Combinatorial relative algebraic K-theory and exterior power operations

Abstract

Grayson has presented a conjectural combinatorial presentation of the higher relative algebraic K-groups. We prove this description to be correct. In proving this result, we provide a concrete description of Grayson's isomorphism between his combinatorial presentation of K_1 and the classical definition of K_1. We construct exterior power operations on Grayson's presentation of the relative algebraic K-groups, and prove that these operations satisfy the axioms of a λ-ring modified to allow nonunital rings. Kasprowski and Winges have provided a proof that the canonical homomorphism from Nenashev's presentation of K_1 to Grayson's presentation of K_1 is an isomorphism. We prove this isomorphism is compatible with the isomorphisms from these groups to the classical definition of K_1. Harris has provided an elementary proof that the homomorphism from Bass' presentation of K_1 to Grayson's presentation of K_1 is an isomorphism in certain circumstances. We use this result to prove the analogous result for relative K_0.

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Identifiers

ORCID for Jane Toni Joy Turner: orcid.org/0000-0003-2359-7770

ORCID for Bernhard Koeck: orcid.org/0000-0001-6943-7874

ORCID for Nick Wright: orcid.org/0000-0003-4884-2576

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