Slices of co-operations forKGL

Abstract

We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic K-theory. the notation that E ⊗ A * denotes q (T q ∧ E) ⊗ A q for a motivic spectrum E and a graded ring A * . In this notation, we prove:Theorem 0.1. Assume that S is smooth over a perfect field. Then there is an isomorphism of cosimplicial ring spectraRemark. If S is over a field of characteristic 0, it seems likely that Theorem 0.1 would follow from the work of Spitzweck [16] on Landweber exact spectra. Such an approach would depend heavily on the Hopkins-Morel-Hoyois theorem [7].Conjecture 8 is intertwined with Voevodsky's Conjectures 1, 7 and 10 in [19], that HZ ← 1 → KGL induces isomorphismsand thus that Hom SH(S) (s 0 (1), s 0 (1)) ∼ = H 0 (S, Z). These are known to hold when the base S is smooth over a perfect field by the work of Voevodsky and Levine (see [9, 10.5.1 and 11.3.6; 21]), or singular over a field of characteristic 0 (see [13]).Here is our main result, which evidently implies Theorem 0.1.Theorem 0.2. Let S be a finite-dimensional separated Noetherian scheme. Then we have the following.(a) There are isomorphisms for all n 0:These isomorphisms commute with all of the coface and codegeneracy operators except possibly ∂ 0 and σ 0 .(b) Assume in addition that s 0 (1) → s 0 (KGL) is an isomorphism in SH(S), and that Hom SH(S) (s 0 (1), s 0 (1)) is torsion-free. Then the maps in (a) are the components of an isomorphism of graded cosimplicial motivic ring spectra:The case n = 0 of Theorem 0.2(a), that s 0 (KGL) ⊗ π 2 * KU ∼ = s * KGL, is immediate from the periodicity isomorphism T ∧ KGL ∼ = KGL defining the motivic spectrum KGL and the formula π 2 * KU ∼ = Z[u, u −1 ]. The need to pass to slices is clear at this stage, because π 2n,n KGL ∼ = K n (S) for S smooth over a perfect field.The left side of Theorem 0.2 is algebraic in nature, involving only the cosimplicial ring π 2 * (KU ∧n+1 ) and H = KU 0 (KU ). In fact, π 2 * (KU ∧n+1 ) is the cobar construction C • Γ (R, R) ∼ = KU * ⊗ H ⊗n for the Hopf algebroid (R, Γ) = (KU * , KU * KU ). We devote the first four sections to an analysis of this algebra, focussing on the rings F = KU 0 (CP ∞ ) and H. Much of this material is well known, and due to Frank Adams [1].If E is an oriented motivic spectrum, the projective bundle theorem says that E ∧ P ∞ ∼ = E ⊗ F and hence that E ∧ P ∞∧n ∼ = E ⊗ F ⊗n . Using this, we establish a toy version of Theorem 0.2 in Propositions 5.8 and 6.4 that, as cosimplicial spectra, KGL ∧ P ∞∧•+1 is KGL ⊗ F ⊗•+1 (the cobar construction on F ) and s * (KGL ∧ P ∞∧•+1 ) ∼ = s * (KGL) ⊗ F ⊗•+1 ∼ = s 0 (KGL) ⊗ KU * (CP ∞∧n ). (0.3) Remark 4.5. Lemma 4.4 fails dramatically if u = 0, for example when E = HZ.