Patching and Quillen K-Theory

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 prod...

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Bibliographic Details
Published in:arXiv.org 2014-03
Main Author: McFaddin, Patrick
Format: Article
Language:English
Subjects:
Isomorphism
Patching
Online Access:Get full text
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Summary: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.
ISSN:2331-8422