λ-rings and algebraic K-theory

AbstractLet A be a commutative ring with identity. Loday [14] and others have described the multiplicative structure (both graded and ungraded) on the higher algebraic K-theory of A. In [19] and [24], Quillen has indicated the existence of a λ-ring structure on the “K-cohomology” groups K(X, A)=[X, K0(A)×BGL(A)+].The purpose of this paper is to develop some of this structure and show how it can be useful in obtaining information about the K-groups themselves. In particular, standard techniques will then allow one to construct Adams operationsψk in algebraic K-theory. These operations enjoy some of the attractive properties of their counterparts in the K-theory of complex vector bundles over a finite complex. Indeed, with this machinery and the étale Chern classes of Soulé [27], one can hope to lift other results from the topological to the algebraic setting. We now give a more detailed description of the contents of this paper.In Section 1 we rapidly review the fundamentals of the Quillen K-theory and construct a natural transformation q connecting representations and K-theory using certain cohomological input from [24]. In Section 2 we demonstrate an important universal property that q enjoys and indicate its usefulness in producing maps between algebraic K-groups. This result is then applied in Section 3 to construct exterior power operations on K(X, A) that make it into a special λ-ring (in the sense of Atiyah and Tall [4]). Various generalities about such λ-rings are also discussed. Section 4 proves the central technical result that K(X, A) as a special λ-ring possesses a locally nilpotent γ-filtration. This is then used to decompose the rationalized K-groups into eigenspaces of ψk. We also mention the relevance of what we call “nilpotent induction” for analyzing such λ-rings. Section 5 is devoted to a proof that the K-theory of a perfect ring of characteristic p>0 is uniquely p-divisible by exploiting both nilpotent induction and the relation between Adams operations and Frobenius. In Section 6 we prove another general result on localizing a λ-ring with a locally nilpotent γ-filtration at a prime p and use it to understand the relation between fixed points of ψp and localization. Finally, in Section 7 we see how the technique of Section 6 sheds some light on the K-theory of algebraically closed fields in characteristic p>0. Also a brief appendix is included describing the computation of Adams operations for certain K-groups. In particular, we show ψk respects the graded ring structure on K∗(A) and, in the case of finite fields, respects the Brauer lifting to complex K-theory