Exterior power operations on higher K-groups via binary complexes

Exterior Power Operations on Higher K-groups via Binary Complexes

We use Grayson’s binary multicomplex presentation of algebraic K-theory to give a new construction of exterior power operations on the higher K-groups of a (quasi-compact) scheme. We show that these operations satisfy the axioms of a λ-ring, including the product and composition laws. To prove the composition law we show that the Grothendieck group of the exact category of integral polynomial f...

Exterior Power Operations on Higher K-theory

Algebraic K-theory via binary complexes

In [11], Quillen defines the notion of exact category. Each such category N is an additive category together with a suitable collection of sequences called short exact sequences. Quillen constructs a space by gluing simplices to each other, using the exact sequences of N to determine the simplices and the gluing instructions. Then for each n ∈ N he defines KnN as a homotopy group of the space. ...

Binary operations and groups

Example 1.3. The examples are almost too numerous to mention. For example, using +, we have (N,+), (Z,+), (Q,+), (R,+), (C,+), as well as vector space and matrix examples such as (Rn,+) or (Mn,m(R),+). Using subtraction, we have (Z,−), (Q,−), (R,−), (C,−), (Rn,−), (Mn,m(R),−), but not (N,−). For multiplication, we have (N, ·), (Z, ·), (Q, ·), (R, ·), (C, ·). If we define Q∗ = {a ∈ Q : a 6= 0}, ...

Adams Operations on Higher Arithmetic K-theory

We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The definition re...