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M n = M with quotients Mi /Mi−1 all in B. Then BQi : BQ(B) → BQ(A) induces an isomorphism i∗ : Kp (B) → Kp (A) for all p. 8 ([30, Theorem 3, §4]). Let i : E0 → E1 be a full exact subcategory of an exact subcategory E1 , with E0 closed under extensions A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 45 in E1 . Suppose that each object M of E1 admits a finite resolution 0 → P n → . . → P 1 → P0 → M → 0 with the Pi in E0 . Then BQi : BQ(E0 ) → BQ(E1 ) induces an isomorphism i∗ : Kp (E0 ) → Kp (E1 ) for all p.

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