## Shortening binary complexes and commutativity of $$K$$-theory with infinite products.(English)Zbl 07183274

Summary: 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.

### MSC:

 19D06 $$Q$$- and plus-constructions 18E10 Abelian categories, Grothendieck categories
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### References:

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