Exterior powers via binary complexes

During the last year I have been writing a paper with Bernhard Köck and Lenny Taelman about a new approach to exterior power operations on higher algebraic K-groups. Yesterday we uploaded a preprint of the article to the arXiv: Exterior power operations on higher K-groups via binary complexes. Here I’ll try to explain a little bit of what it’s about.

If X is a scheme, then its Grothendieck group K_0(X) comes equipped with the structure of a \lambda-ring induced from the exterior powers of bundles over X. This structure also induces Adams operations, and a descending \gamma-filtration F_i K_0(X), which often yield strong results about the structure of K_0(X) (at least, up to torsion). For instance, Grothendieck proved that if X is smooth over a field, then the quotient F_i K_0(X) / F_{i+1} K_0(X) is isomorphic (rationally) to the Chow group \mathrm{CH}^i (X). These structures are the heart of Grothendieck–Riemann–Roch theory.

Following Quillen’s invention of higher algebraic K-groups, exterior power operations on K_n(X) were defined in the affine case by Hiller and Quillen, and the theory was extended to schemes by Kratzer and Soulé.1 Other approaches to the \lambda-ring structure on K_n(X) have been given by Grayson and Levine. Common to all of these definitions is the use of homotopy theory. Indeed, until recently it was assumed necessary to use homotopy theory, since all constructions of higher algebraic K-groups were homotopical in nature.

In a 2012 paper Grayson gave the first purely algebraic description of the higher K-groups, in terms of acyclic binary multicomplexes.2 My PhD work was an investigation of this new approach to algebraic K-theory. In the final chapter of my thesis I started the work of constructing the \lambda-ring structure on K_n(X) entirely algebraically, and this is the work that this paper completes. Let K_* (X) be the graded ring of all K-groups of X. Then we have:

Theorem. Let X be a quasi-compact scheme. Then exterior powers of acyclic binary multicomplexes endow K_{*} (X) with the structure of a \lambda-ring.

What is new about our work is that our \lambda-operations are entirely explicit on generators, leaving the door open to new calculations.

In my thesis I used the Dold—Kan correspondence to construct the required operations on binary multicomplexes and show that they induce a pre-\lambda-ring structure on K_{*} (X) in the affine case. With Bernhard’s help I extended this to general quasi-compact schemes and proved the \lambda-ring axiom relating to products. Bernhard noticed that the final \lambda-ring axiom, which relates to compositions, would follow from my work in the first half of the paper if we could solve a problem related to the well-known plethysm problem in representation theory. However such problems are usually intractable and we were not optimistic.

But we were lucky, and Lenny Taelman saw a way to solve our problem by calculating the Grothendieck group of the exact category of integral polynomial functors, using methods of Krause, Serre, and Green.

Theorem. K_0 \mathrm{Pol}(\mathbb{Z}) is isomorphic to the free \lambda-ring on one variable.

Lenny joined us as a third author on the paper, and we had an enjoyable few days all working together on these ideas in Southampton. We also received some useful feedback on early drafts from Dan Grayson and Marco Schlichting.

We have some work in progress towards another paper devoted to examples of calculations with our operations. I hope to develop some good results in this direction in the coming year.

Update June 2017: This paper now published in Annals of K-Theory. Published version here.

  1. See: Howard L. Hiller, \lambda-rings and algebraic K-theory; C. Kratzer, \lambda-structure en K-théorie algébrique; Christophe Soulé, Opérations en K-théorie algébrique↩︎

  2. Daniel R. Grayson, Algebraic K-theory via binary complexes↩︎

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