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arXiv · 2207.08378

Non-unital algebraic $K$-theory and almost mathematics

Yuki Kato
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Abstract

The Gersten conjecture is still an open problem of algebraic $K$-theory for mixed characteristic discrete valuation rings. In this paper, we establish non-unital algebraic $K$-theory which is modified to become an exact functor from the category of non-unital algebras to the stable $\infty$-category of spectra. We prove that for any almost unital algebra, the non-unital $K$-theory homotopically decomposes into the non-unital $K$-theory the corresponding ideal and the residue algebra, implying the Gersten property of non-unital $K$-theory of the the corresponding ideal.

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Yuki Kato. 2023-02-27. Non-unital algebraic $K$-theory and almost mathematics. https://arxiv.org/abs/2207.08378

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