Exterior power operations on higher K-groups via binary complexes

Abstract

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 functors is the universal λ-ring on one generator.

Two formulae for exterior power operations on higher K-groups

“…Definition 1.2. Given binary multicomplexes P ∈ B m (X) and Q ∈ B n (X), we form their external tensor product P ⊠ Q ∈ B m+n (X) as follows: if n = 0, i.e., if Q consists of a single object Q, the multicomplex P ⊠ Q is obtained from P by tensoring every object in P with Q and every differential with 1 Q , see also [HKT17,Section 7]. If n ≥ 1, we write Q = (Q * , d n Q ,d n Q ) with Q i ∈ B n−1 (X) and recursively define:…”

“…Let X be a quasi-compact scheme. Following Grayson's algebraic description [Gra12] of higher K-groups in terms of explicit generators and relations, Taelman and the authors of this paper have, in [HKT17], constructed an exterior power operation λ r on K n (X) for every r ≥ 1 and n ≥ 0 by assigning an explicit element λ r (P) to each of Grayson's generators P. While our construction is purely algebraic (i.e., unlike all previous constructions does not use any homotopy theoretic methods) and λ r (P) can in principle be written down, this is normally too combinatorially complex to be done by hand. The purpose of this paper is to augment our construction by two formulae that help to compute λ r .…”

Operations on Categories of Modules are Given by Schur Functors

“…I am very grateful to Alexandru Chirvasitu, who made major contributions to this paper in its early stages, particularly to Section 4. I also would like to thank Ingo Blechschmidt for sharing his beautiful constructive proofs appearing in Section 6, Oskar Braun for his careful proofreading of the preprint, Bernhard Köck for drawing my attention to [18], as well as Antoine Touzé and Ivo Dell'Ambrogio for helpful discussions on the subject at the University of Lille. Finally I would like to thank the anonymous referee for making several suggestions for improvement.…”

“…A similar classification therefore also holds for strict polynomial functors [7,15,20,28] over a commutative Q-algebra k, which may be identified with operations Mod fg,proj → Mod fg,proj over k between categories of finitely generated projective modules. The classification in terms of Schur operations does not work for k = Z, but in this case one can at least calculate the K 0 -ring of the category of strict polynomial functors [18,Theorem 8.5].…”

Comparison of Exterior Power Operations on Higher K-Theory of Schemes

“…This will culminate in the proof of a conjecture [Köc98, Conjecture (2.7)] by the first author about composition of exterior power operations in the equivariant context, completing the proof that higher equivariant K-groups satisfy all axioms of a λ-ring. The construction by Harris, the first author and Taelman in [HKT17] is different from all other constructions as it defines exterior power operations on higher K-groups purely combinatorially and does not resort to any homotopy theoretical methods. This construction has been made possible by Grayson's groundbreaking description of higher K-groups in terms of generators and relations, see [Gra12,Corollary 7.4].…”