Patching and Quillen K-Theory

Abstract

This paper provides an isomorphism $K_n (\mathscr{A}) \cong K_n (\mathscr{A}_1) \times_{K_n(\mathscr{A}_0)} K_n(\mathscr{A}_2)$ of $K$-groups, i.e., an exact sequence $0 \to K_n(\mathscr{A}) \to K_n(\mathscr{A}_1)\times K_n(\mathscr{A}_2) \to K_n(\mathscr{A}_0)$ 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 $F_1, F_2 \leq F_0$ 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.