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With some technical fiddling, one can define f∗ : Gp (Y ) → Gp (X) if f is projective. 1. Devissage, resolution and localization. The main theorems of the previous section have the following applications for Kp (X) and Gp (X). 2. Let i : Z → X be the inclusion of a closed subscheme, giving the full embedding of abelian categories i∗ : MZ → MX (Z). Then i∗ : Gp (Z) → Kp (MX (Z)) is an isomorphism for all p. Indeed, each coherent sheaf F supported on Z has a finite filtration with quotient sheaves OZ -modules.

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Algebraic K-theory of generalized free products. III, IV. Ann. of Math. (2) 108 (1978), no. 2, 205–256. [43] Wiles, A. The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540.

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