim X . Let 2 Z 0 (X; P 1 ) correspond to the e ective cycle X f[1 : 0 : ....

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....(resp. covariant) for proper maps and covariant (resp. contravariant) for flat maps. We recall the bivariant theory introduced in [14] which is closely related to a construction in [10] and which is an algebraic version of the bivariant morphic cohomology introduced by Friedlander and Lawson in [11]: A r;i (Y; X) j H Gammai cdh (Y; C (z equi (X; r) cdh ) This bivariant theory is used in x5 when considering the duality relationship between motivic cohomologies and homologies. We conclude this section with a proposition, proved by Voevodsky, which interprets this bivariant theory in the ....

....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 moving lemma, we show that a theorem of Suslin implies that Bloch s ....

E. Friedlander and H.B. Lawson. A theory of algebraic cocycles. Annals of Math, 136 (1992), 361--428.

....a topology on the set Mor(X; Y ) of morphisms between two complex algebraic varieties seems unnatural. 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 ....

....of a proper, constructible presentation of a functor (cf. Definition 2.1) a property which provides a natural topological realization of a contravariant functor on smooth curves. This point of view facilitates (cf. Theorem 2. 6) a careful proof of the continuity of the slant product pairing of [FL 1] and the cap product pairing relating Lawson homology and morphic cohomology which plays a central role in [F 3] Indeed, our techniques provide, not merely a pairing on the level of homology groups, but pairings (in the derived category) of the presheaves of chain complexes used to define Lawson ....

[Article contains additional citation context not shown here]

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....established by A. Grothendieck between K 0 (X) and A (X) More recently, Lawson homology theory for complex algebraic varieties has 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) ....

....cosuspension isomorphisms, and a Gysin exact triangle. We also provide three pairings for our theory: the rst is the direct analogue of the pairing that gives operations in Lawson homology [11] the second is a multiplicative pairing inspired by the multiplicative structure in morphic cohomology [7], and the nal one is a composition product viewed in the context of Lawson homology as composition of correspondences. Because of its good properties and its relationship to higher algebraic K theory, one frequently views the higher Chow groups CH n r (X; i) as at least one version of ....

[Article contains additional citation context not shown here]

Eric M. Friedlander and H. Blaine Lawson. A theory of algebraic cocycles. Ann. of Math., 136:361-428, 1992.

.... complex varieties: the author and Barry Mazur introduced operations in Lawson homology which led to interesting filtrations in (singular) homology [F Mazur] the author and Blaine Lawson introduced a bivariant theory with the purpose of constructing a cohomology theory associated to cycle spaces [F Lawson]; Paulo Lima Filho [Lima Filho] see also [F Gabber] extended Lawson homology to quasi projective varieties; and the author and Ofer Gabber established an intersection theory in Lawson homology [F Gabber] A related theory, the algebraic bivariant cycle complex introduced in [F Gabber] is ....

E. Friedlander and H. B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....Contents Introduction 0. Conventions and terminology 1. Cocycles on projective varieties 2. Duality map 3. Duality theorems 4. Compatibility with Poincar e duality 5. Applications C. Cocycles and the compact open topology M. The Moving Lemma for Families T. Tractable monoids Introduction In [FL 1 ] the authors introduced the notion of an effective algebraic cocycle on an algebraic variety X with values in a variety Y , and developed a bivariant morphic cohomology theory based on such objects. The theory was shown to have a number of intriguing properties, including Chern classes for ....

....and for smooth varieties it intertwines certain Gysin maps. It is also compatible with the s operations of [FM] which act on both theories. This shows that for smooth varieties Poincar e duality preserves the filtrations induced by these operations on singular theory (with Z coefficients) FM] FL 1 ] The basic results have a wide range of applications. For example it is shown that for generalized flag manifolds X;Y (smooth varieties with cell decompositions) there is an isomorphism Mor(X;Z r (Y ) H 2(m r) X Theta Y ; Z) where m = dim(X) Furthermore, for any smooth m ....

[Article contains additional citation context not shown here]

E. Friedlander and H.B. Lawson, A theory of algebraic cocycles, Annals of Math. 136 (1992), 361-428.

....and characteristic classes. Using results in this paper the authors have succeeded in calculating the coefficients of the new equivariant cohomology theories established in [LLM 1 ] The results also play a role in defining cohomology operations on the morphic cohomology groups introduced in [FL 1 ] Since writing this paper the authors have learned of independent results of Jacob Mostovoy on quaternion cycles. Among other things he has obtained Theorem 6.4 over the rationals. x2. Suspension to the regular representation. Consider a finite group G of order fl, and let V be a finite ....

Friedlander, E. and H.B. Lawson, Jr., A theory of algebraic cocycles, Annals of Math., 136 (1992), 361-428.

No context found.

E.M. Friedlander and H. B. Lawson, Jr., A theory of algebraic cocycles, Ann. of Math., 136 (1992), 361 - 428.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC