Algebraic K-theory of schemes

Remarks

The group $K_0(X)$ is the Grothendieck group of vector bundles. It is the quotient of the free abelian group on isomorphism classes $[E]$ of vector bundles by the following relations. For every short exact sequence $0 \to E' \to E \to E'' \to 0$, one sets $[E] = [E'] + [E'']$. The tensor product makes $K_0(X)$ a commutative ring. Example: for $X = \operatorname{Spec} k$ with $k$ a field, every exact sequence of finite-dimensional vector spaces splits, and $K_0(X) \cong \Z$ through the dimension; also $K_1(X) \cong k^{\times}$. For $X = \mathbb{P}^n_k$, $K_0(X)$ is free of rank $n + 1$ with basis $[\mathcal{O}], [\mathcal{O}(-1)], \ldots, [\mathcal{O}(-n)]$. Non-example: using all coherent sheaves instead of vector bundles gives the groups $G_n(X)$, which differ in general. For a regular separated Noetherian scheme the natural maps $K_n(X) \to G_n(X)$ are isomorphisms, but for a singular one they need not be. For general schemes, Thomason and Trobaugh gave a better-behaved version using perfect complexes, which agrees with this one when $X$ has an ample family of line bundles.