Riemann-Roch theorems for higher algebraic 𝐾-theory

Gillet, Henri

doi:10.1090/s0273-0979-1980-14828-4

Cited by 22 publications

(45 citation statements)

References 9 publications

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“…Now fix a bounded below graded complex of sheaves F * ( * ) of abelian groups, giving a twisted duality cohomology theory in the sense of Gillet [10]. Assuming an extra property for F * ( * ), we will be able to reduce the comparison of maps at the level of the simplicial schemes B · GL N to the Grassmannian schemes.…”

Section: Assumptions On the Cohomology Theorymentioning

confidence: 99%

“…In [10], Gillet constructed Chern classes for higher K -theory starting from the universal Chern classes on B · GL N . More specifically, let the c j,N ∈ H 2j ZAR (B · GL N , F * (j)), j ≥ 1, be given as in [10].…”

Section: The Uniqueness Of Characteristic Classesmentioning

confidence: 99%

“…More specifically, let the c j,N ∈ H 2j ZAR (B · GL N , F * (j)), j ≥ 1, be given as in [10]. These induce a compatible system of pointed maps…”

Section: The Uniqueness Of Characteristic Classesmentioning

confidence: 99%

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On uniqueness of characteristic classes

Feliu

2011

Journal of Pure and Applied Algebra

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a b s t r a c tWe give an axiomatic characterization of maps from algebraic K -theory. The results apply to a large class of maps from algebraic K -theory to any suitable cohomology theory or to algebraic K -theory. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Adams operations and λ-operations on higher algebraic K -theory. We show that the Adams operations and λ-operations defined by Grayson agree with the ones defined by Gillet and Soulé.In this paper we address the problem of comparing maps from the algebraic K -groups of a scheme to either algebraic K -groups or suitable cohomology theories. This type of question often arises when one constructs a map that is supposed to induce, in some particular cohomology theory, a specific regulator or known map, and one needs to show that the map is indeed the expected one. Examples of this situation are found in the construction of the Beilinson regulator given by Burgos and Wang in [6], in the regulator defined by Goncharov in [13] and in the definition of the Adams operations given by Grayson in [14].The various natures of the constructions usually mean that direct comparison is not an available option and one is forced to turn to theoretical tricks. In this work, we identify sufficient conditions for two maps to agree, thus obtaining an axiomatic characterization of maps from K -theory. These aim to extend known characterization theorems for characteristic classes at K 0 , such as the characterization of the Chern classes [17, Th1] and the splitting principle for Adams operations [1].As a main consequence, we give a characterization of the Adams operations and λ-operations on higher algebraic K -theory and of the Chern character and Chern classes on a suitable cohomology theory (see Sections 4.2, 4.3 and 5.3).In particular, we show that the Adams operations defined by Grayson in [14] agree with the ones defined by Gillet and Soulé in [12], for all noetherian schemes of finite Krull dimension. This implies that for this class of schemes, the operations defined by Grayson satisfy the usual identities for the Adams operations in a (special) λ-ring (cf. Section 4.1).A similar conclusion is reached with the λ-operations defined by Grayson in [15]. We show that his operations agree with the ones defined by Gillet and Soulé on the algebraic K -groups of degree higher than 0. Since Grayson had already shown that his operations are the usual ones on degree 0, we conclude that they agree at all degrees. This implies that these λ-operations satisfy the usual identities of a special λ-ring as well.The second specific application of this work is a proof that the regulator defined by Burgos and Wang in [6] is the Beilinson regulator. The proof provided here is simpler than the one given in [6], where delooping in K -theory was required.The results of this paper are further exploited in the paper by the author [8], where an explicit chain morphism representing the Adams operations on higher algebraic K -theory with rational coef...

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Section: Assumptions On the Cohomology Theorymentioning

confidence: 99%

Section: The Uniqueness Of Characteristic Classesmentioning

confidence: 99%

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On uniqueness of characteristic classes

Feliu

2011

Journal of Pure and Applied Algebra

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“…The K-theory of coherent sheaves (G-theory), or the homology theory corresponding to a cycle complex in the sense of Rost, [28], and in particular the homology theory associated to the Gersten complexes [11], all satisfy properties (1.) to (5.…”

Section: Homological Descentmentioning

confidence: 99%

Motivic weight complexes for arithmetic varieties

Gillet¹,

Soulé²

2009

Journal of Algebra

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We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne-Mumford stacks; this extends the results in the paper [H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996) 127-176], where a similar result was proved for varieties over a field of characteristic zero. We use K 0 -motives with rational coefficients, rather than Chow motives, because we cannot appeal to resolution of singularities, but rather must use de Jong's results. In addition, for varieties over a field we prove a general result on contravariance of weight complexes, in particular showing that any morphism of finite tor-dimension between projective varieties induces a morphism of weight complexes.

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“…On pourrait remplacer la cohomologie motivique, représentée par H Z , par une théorie cohomologique orientée plus générale, pourvu que celle-ci soit représentée par un objet de SH(k) ; on peut donc obtenir des caractères de Chern comme dans [2].…”

Section: Remarqueunclassified

Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas

Riou

2006

Comptes Rendus. Mathématique

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Riemann-Roch theorems for higher algebraic 𝐾-theory

Cited by 22 publications

References 9 publications

On uniqueness of characteristic classes

On uniqueness of characteristic classes

Motivic weight complexes for arithmetic varieties

Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas

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