Patching and Quillen K-Theory

Abstract

This paper provides an isomorphism $K_n(\mathcal{A})\cong K_n({\mathcal{A}}_1){\times }_{K_n({\mathcal{A}}_0)}{K}_n({\mathcal{A}}_2)$ of $K$-groups, i.e., an exact sequence $0\to K_n(\mathcal{A})\to K_n({\mathcal{A}}_1)\times K_n({\mathcal{A}}_2)\to K_n({\mathcal{A}}_0)$ corresponding to a 2-fiber product of abelian categories, taken with respect to exact functors. Using recent patching results of D. Harbater, J. Hartmann and D. Krashen, given fields $F_1,F_2\le F_0$ and $F=F_1\cap F_2$ which satisfy a simple matrix factorization criterion, our isomorphism relates the $K$-groups of the fields $F$ and $F_i$ ($i$ = 0, 1, 2). In particular, we establish a local-global principle for $K$-theory of function fields of curves defined over a complete discretely valued field.

arXiv commentThis paper has been withdrawn by the author due to an error in Lemma 4.2