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Journal of Pure and Applied Algebra Adams Operations and Λ-Operations View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 213 (2009) 409–420 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa Adams operations and λ-operations on classifying oriented cohomology theories José Malagón-López Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada article info a b s t r a c t Article history: Adams operations on algebraic cobordism with rational coefficients are defined and shown Received 19 October 2007 to descend to the oriented cohomology theories with rational coefficients which are Received in revised form 21 June 2008 universal with respect to their group law. With the induced structure of special λ-ring Available online 30 August 2008 ∗ we show that we can regard the γ -filtration on CH as a deformation of the classical γ - Communicated by C.A. Weibel Q filtration on K0 . Q MSC: ' 2008 Elsevier B.V. All rights reserved. 14F43 14C15 19A99 1. Introduction In [6] Levine and Morel introduced the notion of an oriented cohomology theory (OCT) on the category of smooth quasi- projective varieties over a field k. An important feature of these theories is the existence of a formal group law describing the behaviour of the first Chern class with respect to the tensor product of line bundles. When char.k/ D 0, in [6] the theory X 7! Ω∗.X/, called algebraic cobordism, is constructed as the universal OCT in the sense that for any such theory A∗, there is a unique map Ω∗ ! A∗ of OCTs so that the induced map on the coefficient rings is precisely the map classifying the ∗ ∗ formal group law of A . Given a formal group law .R; µ), the theory Ω ⊗Ω∗.k/ R is the universal OCT with respect to its formal group law. Other important results from [6] are that the Chow ring CH∗ is the universal additive theory, and that the ``extended'' algebraic K-theory K0Tβ; β−1U, where β is a variable of degree -1, is the universal multiplicative periodic theory. From another side, given a scheme X, part of the algebraic structure of K0.X/ is the presence of an exterior power product. In [5], Grothendieck gave a general treatment of such rings by introducing the notion of special λ-ring and λ-operations. These operations are non-additive; the associated Adams operations, n, being ring morphisms, are thus easier to handle. In fact, with Q-coefficients, one can recover the λ-operations from the Adams operations. ∗ In the present work we propose a -structure on Ω that descends to the OCTs with rational coefficients that are Q universal with respect to their formal group law. We proceed by using the characterization of the classical Adams operations by their action on the first Chern class of a line bundle. Looking at the classical Adams–Riemann–Roch in the context of OCTs, we have that the nth Bott's cannibalistic class corresponds to the inverse Todd genus associated to the classical nth Adams operation. This genus can be described in terms of the group law, so we can extend the definition to any OCT. ∗ ⊗ T U Ω Further, the ``universal'' Todd genus will induce a twisting on Ω Z Z 1=n . Thus, by universality we obtain a map n of cohomology theories from Ω∗ to the twisted theory. Such map will be our nth Adams operation. The main results obtained are summarized in the following. Theorem 1. Let k be a field of characteristic zero. Then ∗ (1) Ω is endowed with a structure of -ring, so we have an induced structure of special λ-ring. Moreover, by taking the Q Levine–Morel degree homomorphism as augmentation we have a γ -filtration on Ω∗ . Q E-mail address: [email protected]. 0022-4049/$ – see front matter ' 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2008.07.017 410 J. Malagón-López / Journal of Pure and Applied Algebra 213 (2009) 409–420 (2) For any n ≥ 1, the nth Adams operation on Ω∗ is an isomorphism. Q ∗ ∗ (3) Given a formal group law .R; µ), the Adams operations descend to Ω .k/-linear endomorphisms on Ω ⊗ ∗ R. In particular Q Ω .k/ we have the same induced structures as on Ω∗ . Q For the case of K0Tβ; β−1U, we check that our Adams operations generalize the classical ones as well that the degree zero part of the graded ring induced by the γ -filtration is the corresponding graded ring in the classical case. We also see that the nth Adams operation on CHq is just multiplication by nq and that the qth part of the γ -filtration on CH∗ is given by the Q sum of the pieces of codimension ≥ q. Finally, we see how our results imply that we have a deformation from the classical γ -filtration on K0 to the filtration by degree on CH∗ . Q Q In Section 2 we recall the basic properties of λ-rings and Adams operations. In the next section we give a rough description of OCTs and algebraic cobordism, and touch on the properties of such theories that we need for the present work. We review in Section 4 the technique of twisting a theory, which is needed to construct cohomological operations. In the last section we give our construction of Adams operations and prove the results mentioned above. Conventions. Throughout the present work a ring will mean a commutative ring with unit. Given a field k, a separated scheme of finite type over k will be referred to as a k-scheme. Given a vector bundle E over a k-scheme X, let E be the sheaf of sections of E. The projective bundle associated to E, Proj Sym∗ .E/, will be denoted by .E/. From now on, we will make OX P no distinction between a vector bundle and its corresponding locally free OX -sheave. 2. λ-structures and -structures Let R be a ring and denote by Wt .R/ the multiplicative abelian group of formal power series in t with constant term 1 and V ! 7! C C P n coefficients in R. We say that R is a λ-ring if there is a group morphism λt R Wt .R/ of the form x 1 xt n>1 ant . D P n 2 f V ! g Set λt .x/ n≥0 λn.x/t , for any x R. The set of maps λn R R n≥0 is called a λ-structure on R.A λ-morphism is a ring morphism commuting with the λ-operations. The group Wt .R/ has a structure of λ-ring ([1, Section 1]), and we say that R is a special λ-ring if λt is a λ-morphism. The morphisms of special λ-rings are just λ-morphisms. An ideal I ⊆ R is called a λ-ideal if λn.I/ ⊆ I for all n ≥ 0. Let S be a λ-ring, we say that R is an augmented λ-ring over S if there is a surjective λ-morphism " V R ! S called the augmentation. We have that ker."/ ⊆ R is a λ-ideal. A λ-ring is called x binomial if λt .x/ D .1 C t/ , for any x 2 R. Examples of binomials λ-rings are Z and Q. In what follows, when we refer to these rings as λ-rings, we consider them with the binomial structure. A ring R together with a collection of endomorphisms f n V R ! Rgn≥1 is called a -ring if 1 is the identity and n ◦ m D nm for all integers n; m ≥ 1. The collection of endomorphisms f ngn≥1 is called a -structure on R. For any n ≥ 1, n is called nth Adams operation.A -morphism is a ring morphism that commutes with the Adams operations. Proposition 2. Let R be a commutative ring containing a subring isomorphic to Q. Then the existence of a special λ-structure on R is equivalent to the existence of a -structure on R. More precisely, for all n ≥ 1, n−2 n−1 nλn D 1λn−1 − 2λn−2 C···C .−1/ n−1λ1 C .−1/ nλ0 n nC1 n D n−1λ1 − n−2λ2 C···C .−1/ 1λn−1 C .−1/ nλn: Proof. See [10] or [7, Thm. 1.3.10]. 0 Example 3. Let X be an scheme over a field k. Let K .X/ be the Grothendieck group of locally free OX -sheaves. It has a n 0 structure of special λ-ring by setting λn.E/ D T^ .E/U, for all E 2 K .X/. The rank map makes it augmented over Z. Also we 0 ⊗n have that the nth Adams operations on K .X/ are characterized by n.TLU/ DTL U, with L ! X a line bundle. For the rest of the section we assume R to be a special λ-ring. Set s D t=.1−t/ and define γt VD λs. This induced a λ-structure fγng on R. The map γn is referred as the nth γ -operation. Assume that " V R ! S is a splitting augmentation, with S a binomial special λ-ring. Let I D ker."/. For all integer n ≥ 0, define the λ-ideal * + Xr R D γ .a / ··· γ .a / a 2 I; n ≥ n : n n1 1 nr r i i iD1 The γ -filtration on R is defined by the descending sequence of ideals · · · ⊂ Rn ⊂ · · · ⊂ R1 D I ⊂ R0 D R: ⊗ V ! Lemma 4.
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