## An algebraic geometric realization of the Chern character

Citations: | 5 - 1 self |

### BibTeX

@MISC{Cohen_analgebraic,

author = {Ralph L. Cohen and Paulo Lima-Filho},

title = {An algebraic geometric realization of the Chern character},

year = {}

}

### OpenURL

### Abstract

### Citations

840 | Symmetric functions and Hall polynomials - Macdonald - 1979 |

427 |
Characteristic Classes
- Nlilnor, Stasheff
- 1974
(Show Context)
Citation Context ...n ) ×d ; Q) Sq under the action of Sq. On the other hand, it is well-known that F n q also induces an isomorphism (F n q ) ∗ : Hk (Gr q n; Q) ∼ = −→ Hk (P(Cn ×d Sq ) ; Q) , whenever q(n − 1) > k; cf. =-=[MS74]-=-. Using the isomorphisms H k (Gr q n /Sq; Q) ∼= −−−→ (ρ q n) ∗ H k (Gr q n ; Q)Sq ∼ = H k (Gr q n ; Q), exhibited in the proof of Proposition 2.8, together with ρ q n ◦F q n = f q n ◦tn q , one conclu... |

145 |
Young tableaux
- Fulton
- 1997
(Show Context)
Citation Context ...∗ ( ∏ j≥1 K(Z,2j); Q). Let Λ = ⊕ ∞ n=0 Λn denote the ring of symmetric functions p(x1,x2,... ) on infinitely many variables, where Λn denotes the functions of degree n. Here we follow the notation of =-=[Ful97]-=-. Let ek = ∑ i1<···<ik xi1 · · · xik be the k-th elementary symmetric function and pk = ∑ i xki be the k-th Newton power sum. It is well-known that Λ is a polynomial ring over Z in the variables {e1,e... |

112 | Algebraic Geometry: A First Course. Graduate Texts in Mathematics - Harris - 1992 |

85 |
Quasifaserungen und unendliche symmetriche produkte
- Dold, Thom
- 1958
(Show Context)
Citation Context ...k ). Therefore, they induce a canonical filtration-preserving splitting homomorphism (19) sp : SP∞(P(C ∞ )) → ∏ SP∞(S 2k ). We use SP∞(S 2j ) as our model for the Eilenberg-MacLane space K(Z,2j) (cf. =-=[DT56]-=-), and denote by ı2j ∈ H 2j (SP∞(P(C ∞ )); Z) the class represented by the composition (20) SP∞(P(C ∞ )) sp −→ ∏ k≥1 k≥1 SP∞(S 2k ) prj −−→ SP∞(S 2j ), where prj is the projection. Let i2j ∈ H 2j (SP∞... |

58 | Foundations of Hyperbolic manifolds, Graduate Texts - Ratcliffe - 2006 |

55 |
A theory of algebraic cocycles
- Lawson
- 1992
(Show Context)
Citation Context ...tegory VarC of projective algebraic varieties and algebraic maps. We will then explain the relevance of this fact for the study of the morphic cohomology introduced by E. Friedlander and B. Lawson in =-=[FL92]-=- and of holomorhic K-theory performed by Cohen and Lima-Filho in [CLF]. We expect that the constructions made here, as well as in [CLF], can be extended to a broader context, such as [Fri97], once app... |

52 |
Algebraic cycles and homotopy theory
- Lawson
- 1989
(Show Context)
Citation Context ... cycles in projective spaces, and their relation to the Chern classes in the present context. We start considering the case X = {pt}, and this is essentially a survey of fundamental results of Lawson =-=[Law89]-=-, Lawson and Michelsohn [LM88], Boyer, Lawson, Lima-Filho, Mann and Michelsohn [BLLF+ 93] and Lima-Filho [LF99]. These form a directed system {C q n,d ;tq n,(d,e) ,ǫ(q,k) d,n ,sq (n,m),d}, whose colim... |

41 |
Classifying spaces and fibrations
- May
- 1975
(Show Context)
Citation Context ... satisfying the following properties.40 R.L. COHEN AND P. LIMA-FILHO 1. If (A,M,B) is such a triple and M acts trivially on C, then B(C × A,M,B) = C × B(A,M,B), where M acts diagonally on C × A; cf. =-=[May75]-=-. 2. B(∗,M, ∗) = BM is the classifying space of M and the map EM := B(M,M, ∗) → BM induced by the obvious map of triples (M,M, ∗) → (∗,M, ∗) is the universal quasifibration for M; cf. [May75]. 3. If M... |

40 |
Algebraic cycles, Chow varieties, and Lawson homology
- Friedlander
- 1991
(Show Context)
Citation Context ...oid M is homotopy equivalent to the colimit lim −→ α∈Λ Mα, where the Mα’s are connected components of M, and Λ is a collection contained countably infinitely many copies of each element in π0(M); cf. =-=[Fri91]-=-. This argument implies the following lemma.AN ALGEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 11 Lemma 2.9. Let M be an abelian topological monoid, equipped with a continuous monoid augmentat... |

33 | Duality relating spaces of algebraic cocycles and cycles’, Topology 36
- Friedlander, Lawson
- 1997
(Show Context)
Citation Context ... + → quotient map, and let Mor(X,SP∞(S 2n )) + the natural homotopy qq,n : Mor(X,SP∞(P q )) + → Mor(X,SP∞(S 2n )) +18 R.L. COHEN AND P. LIMA-FILHO denote the composition ψn ◦ ρq,n ∗ . It is shown in =-=[FL92]-=- that the map (17) Ψ q : Mor(X,SP∞(P q )) + −→ q∏ j=1 Mor(X,SP∞(S 2j )) + defined as Ψ q := ∏ q j=1 qj,q, is a homotopy equivalence. Definition 3.5. Let X be an algebraic variety. The colimit of the m... |

21 |
Lawson homology for quasi-projective varieties
- Lima-Filho
- 1992
(Show Context)
Citation Context ...ty F, i.e. a compact homogeneous space of the form F = G/P, where is a complex algebraic group and P < G is a parabolic subgroup. It follows from the duality results in [FL92] and the computations in =-=[LF92]-=- that the forgetful functor Mor(F,SP∞(P(C ∞ ))) + → Map(F,SP∞(P(C ∞ ))) + is a homotopy equivalence. As a consequence one has the following. Proposition 3.10. If G is a finite group of automorphisms o... |

19 | Completions and fibrations for topological monoids - Lima-Filho - 1993 |

18 |
Algebraic cycles, Bott periodicity, and the Chern characteristic map
- Lawson, Michelsohn
- 1988
(Show Context)
Citation Context ...nd their relation to the Chern classes in the present context. We start considering the case X = {pt}, and this is essentially a survey of fundamental results of Lawson [Law89], Lawson and Michelsohn =-=[LM88]-=-, Boyer, Lawson, Lima-Filho, Mann and Michelsohn [BLLF+ 93] and Lima-Filho [LF99]. These form a directed system {C q n,d ;tq n,(d,e) ,ǫ(q,k) d,n ,sq (n,m),d}, whose colimit C has a canonical splitting... |

15 |
Spaces of algebraic cycles
- Lawson
- 1995
(Show Context)
Citation Context ...t}. Definition 4.1. Given n > 0 and q ≥ 0, let C q ( ) n,d = Chowq d P(C∨n ⊗ Cq ) be the Chow variety consisting of the effective algebraic cycles of codimension q and degree d in P(C ∨n ⊗ C q ); cf. =-=[Law95]-=-. The formal addition of cycles + : C q n,d × Cq n,d ′ → C q n,d+d ′ is an algebraic map which makes C q n,∗ := ∐d≥0C q n,d , into a graded abelian topological monoid, called the Chow monoid of effect... |

11 | Algebraic cycles and infinite loop spaces - Boyer, Lawson, et al. - 1993 |

11 | Algebraic cycles and equivariant cohomology theories
- Lawson, Lima-Filho, et al.
- 1996
(Show Context)
Citation Context ...GEBRAIC GEOMETRIC REALIZATION OF THE CHERN CHARACTER 33 b: It is shown in [LF99] that cX is a map of spectra from the holomorphic K-theory spectrum of X to its morphic spectrum, in the terminology of =-=[LLFM96]-=-. c: Under the forgetful functor Mor(,) → Map(,) one obtains a commutative diagram ˜K −i hol (X) ⏐ ↓ ˜K −i top (X) ci X −−−→ ∏ p≥1 Lp H 2p−i (X) ⏐ ↓ c −−−→ ∏ p≥1 H2p−i (X; Z) where ˜ K −i top (X) is t... |

7 |
Holomorphic K-theory, algebraic cocycles and loop groups
- Cohen, Lima-Filho
(Show Context)
Citation Context ...ll then explain the relevance of this fact for the study of the morphic cohomology introduced by E. Friedlander and B. Lawson in [FL92] and of holomorhic K-theory performed by Cohen and Lima-Filho in =-=[CLF]-=-. We expect that the constructions made here, as well as in [CLF], can be extended to a broader context, such as [Fri97], once appropriate facts in motivic rational homotopy theory are in place. The c... |

5 | Motivic complexes of Suslin and Voevodsky
- Friedlander
- 1996
(Show Context)
Citation Context ... Lawson in [FL92] and of holomorhic K-theory performed by Cohen and Lima-Filho in [CLF]. We expect that the constructions made here, as well as in [CLF], can be extended to a broader context, such as =-=[Fri97]-=-, once appropriate facts in motivic rational homotopy theory are in place. The constructions of classifying spaces made, together with their rationalizations, involve three different algebraic geometr... |

3 |
Resultants and the algebraicity of the join pairing on Chow varieties
- Plümer
- 1997
(Show Context)
Citation Context ... by linearity. The join satisfies the following properties: Facts 4.3. a. The join is a strictly associative operation. b. Its restriction to the connected components yields an algebraic map (38) cf. =-=[Plü97]-=- and [Bar91]. ♯ : C q n,d × Cq′ n,e → C q+q′ n,de ; c. In the particular case of cycles of degree one (d = d ′ = 1), the join coincides with the usual direct sum operation (39) ⊕ : Gr q n × Gr q′ n → ... |

2 |
Notes on the “joint theorem
- Barlet
- 1991
(Show Context)
Citation Context ...y. The join satisfies the following properties: Facts 4.3. a. The join is a strictly associative operation. b. Its restriction to the connected components yields an algebraic map (38) cf. [Plü97] and =-=[Bar91]-=-. ♯ : C q n,d × Cq′ n,e → C q+q′ n,de ; c. In the particular case of cycles of degree one (d = d ′ = 1), the join coincides with the usual direct sum operation (39) ⊕ : Gr q n × Gr q′ n → Gr q+q′ n ; ... |

2 | Semi-topological K- theory using function complexes, preprint - Friedlander, Walker - 1999 |

1 |
Algebraic Families of E∞-spectra, Topology and its Applications
- Lima-Filho
- 1999
(Show Context)
Citation Context ...ng the case X = {pt}, and this is essentially a survey of fundamental results of Lawson [Law89], Lawson and Michelsohn [LM88], Boyer, Lawson, Lima-Filho, Mann and Michelsohn [BLLF+ 93] and Lima-Filho =-=[LF99]-=-. These form a directed system {C q n,d ;tq n,(d,e) ,ǫ(q,k) d,n ,sq (n,m),d}, whose colimit C has a canonical splitting homotopy equivalence C ≃ ∏∞ j=1 K(Z,2j); cf. [Law89]. An important fact is that ... |

1 |
On the group completion of a simplicial monoid, Appendix to Mem
- Quillen
- 1994
(Show Context)
Citation Context ...e system π0(M) of the Pontrjagin ring H∗(M) is sent by α to the multiplicative subgroup π0(N + ) of the units of H∗(N + ). Recall that H∗(M + ) is isomorphic to the localization H∗(M)[π0(M)] −1 ; cf. =-=[Q]-=-. Therefore, there is a unique ring homomorphism α+ : H∗(M + ) → H∗(N + ) satisfying α+ ◦ u∗ = α∗. Since both M + and N + are 0-local abelian topological monoids, they are a product of rational Eilenb... |

1 |
Triangulation of stratified fibre bundles, Manuscripta Math. 30
- Verona
- 1980
(Show Context)
Citation Context ...hen X/G is simply connected. Proof. The first part of the theorem is well-known and follows from standard transfer arguments. Consider a fixed point x ∈ X G and denote x = ρ(x) ∈ X/G. It follows from =-=[Ver80]-=- that one can find an equivariant triangulation of X in which x is a vertex, and such the quotient ρ : X → X/G becomes a simplicial map for the quotient triangulation of X/G. In particular, given any ... |