K₀ and K₁ of Categories, Negative K-Theory

Abstract

For many of the applications of K-theory, it is useful to have the notion of K-theory for categories and not just for rings. In this more general context, the K-theory of a ring Ris just the K-theory of the category Proj R of finitely generated projective modules over R. Another natural example is the topological K-theory of a compact space X, which is the K-theory of the category Vect X of (locally trivial, real, or complex) vector bundles over X. The identification of this with the K-theory of the ring R= C(X) then follows from an equivalence of categories Proj R ≅ Vect X, But there axe also many examples that don’t come so directly from rings; for instance, if X is a projective algebraic variety, one can consider in a similar way the category Vect Xof algebraic vector bundles over X.We will see many more examples shortly