On uniqueness of characteristic classes

Abstract

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...

A chain morphism for Adams operations on rational algebraic K-theory

“…The fact that our construction induces indeed the Adams operations defined by Gillet and Soulé in [6] and the ones defined by Grayson in [7] follows from a general result on the comparison of morphisms from algebraic K-theory to itself, given in [4].…”

“…The composition of the transgression morphism with the Adams operations for split cubes gives a chain morphism representing the Adams operations for any regular noetherian scheme of finite Krull dimension (Theorem 4.2). The fact that our construction induces indeed the Adams operations defined by Gillet and Soulé in [6] and the ones defined by Grayson in [7] follows from a general result on the comparison of morphisms from algebraic K-theory to itself, given in [4].…”

“…the Adams operations derived from the lambda structure coming from the exterior product of locally free sheaves. Then, the statement follows from corollary 4.4 in [4].…”

The Beilinson regulator is a map of ring spectra

“…Similarly, the map H) induced by reg ε can be considered as such a map. Then by [Fel11,Thm. 5.6] both regulators coincide if they induce the same map…”

Fredholm modules and the Beilinson-Bloch regulator

“…Second formula. Alternatively, we can give a formula for R{p, q} based on formula (16). In terms of the standard generators of the Steinberg group, the Steinberg symbol {M (p), M (q)} has the expression…”