Two formulae for exterior power operations on higher \(K\)-groups (Q2001433)

This paper is a continuation of [the authors, Ann. \(K\)-Theory 2, No. 3, 409--450 (2017; Zbl 1369.19004)], in which the authors construct an operation of exterior powers on all algebraic \(K\)-groups using Grayson's purely algebraic description of \(K\)-theory in terms of acyclic binary complexes. Since Grayson's description does not use any homotopy theory, this construction of the paper under review does not either. The current paper extends the original paper of Harris, Köck, and Taelman by deriving two formulae for the exterior product. These formulae are important as they aid in computations. This is quite a natural thing to do. Indeed, to put things in perspective, while Grayson's algebraic description of \(K\)-theory is amazing, it is one of the hardest to compute with. For example, even computing the lower or classical \(K\)-groups from Grayson's description is fiendishly difficult. Without further ado, let us describe the two formulae given by the present authors. The first formula concerns their exterior power operation \(\lambda^r:K_n(X)\to K_n(X)\) where \(X\) is any quasicompact scheme and \(r\geq 1\). Their formula describes how \(\lambda^r\) behaves with respect to an external product denoted \(\smallsmile\) which is derived from a type of tensor product on acyclic binary complexes. They prove that \[ \lambda^r(x\smallsmile y) = (-1)^{r-1}r\lambda^r(x)\smallsmile\lambda^r(y). \] This combinatorial formula should be useful and as the authors note, it already gives a corresponding formula for the \(r\)th Adams operator. The second formula concerns how the \(r\)th exterior power \(\lambda^R\) applies to an \(n\)-cube; that is, a multicomplex supported on \([0,1]^n\). They use this to prove that their exterior power operation agrees with a definition of Hiller who was concerned only with \(K_1\). Since the formula involves a bit of notation probably too verbose for a short review, I invite readers to consult the original paper. Here are the main consequences of this second formula: the previously mentioned agreement with Hiller's definition, and that a certain modified Milnor \(K\)-group is invariant under the exterior product. In the grand scheme of things, this paper contributes to a growing body of literature telling us how to work with Grayson's construction of all the \(K\)-groups. I hope that one day soon this will lead to new computational advances in the computation of these groups for certain well-known rings.