This paper by Daniel Quillen aims to develop a higher K-theory for additive categories with exact sequences, extending the existing theory of the Grothendieck group. The approach involves forming a new category \( Q(M) \) from an additive category \( M \) embedded in an abelian category \( A \), where morphisms are defined through subquotients. The fundamental group of the classifying space \( BQ(M) \) is shown to be canonically isomorphic to the Grothendieck group of \( M \), leading to the definition of a sequence of K-groups \( K_1(M), K_2(M), \ldots \). The first part of the paper covers the general theory of these K-groups, including tools for working with the classifying space and results on the homotopy-theoretic fiber of a functor. The second part applies this theory to rings and schemes, defining K-groups for rings and schemes using modules and vector bundles, respectively. Key results include formulas for K-groups of polynomial rings and localization theorems. The paper also discusses the computation of K-groups for projective bundles and Severi-Brauer schemes, and provides a spectral sequence for coherent sheaves on regular schemes, leading to Bloch's formula for cycles on regular varieties.This paper by Daniel Quillen aims to develop a higher K-theory for additive categories with exact sequences, extending the existing theory of the Grothendieck group. The approach involves forming a new category \( Q(M) \) from an additive category \( M \) embedded in an abelian category \( A \), where morphisms are defined through subquotients. The fundamental group of the classifying space \( BQ(M) \) is shown to be canonically isomorphic to the Grothendieck group of \( M \), leading to the definition of a sequence of K-groups \( K_1(M), K_2(M), \ldots \). The first part of the paper covers the general theory of these K-groups, including tools for working with the classifying space and results on the homotopy-theoretic fiber of a functor. The second part applies this theory to rings and schemes, defining K-groups for rings and schemes using modules and vector bundles, respectively. Key results include formulas for K-groups of polynomial rings and localization theorems. The paper also discusses the computation of K-groups for projective bundles and Severi-Brauer schemes, and provides a spectral sequence for coherent sheaves on regular schemes, leading to Bloch's formula for cycles on regular varieties.