This paper by R. W. Thomason and Thomas Trobaugh presents a localization theorem for the $K$-theory of commutative rings and schemes, which is a significant advancement in the field. The theorem, Theorem 7.4, relates the $K$-groups of a scheme, an open subscheme, and the category of perfect complexes on the scheme that are acyclic on the open subscheme. This localization theorem, inspired by Quillen's work on $K'$- and $G$-theory, has been a long-standing gap in the development of $K$-theory since 1973. The authors' theorem enables the derivation of numerous new results, including the "Bass fundamental theorem," Zariski (Nisnevich) cohomological descent spectral sequence, Mayer-Vietoris theorem for open covers, invariance under polynomial extensions, Vorst-van der Kallen theory, and comparisons between algebraic and topological $K$-theory. Additionally, the paper develops higher $K$-theory for derived categories, building on the foundational work of Waldhausen. The key concepts, such as perfect complexes, are rooted in Grothendieck's original ideas, which have been revitalized and extended by the authors. The paper dedicates its findings to Alexander Grothendieck on his 60th birthday, highlighting the enduring relevance and potential of his ideas in modern mathematics.This paper by R. W. Thomason and Thomas Trobaugh presents a localization theorem for the $K$-theory of commutative rings and schemes, which is a significant advancement in the field. The theorem, Theorem 7.4, relates the $K$-groups of a scheme, an open subscheme, and the category of perfect complexes on the scheme that are acyclic on the open subscheme. This localization theorem, inspired by Quillen's work on $K'$- and $G$-theory, has been a long-standing gap in the development of $K$-theory since 1973. The authors' theorem enables the derivation of numerous new results, including the "Bass fundamental theorem," Zariski (Nisnevich) cohomological descent spectral sequence, Mayer-Vietoris theorem for open covers, invariance under polynomial extensions, Vorst-van der Kallen theory, and comparisons between algebraic and topological $K$-theory. Additionally, the paper develops higher $K$-theory for derived categories, building on the foundational work of Waldhausen. The key concepts, such as perfect complexes, are rooted in Grothendieck's original ideas, which have been revitalized and extended by the authors. The paper dedicates its findings to Alexander Grothendieck on his 60th birthday, highlighting the enduring relevance and potential of his ideas in modern mathematics.