# Shortening binary complexes and commutativity of $K$-theory with infinite products

## TL;DR

This paper demonstrates that binary complexes of length two generate higher algebraic K-theory groups in Grayson's model and proves the isomorphism of K_1 models, establishing that algebraic K-theory commutes with infinite products.

## Contribution

It shows that shorter complexes suffice in Grayson's model and confirms the equivalence of K_1 models, ensuring K-theory's compatibility with infinite products.

## Key findings

- Binary complexes of length two generate the entire K-theory group.
- The comparison map for K_1 models is an isomorphism.
- Algebraic K-theory commutes with infinite products.

## Abstract

We show that in Grayson's model of higher algebraic $K$-theory using binary acyclic complexes, the complexes of length two suffice to generate the whole group. Moreover, we prove that the comparison map from Nenashev's model for $K_1$ to Grayson's model for $K_1$ is an isomorphism. It follows that algebraic $K$-theory of exact categories commutes with infinite products.

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Source: https://tomesphere.com/paper/1705.09116