Patching and Quillen K-Theory

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This paper provides an isomorphism Kₙ (A) Kₙ (A₁) ×Kₙ(A₀) Kₙ(A₂) of K-groups, i.e., an exact sequence 0 → Kₙ(A) → Kₙ(A₁)× Kₙ(A₂) → Kₙ(A₀) 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 F₁, F₂ ≤ F₀ and F= F₁ ∩ F₂ which satisfy a simple matrix factorization criterion, our isomorphism relates the K-groups of the fields F and Fᵢ (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.