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Exact sequences in algebraic K-theory. (English) Zbl 0629.18010
Suppose P and M are exact categories (in the sense of Quillen), and that $$F: P\to M$$ is an exact functor. If a certain “cofinality”-type criterion is satisfied by F, it follows that a particular square of spaces is homotopy-Cartesian, giving rise to a long exact sequence of algebraic K-groups which includes the induced maps $$K_ iP\to K_ iM.$$
It turns out that the criterion on F is often satisfied, and that the third term in the resulting exact sequence can often be identified. This general machine, therefore, produces new proofs, which share a common core, of the following theorems of algebraic K-theory: (1) the cofinality theorem of Grayson and Waldhausen; (2) Quillen’s dévissage theorem; (3) Quillen’s resolution theorem; and (4) Quillen’s theorem on localization for projective modules.
The author also uses his construction to produce new definitions for the higher algebraic K-theory of an exact category.
Reviewer: M.Stein

##### MSC:
 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 18E10 Abelian categories, Grothendieck categories