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arXiv · 1306.3688

On the Negative $K$-theory of Singular Varieties

Justin Shih
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Abstract

Let $X$ be an $n$-dimensional variety over a field $k$ of characteristic zero, regular in codimension 1 with singular locus $Z$. In this paper we study the negative $K$-theory of $X$, showing that when $Z$ is sufficiently nice, $K_{1-n}(X)$ is an extension of $KH_{1-n}(X)$ by a finite dimensional vector space, which we compute explicitly. We also show that $KH_{1-n}(X)$ almost has a geometric structure. Specifically, we give an explicit 1-motive $[L \rightarrow G]$ and a map $G(k) \rightarrow KH_{1-n}(X)$ whose kernel and cokernel are finitely generated abelian groups.

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Justin Shih. 2013-06-16. On the Negative $K$-theory of Singular Varieties. https://arxiv.org/abs/1306.3688

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