Tensor Product K-theory is Rational Algebraic K-theory
Abstract
For a commutative ring R with unity, its algebraic K-theory space K(R) may be obtained by group-completing the symmetric monoidal category of finitely generated free R-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of K(R), up to π0. We also discuss how a similar group-completion would give the p-perfection and, more generally, the localization of K(R) at any non-trivial multiplicatively closed subset S ⊆ ℤ> 0. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.
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BibTeX
@misc{dubey2026tensor,
title = {{Tensor Product $K$-theory is Rational Algebraic $K$-theory}},
author = {Amartya Shekhar Dubey and Mattie Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.11412},
eprint = {2606.11412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11412},
}
RIS
TY - UNPB
TI - Tensor Product K-theory is Rational Algebraic K-theory
AU - Amartya Shekhar Dubey
AU - Mattie Ji
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2606.11412
UR - https://arxiv.org/abs/2606.11412
AB - For a commutative ring R with unity, its algebraic K-theory space K(R) may be obtained by group-completing the symmetric monoidal category of finitely generated free R-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of K(R), up to π_0. We also discuss how a similar group-completion would give the p-perfection and, more generally, the localization of K(R) at any non-trivial multiplicatively closed subset S ⊆ ℤ_> 0. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.
ER -